# Patent

## Title
Efficient and robust routing independent of traffic pattern variability

## Abstract

A scheme for routing packets of traffic to their destination after ensuring that they pass through one or more pre-determined intermediate nodes, thereby permitting all permissible traffic patterns to be handled without knowledge of the traffic matrix, subject to edge-link capacity constraints. In one embodiment, a request for a path with a service demand for routing data between the ingress point and the egress point is received. A set of two or more intermediate nodes between the ingress point and the egress point is selected. Based on a bandwidth of the network, respective fractions of the data to send from the ingress point to each node of the set of intermediate nodes are determined. The data is routed in the determined respective fractions from the ingress point to each node of the set of intermediate nodes, and routed from each node of the set of intermediate nodes to the egress point.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims priority to co-pending U.S. Provisional Application Ser. No. 60/575,350, filed on May 28, 2004, the teachings of which are incorporated herein by reference.


BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to routing in telecommunication systems, and, more particularly, to determining paths through nodes of a network for routing with guaranteed service levels.
2. Description of the Related Art
In packet-based communications networks, such as the Internet, each stream of data packets, called a packet flow, is transferred through the network over a network path from a source to a destination. Each network path is defined by a set of nodes, interconnected by a set of links. A node may include one or more routers, which are devices in the network that handle data transfer between computers.
A communications system may be structured such that different-sized networks are interconnected and may alternatively or additionally include one or more peer structures in which equivalent-sized networks are interconnected. A packet network may connect to another packet network through nodes referred to as the ingress and egress points. The terms ingress point and egress point may refer to a node of a packet network that connects to another packet network, or alternatively, these terms may refer to the connecting nodes of the other packet network. Packet networks with high capacity that transfer packets between two or more other packet networks are commonly referred to as “backbone” networks.
FIG. 1 shows a backbone network 100 of the prior art having nodes n1-n9 interconnected through links 101, which enable communication between packet networks 102-104. One of the ingress points of backbone network 100 is node n1, which receives packets from a source (i.e., packet network 102), and one of the backbone network's egress points is node n4, which transmits packets to a destination (i.e., packet network 104). Backbone network 100 may support an interior routing protocol to distribute network topology information and route packets between ingress and egress points based on best-effort routing (e.g., destination-based shortest-path routing) through nodes n1-n9. A centralized network management system 105 may be employed to (i) provision virtual circuits, or packet flows, through backbone network 100; (ii) monitor capacity and utilization of links 101; and (iii) coordinate calculation and installation of provisioned paths. Forwarding tables are used by each node to forward each received packet to the next node toward its destination. In addition, centralized network management system 105 may also be employed to collect and distribute network topology information.
An interior routing protocol is employed to determine forwarding of packets between a source and destination pair along a path through the nodes of the backbone network. Packets received by a node are forwarded to other nodes based on a forwarding table constructed in accordance with the interior routing protocol or routes installed with explicit route provisioning. An interior routing protocol may also specify the exchange of network topology and link-state information (“network topology information”) among nodes to allow a node to construct the corresponding forwarding table. In addition, some routing protocols associate a link “cost” with each link between nodes. This link cost may be associated with, for example, average link utilization or revenue generated by the link, as well as link importance in the network. When link-state information or link-bandwidth (e.g., connectivity or available bandwidth) is exchanged between routers, each node in the network has a complete description of the network's topology. An example of a widely used, interior routing protocol for “best-effort” routing is the Open Shortest Path First (OSPF) protocol.
Routing protocols, in addition to providing connectivity, may also enable traffic management. The Multi-Protocol Label Switched (MPLS) standard, for example, allows such routing protocols in backbone networks. The MPLS standard may be employed for networks having virtual circuits (packet flows) with provisioned service levels (also known as guaranteed quality-of-service (QoS)).
A provisioned service level may be, for example, a guaranteed minimum bandwidth for the path of a packet flow through the backbone network. This path having a guaranteed level of service between ingress and egress points may be referred to as a Network Tunnel Path (NTP). As would be apparent to one skilled in the art, specific implementations of NTPs exist for different types of networks. As examples of NTPs, virtual circuits may be established for packet flows in TCP/IP networks, virtual circuits may be established for cells in Asynchronous Transfer Mode (ATM) networks, and label-switched paths (LSPs) may be established for packets in MPLS networks. Packets of a signaling protocol, such as RSVP (Reservation Protocol for IP and MPLS networks) or LDP (Label Distribution Protocol for MPLS networks), may be used to reserve link bandwidth and establish an NTP, once routing for the NTP is calculated. An NTP may be provisioned as an explicit route along a specific path between nodes of the backbone network, i.e., when an NTP is provisioned for a packet flow, all intermediate nodes between the ingress and egress points of the NTP may be specified through which each packet of the flow passes.
In MPLS networks, packets are encapsulated by appending to the packet, or forming from the packet, additional information when the packet is received at an ingress point. The additional information, called a label, is used by routers of the backbone network to forward the packets. FIG. 2 shows such an encapsulated packet 200 having a label 201 appended to packet 202. The label summarizes information in the packet header. The summary may be based on the header field and include an origination (source) address field (o) 210 identifying the address of the ingress point and a termination (destination) address field (t) 211 identifying the address of the egress point(s). In some cases, the label may simply be a pointer that identifies or is otherwise related to specific origination and termination address fields in the header of the received packet. The label also includes one or more service-level fields (bd) 212. Service-level field 212 may identify a desired service level for the virtual circuit (called a “demand”), such as minimum bandwidth required. In some networks, the service-level field is implied from the label itself. Other fields 213 may be included in label 201, such as MPLS standard version, interior routing protocol version, maximum delay, or other types of service-level parameters. Label 201 may alternatively be inserted into packet header (PH) 214 of packet 202, so the order of fields shown in FIG. 2 is exemplary only. Backbone networks may employ labels to group encapsulated packets having similar LSPs into classes (equivalence classes), and methods for forwarding equivalence classes may be employed to simplify calculation of routing for LSPs.
To generate a forwarding table, a set of preferred paths through the network nodes is computed, and weights may be used to calculate the set of preferred paths. Each preferred path has a minimum total weight between nodes (the total weight of a path being the summation of the weights of all links in the path), which is employed in a technique known in the art as shortest-path routing. The resulting set of preferred paths may be defined with a shortest-path tree (SPT). The forwarding table with routing information (e.g., source-destination pair, source ports, and destination ports) is generated from the SPT. The routing information is then used to forward a received packet to its destination along the shortest path of the SPT. The SPT may be calculated using an algorithm such as Dijkstra's algorithm, described in E. Dijkstra, “A Note: Two Problems In Connection With Graphs,” Numerical Mathematics, vol. 1, 1959, pp. 269-271, the teachings of which are incorporated herein by reference.
A common shortest-path routing algorithm employed by routers to generate routing of an LSP is the min-hop algorithm. In the min-hop algorithm, each router calculates a path through the backbone network for the stream of packets (packet flow) between the ingress and egress points. Each router constructs a path for routing the packet flow from the ingress point to the egress point with the least number (“min”) of feasible links (“hops”) (a feasible link is a link that has sufficient capacity to route the packet flow). Routing schemes of the prior art, such as shortest-path routing, forward packets based only on destination addresses and use only static and traffic-characteristic-independent link weights to calculate paths for routing tables. Some links on the shortest path between certain pairs of ingress and egress points may be congested, while other links on alternative paths are under-utilized.
A signaling mechanism, such as RSVP or LDP, may be employed to both reserve and establish a connection through the network for a packet flow. The signaling mechanism may specify quality-of-service attributes for the LSP traversing the backbone network. Link congestion caused by shortest-path routing of multiple LSPs may cause rejection of reservation requests by signaling mechanisms, even though sufficient levels of service (quality-of-service guarantees) for the LSP may exist in alternative, under-utilized paths that are only slightly longer. Available network resources are not always efficiently utilized when shortest-path routing is employed.
The Border Gateway Protocol (BGP) is an interautonomous system routing protocol. An autonomous system is a network or group of networks under a common administration and with common routing policies. An interautonomous system routing protocol is used to route data between autonomous systems. BGP is used to exchange routing information for the Internet and is the protocol used between Internet service providers (ISPs). Customer networks, such as universities and corporations, usually employ an Interior Gateway Protocol (IGP), such as Routing Information Protocol (RIP) or Open Shortest Path First (OSPF), for the exchange of routing information within their networks. Customers connect to ISPs, and ISPs use BGP to exchange customer and ISP routes. BGP can be used between autonomous systems, or a service provider can use BGP to exchange routes within an autonomous system.
A major problem in networks is BGP-induced traffic variation. Extreme network traffic fluctuations can happen for a variety of reasons. For example, in the case of a large Internet service provider exchanging traffic with several other providers, the traffic exchange between carriers is typically specified by total traffic volumes over long time periods and possibly a peak rate limit (usually just determined by physical link capacities). The actual distribution of traffic entering at an ingress point to the various network egress points might not be known a priori and can change over time. This is because the distribution is determined by many factors, such as intrinsic changes in traffic to different destination prefixes, and by routing changes either made locally by the carrier or due to changes made in other autonomous systems over which the carrier has no control. Intrinsic changes in traffic distribution can be caused by many factors, such as the sudden appearance of flash crowds responding to special events. An example of local routing changes that can affect the traffic distribution is IGP weight changes combined with “hot-potato” routing, which can change the network egress point that traffic destined to a set of prefixes would otherwise choose. “Hot-potato” routing is a form of routing in which the nodes of a network have no buffers to store packets in before they are moved on to their final predetermined destination, such that each packet that is routed is constantly transferred until it reaches its final destination. Thus, the packet is bounced around like a “hot potato,” sometimes moving further away from its destination because it has to keep moving through the network. Another example is the change in BGP when a Multi-Exit Discriminator (MED) is employed. An MED, also referred to as the “external metric” of a route, is a suggestion to external neighbors about the preferred path into an autonomous system that has multiple entry points. While local routing changes are under a carrier's control and hence change traffic patterns only at planned instances, unpredictable traffic shifts can happen when routing changes in other autonomous systems affect downstream autonomous systems. Due to widespread use of hot-potato routing, IGP weight changes (which can be due to new links being added, maintenance, traffic engineering, etc.) in an autonomous system can cause significant shifts in traffic patterns. Changes in IGP costs can affect the BGP route for a significant percentage of the prefixes, and the affected prefixes can account for a significant percentage of the traffic. Thus, significant shifts in traffic may happen at a carrier due to changes elsewhere in the network.
Another reason that high traffic variability should be considered is that users or carriers entering peering agreements might not be able to characterize their traffic to various sites well. It is much easier to estimate only the total aggregate bandwidth that is either received or sent. Hence, it is preferable to avoid having to rely on knowing the exact traffic matrix and instead use only a partial specification of the traffic matrix. Also, even when the traffic matrix is known, it is often difficult to detect changes in the traffic distribution.
Network congestion typically occurs either due to loss of capacity (upon router or link failures) or due to increased capacity demand (caused by large increases in traffic). In response to these uncontrollable events, carriers should and repeatedly adapt their intra-domain routing to avoid network congestion or have sufficient capacity set aside a priori to accommodate the different traffic and failure patterns that can occur without resorting to routing changes. It is preferable to avoid frequent intra-domain routing changes due to operational complexity and costs, and due to the risk of network instability if changes are not implemented correctly. Moreover, as discussed above, changes in one autonomous system may cause cascading traffic changes in other autonomous systems, thereby affecting the overall stability of many Internet paths. The trade-off in avoiding large routing changes is the significant capacity overprovisioning that must be done to accommodate failures or changing traffic patterns. Ideally, providers would prefer to use an almost-fixed routing scheme that (i) does not require traffic-dependent dynamic adaptation of configuration parameters, (ii) minimizes dynamic capacity re-allocation after failures, and (iii) is minimal in its overprovisioning needs.
Another application where the traffic matrix is unknown a priori is the provision of network-based virtual private network (VPN) services to enterprise customers. Here, a service-level agreement with each customer specifies the amount of traffic that can be sent or received by each site belonging to a VPN. In this scenario, users do not know their traffic matrices and specify to the carrier only the total traffic volume and the peak rate. It is the carrier's task to transport all of the offered VPN traffic to the network and carry that traffic without introducing too much delay. The actual traffic distribution from each site to the other sites is typically unknown and could vary by time-of-day. The carrier network is tasked to carry all of the offered VPN traffic without experiencing network congestion upon traffic-pattern changes or upon node or link failures.
Networks for grid computing provide a further scenario in which traffic variations can be extreme, and the traffic matrix is not known a priori. In grid computing, a complex computational task is partitioned amongst different computing nodes that can be geographically distributed and are connected by a network. The communication patterns amongst grid-computing nodes are highly unpredictable and also can experience high burst rates. Since the traffic matrix is not known a priori, one option is to dynamically reserve capacity over an underlying network, but this approach will be too slow for many grid-computing applications. Because of the high variability in destinations and the bursty nature of the traffic, overprovisioning the network leads to very poor capacity usage most of the time.
To provide good service when traffic patterns can change uncontrollably, carriers should either quickly and repeatedly adapt their intra-domain routing to avoid network congestion or have sufficient capacity set aside a priori to accommodate the different traffic patterns that can occur without resorting to routing changes. Service providers prefer to avoid frequent intra-domain routing changes due to (i) operational complexity and costs and (ii) the risk of network instability if link metric changes are not implemented correctly. Moreover, changes in one autonomous system in a BGP application may cause cascading traffic changes in other autonomous systems, thereby affecting the overall stability of many Internet paths. The trade-off in avoiding routing changes is the significant capacity overprovisioning that can be done to accommodate changing traffic patterns while keeping the routing fixed. Ideally, providers would like to use a fixed routing scheme that does not require traffic-dependent dynamic adaptation of configuration parameters and is parsimonious in its capacity needs.
Moreover, in IP-over-Optical Transport Networks (OTN), routers are connected through a reconfigurable switched optical backbone, or OTN, consisting of optical cross-connects (OXCs) that are typically less expensive than IP router ports. The OXCs are interconnected in a mesh topology using wave-division multiplexing (WDM) links. The core optical backbone consisting of such OXCs takes over the functions of switching, grooming, and restoration at the optical layer. Since the IP traffic flow is carried on an optical-layer circuit (called a “lightpath”), the bypass of router ports for transit traffic creates a basis for huge economies of scale to be reaped by interconnecting IP routers over an optical backbone in IP-over-OTN. By moving transit traffic from the routers to the optical switches, the requirement to upgrade router Point-of-Presence (PoP) configurations with increasing traffic is minimized, since optical switches are more scalable due to their typically increased port count over that of routers. In an IP-over-OTN architecture, a router line card is typically more expensive than an optical switch card, and thus, network cost is typically reduced by keeping traffic mostly in the optical layer. Also, since optical switches are typically much more reliable than routers, their architecture is typically more robust and reliable. Because routers are interconnected over a switched optical backbone, the routing process compromises between keeping traffic at the optical layer and using intermediate routers for packet grooming in order to achieve efficient statistical multiplexing of data traffic.
Dynamic provisioning of bandwidth-guaranteed paths with fast restoration capability is a desirable network service feature for many networks, such as Multi-Protocol Label Switched (MPLS) networks and optical mesh networks. In optical networks, fast restoration is also desirable, since optical transport networks carry a variety of traffic types, each with different, stringent reliability requirements. Similar fast restoration capabilities may be used in MPLS networks in order to provide the needed reliability for services such as packetized voice, critical virtual private network (VPN) traffic, or other quality-of-service (QoS) guarantees.
A connection in a network might be protected at the path level or at the link level. For link restoration (also referred to as local restoration or as fast restoration), each link of the connection is protected by a set of one or more pre-provisioned detour paths that exclude the link being protected. Upon failure of the link, traffic on the failed link is switched to the detour paths. Thus, link restoration provides a local mechanism to route around a link failure. In path restoration, the primary, or working, path of the connection is protected by a “diverse” backup path from source to destination. Upon failure of any of the resources on the working path, traffic is switched to the backup path by the source node. Link restoration might typically restore service much faster than path restoration because restoration is locally activated and, unlike path restoration, failure information need not propagate back through the network to the source.
Service restoration is an important requirement of optical networks. If a network element fails, such as a node (optical switch) or link (optical fiber), the failure causes one or more particular wavelength paths to fail, and affected traffic flow(s) must be restored using an alternative path within a very short interval (e.g., 50 ms). To accomplish relatively rapid restoration times, provisioning identifies, for each wavelength path, two paths through the network: a primary (active) path and a secondary (backup) path. The backup path is link disjoint (active and backup paths do not share links) or node disjoint (active and backup paths do not share either nodes or links) with the primary path. The capacity of links in the backup path may be exclusively assigned to a corresponding primary path (e.g., wavelength), or, for network bandwidth usage efficiency, the capacity may be shared between links of backup paths for different primary paths, depending on the type of restoration desired. Optical network capacity design typically accounts for restoration needs to route disjoint secondary paths with possible sharing.
Robust network routing in the modern, highly dynamic and changing traffic environment can be done only with knowledge of which network routing method(s) an Internet Service Provider should use so as to (i) accommodate users demanding “good” service while being unpredictable in the traffic that they would like to send to different destinations, (ii) minimize the amount of “overprovisioning” that needs to be done in the network in order to make “best-effort networking better” without resorting to sophisticated traffic prediction and management mechanisms, and (iii) operate the network efficiently with mostly static routing configurations and without dynamic routing adjustments, thereby avoiding congestion caused by drastic changes in traffic flows between a network's ingress and egress routers. Achieving these goals has been difficult, resulting instead in networks being very much overprovisioned in order to avoid the management complexity of implementing traffic management schemes that adapt network routing to changed traffic demands.
SUMMARY OF THE INVENTION
The present invention provides a scheme based on the idea of replacing shortest-path Interior Gateway Protocol (IGP) routing within a carrier's domain by a modified routing scheme that routes traffic to the destination after ensuring that it passes through one or more pre-determined intermediate nodes also in the carrier's domain. (The assignment of intermediate nodes is made at the flow level to avoid packet resequencing issues.) Egress nodes are still chosen based on Border Gateway Protocol (BGP)-determined autonomous system paths and auxiliary carrier routing policies such as hot-potato routing. A scheme consistent with one embodiment of the present invention changes the IGP path selection of direct shortest paths to one that passes through one or more a priori-assigned intermediate nodes. In MPLS networks, this routing through one or more pre-determined intermediate nodes can be accomplished using a pre-configured set of MPLS LSPs between the ingress node and a chosen set of one or more intermediate nodes to which flows are assigned according to specified probabilities. In pure IP networks, this routing can be accomplished by tunneling packets to the one or more pre-determined intermediate nodes first. This routing with pre-determined selection of one or more intermediate nodes is sufficient to handle all traffic patterns that are permissible, subject to edge-link capacity constraints, and further, provides protection against router and optical-layer link failures. Moreover, routing adaptations are not needed when the traffic matrix changes, and the scheme is bandwidth efficient.
The present invention further provides a routing scheme that, when applied to IP-over-OTN or other circuit-switched networks, can route packets in the optical layer with packet grooming at only one intermediate router and can provide the desirable statistical multiplexing properties of packet switching with highly variable traffic.
In one embodiment, the present invention provides a method of routing data through a network of nodes interconnected by links and having at least one ingress point and at least one egress point, the method comprising the steps of: (a) receiving a request for a path with a service demand for routing data between the ingress point and the egress point; (b) selecting a set of one or more intermediate nodes between the ingress point and the egress point; (c) determining, based on a bandwidth of said network, respective fractions of the data to send from the ingress point to each node of the set of one or more intermediate nodes; (d) routing the data in the determined respective fractions from the ingress point to each node of the set of one or more intermediate nodes; and (e) routing the data from each node of the set of one or more intermediate nodes to the egress point.
In another embodiment, the present invention provides an apparatus for routing data through a network of nodes interconnected by links and having at least one ingress point and one egress point, the apparatus comprising an input module, a processing module, and a router. The input module is adapted to receive (i) a request for a path with a service demand for routing data between the ingress point and the egress point and (ii) the data associated with the request. The processing module is adapted to determine the path of the request, wherein the processing module determines the path by (a) selecting a set of one or more intermediate nodes between the ingress point and the egress point; and (b) determining, based on a bandwidth of said network, respective fractions of the data to send from the ingress point to each node of the set of one or more intermediate nodes. The router is adapted to transfer the packets from the input module to an output module of the router in accordance with the path of the request, wherein the router is adapted to (c) route the data in the determined respective fractions from the ingress point to each node of the set of one or more intermediate nodes; and (d) route the data from each node of the set of one or more intermediate nodes to the egress point.
In yet another embodiment, the present invention provides a computer-readable medium having stored thereon a plurality of instructions, the plurality of instructions including instructions which, when executed by a processor, cause the processor to implement a method for routing data through a network of nodes interconnected by links and having at least one ingress point and one egress point, the method comprising the steps of: (a) receiving a request for a path with a service demand for routing data between the ingress point and the egress point; (b) selecting a set of one or more intermediate nodes between the ingress point and the egress point; (c) determining, based on a bandwidth of said network, respective fractions of the data to send from the ingress point to each node of the set of one or more intermediate nodes; (d) routing the data in the determined respective fractions from the ingress point to each node of the set of one or more intermediate nodes; and (e) routing the data from each node of the set of one or more intermediate nodes to the egress point.
In a further embodiment, the present invention provides a system for routing data through a network of nodes interconnected by links and having at least one ingress point and one egress point, the system comprising means for receiving (i) a request for a path with a service demand for routing data between the ingress point and the egress point and (ii) the data associated with the request. The system further includes means for determining the path of the request by (a) selecting a set of one or more intermediate nodes between the ingress point and the egress point; and (b) determining, based on a bandwidth of said network, respective fractions of the data to send from the ingress point to each node of the set of one or more intermediate nodes. The system further includes means for transferring the packets, in accordance with the path of the request, by (c) routing the data in the determined respective fractions from the ingress point to each node of the set of one or more intermediate nodes; and (d) routing the data from each node of the set of one or more intermediate nodes to the egress point.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows an exemplary backbone network of the prior art having nodes interconnected through links that allow communication between other packet networks;
FIG. 2 shows an encapsulated packet employed by the backbone network of FIG. 1 to route packets from an ingress point to an egress point;
FIG. 3 shows a network of interconnected nodes that employs a method of routing with service-level guarantees for routing label-switched paths consistent with one embodiment of the present invention;
FIG. 4 shows physical and logical views of an exemplary two-phase routing scheme consistent with one embodiment of the present invention;
FIG. 5 is a flow chart showing an exemplary method for a routing architecture consistent with one embodiment of the present invention;
FIG. 6 shows one step in an exemplary primal-dual linear program consistent with one embodiment of the present invention;
FIG. 7 shows a schematic of a routing guarantee for row sum {circumflex over (R)} and column sum Ĉ in an algorithm consistent with one embodiment of the present invention;
FIG. 8 shows an exemplary 15-node network with 28 bi-directional links in a network topology representative of a carrier backbone network, employed for simulations of exemplary implementations of the present invention;
FIG. 9 is a graph of simulation results comparing scaling factors for the exemplary network of FIG. 8; and
FIG. 10 is a graph of simulation results comparing scaling factors for an exemplary 20-node network.



DETAILED DESCRIPTION
FIG. 3 shows a network 300 of interconnected nodes n1-n10 that employs an exemplary implementation of the routing method with service-level guarantees in accordance with the present invention. The routing method determines a path through network 300 for a request for a network tunnel path, such as a label-switched path (LSP). Each of the nodes n1-n10 includes one or more routers that forward packets based on a forwarding table constructed from paths determined in accordance with a routing method of the present invention. The exemplary routing method routes packets of the requested LSP in two phases, whereby incoming traffic is first sent to a set of one or more intermediate nodes in predetermined proportions, and then from the intermediate nodes to the final destination, to maximize network throughput. Such routing method might not necessarily route packets of a requested LSP along the shortest path, or minimum number of hops, through the network.
While the exemplary embodiments of the present invention are described herein for networks employing the MPLS standard with path requests with associated service levels, such as LSP requests, the present invention is not so limited. The present invention may also be employed in other contexts, such as where requests are received for Network Tunnel Paths (NTPs) having a guaranteed level of service between ingress and egress points. NTPs may be, for example, virtual circuits for packet flows in TCP/IP networks, connections of cells in Asynchronous Transfer Mode (ATM) networks, and LSPs (for packets in MPLS networks). The present invention may also have particular utility in IP-over-OTN (or other circuit-switched networks) in the context of routers that are connected through a reconfigurable switched optical backbone consisting of OXCs, whereby the core optical backbone takes over the functions of switching, grooming, and restoration at the optical layer.
A network of interconnected nodes such as network 300 is defined as G(N, E), where N is the set of nodes n1 through n10, and E is the set of links (arcs) or (directed) edges interconnecting the nodes. While, in the exemplary embodiments described herein, the value for available resources such as service level is bandwidth capacity of a link or path, service-level values in other embodiments may alternatively or additionally include one or more link parameters such as delay, probability of packet loss, revenue, or other quality-of-service parameter. As known in the art, one or more of these various service-level values may be expressed by a quantity termed effective bandwidth. A link eij in the set E of links has two subscripts, i and j (0<i, j≦N), representing the nodes ni and nj connected by link eij. Without loss of generality, each link eij is directional (packet flows go from node ni to node nj).
Sources S1, S2, and S3 in FIG. 3 may be packet networks collectively providing packet flows to routers in nodes n1, n2, n3, n5, and n9, which nodes are potential ingress points for connecting to external networks, e.g., other carriers. Similarly, destinations D1, D2, and D3 may be packet networks collectively receiving packet flows from routers in nodes n3, n4, n8, n9, and n10, which nodes are potential egress points for connecting to external networks. It should be recognized that any node in a network could be an ingress point and/or an egress point. Sources S1, S2, and S3 are connected to ingress points, while destinations D1, D2, and D3 are connected to egress points. Source-destination pairs are defined as (S1, D1), (S1, D2), (S1, D3), (S2, D1), (S2, D2), (S2, D3), (S3, D1), (S3, D2), and (S3, D3), and each node may support one or more sources and/or one or more destinations. Nodes n1-n10 may also have, or have access to, current network topology and link status information (hereinafter referred to as “network topology”), which may be provided and distributed through the network using a distributed protocol (e.g., by control packets conforming to the OSPF protocol).
Sources S1, S2, and S3 generate packets for new or currently provisioned LSPs in network 300, which packets include fields identifying the ingress-egress point pair (e.g., address of either source S1, S2, or S3 and address of either destination D1, D2, and D3). Signaling packets of, for example, RSVP or LDP may be used to communicate quality-of-service (QoS) attributes or guarantees, such as bandwidth, to network elements (e.g., routers or nodes); however, packets of LSPs may also include values for one or more service-level parameters corresponding to QoS attributes or guarantees. These packets of LSPs transferred through network 300 may conform to the MPLS standard and may have a format similar to that shown and described with respect to FIG. 2.
For network 300 shown in FIG. 3, nine potential ingress-egress point pairs (source-destination pairs) exist. For the following discussion, each link (i,j) (also referred to herein as eij) interconnecting nodes ni and nj has an associated available capacity uij (or ue), termed residual bandwidth. Residual bandwidth ue of a link is the difference between the total bandwidth of the link and the sum of the bandwidth demands of LSPs that are currently assigned to that link. Networks may exchange information regarding residual capacity of links (such as in QoS shortest-path first (QoSPF) networks), which may be employed for distributed calculation of routes. Residual bandwidth may commonly be expressed in, for example, kbits/sec or Mbits/sec, or may be expressed as a percentage of the link's total capacity. Each link (i,j) interconnecting nodes ni and nj may also have an associated link cost cij (or ce), i.e., an associated scalar weight that may correspond to a relative usage, importance, or other cost of the particular link. Link cost may also refer to the scalar weight of a particular ingress-egress point pair. Link cost may be assigned to a particular link to allow routing algorithms to favor or disfavor routing through the particular link because of, for example, delay, cost to provide bandwidth, other traffic engineering considerations, or other physical link-layer considerations.
In general, a request arrives at network 300 to provision and route a path between an ingress point o and egress point t having a requested service level of bd (a “demand” bd). For the exemplary network of FIG. 3, this may be an LSP or other form of NTP request to provision a path between a source-destination pair, e.g., (S1, D1), with a requested bandwidth bd Mb/sec. LSP requests may arrive one at a time, with no a priori knowledge of the characteristics of demands for bandwidth by future LSP requests. In addition, no a priori knowledge of (i) the characteristics of QoS attributes or guarantees, (ii) connection arrivals, hold time, or departures, and (iii) other traffic-engineering information is necessarily available. The demand bd may be an “equivalent” or “effective” bandwidth value since the packets of a packet flow may represent a stochastic process with varying bandwidth needs. As is known in the art, service-level (e.g., QoS) attributes or requirements may be translated into an equivalent or effective bandwidth value. The equivalent or effective bandwidth value is a deterministic value approximating the stochastic variable based on, e.g., peak and average packet rate, arrival and hold times, and connection duration.
A routing method in accordance with the present invention evaluates and routes an LSP along one or more paths through the network between ingress-egress point pairs. The set P is the set of specific (distinguished) node ingress-egress point pairs included in the network G(N, E) that are the potential source-destination pairs (S1, D1), (S1, D2), . . . , (S3, D3). An element of the set P is denoted as (s, d) (i.e., (s, d) εP), where s and d correspond, respectively, to a source network and a destination network. Multiple LSPs may be provisioned between an element (s, d).
An LSP request to network 300 may be implemented either through a centralized network management system (not shown in FIG. 3) or by control messages provided to nodes n1-n10 of the network 300 in accordance with a distributed protocol. Either a centralized network management system and/or each network router implements an exemplary routing method for the LSP request to determine a path to be provisioned through the network corresponding to the requested LSP. Provisioning by either the centralized network management system and/or each network router allows RSVP control (e.g., QoS requests of the RSVP signaling protocol) to establish one or more connections (packet flows) with, for example, a demanded bandwidth or other type of service level.
The node-arc incidence matrix M is defined as an (n×e) matrix (where n is equal to the number of elements of the set N, and e is equal to the number of elements of the set E) in which each row corresponds to a different node n of the set N, and each column corresponds to a different link e of the set E. Each column has two non-zero entries (i,j) for the corresponding link eij between nodes ni and nj. The column corresponding to link eij has a “+1” value in the row i, a “−1” value in the row j, and a “0” value in each position corresponding to all other rows.
The total amount of traffic that enters (or leaves) an ingress (or egress) node in the network is bounded by the total capacity of all external ingress (or egress) links (e.g., line cards to customer networks or other carriers) at that node. For any given node i, the upper bounds on the total amount of traffic (e.g., bandwidth or other service level) leaving from node i is Ri, and the upper bounds on the total amount of traffic (e.g., bandwidth or other service level) entering node i is Ci. These link-capacity bounds, which are modeled based on factors such as the maximum capacity of hardware physically located inside a router's chassis, constrain the point-to-point matrix for the traffic in the network. These constraints might be the only known aspects of the traffic to be carried by the network, and knowing these is equivalent to knowing the row and column sum bounds on the traffic matrix, i.e., the maximum possible row sum indicates the maximum possible outgoing traffic, and the maximum possible column sum indicates the maximum possible incoming traffic. Accordingly, any allowable traffic matrix T=<tij> for the network obeys the following equations (1) and (2), where T is an (n×n) matrix (wherein n is the number of nodes), with the (i,j)th entry representing the traffic from node i to node j:




  
    
      
        
          
            
              ∑
              
                j
                :
                
                  
                
                ⁢
                
                  j
                  ≠
                  i
                
              
              n
            
            ⁢
            
              t
              ij
            
          
          =
          
            
              R
              i
            
            ⁢
            
              
            
            ⁢
            
              ∀
              
                i
                ∈
                N
              
            
          
        
        ,
        and
      
    
    
      
        (
        1
        )
      
    
  
  
    
      
        
          
            ∑
            
              j
              :
              
                
              
              ⁢
              
                j
                ≠
                i
              
            
            n
          
          ⁢
          
            t
            ji
          
        
        =
        
          
            C
            i
          
          ⁢
          
            
          
          ⁢
          
            ∀
            
              i
              ∈
              
                N
                .
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




It is sufficient to consider equality (as opposed to ≦) in the foregoing equations (1) and (2) because any matrix T′εT(R,C) whose any row or column sums to a value less than the given bounds can be transformed to a matrix T=T′+T′′εT(R,C) by addition of a matrix T′′ with non-negative (non-diagonal) entries. T(R,C) represents the set of all possible traffic matrices. Thus, any routing scheme that routes T can also route T′.
For given Ri and Ci values, the set T(R,C) of all such matrices that are specified only by their row and column sums may be denoted by the following equation (3):




  
    
      
        
          T
          ⁡
          
            (
            
              R
              ,
              C
            
            )
          
        
        =
        
          
            {
            
              
                
                  〈
                  
                    t
                    ij
                  
                  〉
                
                ⁢
                
                  
                
                ⁢
                where
                ⁢
                
                  
                
                ⁢
                
                  
                    ∑
                    
                      j
                      ≠
                      i
                    
                  
                  ⁢
                  
                    t
                    ij
                  
                
              
              =
              
                
                  
                    R
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  and
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        ≠
                        i
                      
                    
                    ⁢
                    
                      t
                      ji
                    
                  
                
                =
                
                  
                    C
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ∀
                    i
                  
                
              
            
            }
          
          .
        
      
    
    
      
        (
        3
        )
      
    
  




It should be noted that the traffic distribution T could be any matrix in T(R,C) and could change over time. In a routing architecture consistent with certain embodiments of the present invention, it is desirable that the only assumption that needs to be made about T is that it is specified only by row and column sum bounds. Accordingly, a routing strategy consistent with one embodiment of the present invention desirably (i) should permit routing every matrix in T(R,C), (ii) should not require reconfiguration of existing connections, i.e., should be oblivious to changes in the traffic matrix T as long as it belongs to T(R,C), and (iii) should be bandwidth efficient, i.e., should not use much more bandwidth than the conventional strategy of provisioning min(Ri, Cj) amount of demand from node i to node j.

One known model of a method for specifying bandwidth requirements of a VPN is the hose model, as explained in N. G. Duffield, P. Goyal, A. G. Greenberg, P. P. Mishra, K. K. Ramnakrishnan, J. E. van der Merwe, “A flexible model for resource management in virtual private network,” ACM SIGCOMM 1999, August 1999, the teachings of which are incorporated herein by reference. In this model, the traffic matrix is specified only partially, such that, for each VPN endpoint i, only Ri and Ci are specified, Ri being the maximum total bandwidth of traffic that i will send into the network at any time, and Ci being the maximum total bandwidth of traffic that i will ever receive from the network at any time. The network capacity reserved for the VPN should be sufficient for every possible traffic pattern that is consistent with the Ri and Ci values.
A routing scheme consistent with certain embodiments of the present invention permits a network to meet arbitrary (and possibly rapidly changing) traffic demands without requiring sophisticated traffic engineering mechanisms or additional network signaling. In fact, the network does not even need to detect changes in the traffic distribution. The only knowledge about the traffic that might be required is the limits imposed by the total capacity of all line cards that connect to external interfaces at network edges.
With reference now to FIG. 4, a two-phase routing scheme consistent with one embodiment of the present invention is illustrated in both physical and logical views. In phase 1 (401), a predetermined fraction αk of the traffic entering the network at any node i is distributed to one or more intermediate nodes k, independent of the final destination of the traffic. In phase 2 (402), each node k receives traffic destined for different destinations and routes the received traffic to the respective destinations. One method of implementing this routing scheme is to form fixed-bandwidth tunnels between the nodes, some of which carry phase 1 traffic, and others of which carry phase 2 traffic. The two-phase routing strategy works because the bandwidth required for these tunnels depends only on R and C, and not on the individual entries in the traffic matrix. It is noted that, in phase 1, α1, α2, . . . , αn, are such that the following equation (4) is satisfied:

Σi=1nαi=1.  (4)

The two-phase routing method will now be described in further detail. For a given node i with maximum outgoing traffic Ri, node i sends αkRi amount of this traffic to intermediate node k during phase 1, for each kεN. Thus, the demand from node i to node k as a result of phase 1 is αkRi. At the end of phase 1, node k has received αkRi from each node i. It is noted that, since the sum of the row bounds must equal the sum of the column bounds, the total traffic received at node k from all sources i is




  
    
      ∑
      
        i
        =
        1
      
      n
    
    ⁢
    
      
        α
        k
      
      ⁢
      
        R
        i
      
    
  
  =
  
    
      ∑
      
        j
        =
        1
      
      n
    
    ⁢
    
      
        α
        k
      
      ⁢
      
        
          C
          j
        
        .
      
    
  




Out of the traffic received at node k after phase 1, the traffic destined for node j is αktij, assuming that traffic to the same destination is split in predetermined ratios. Thus, the total traffic that needs to be routed from node k to node j during phase 2, i.e., the traffic demand from node k to node j, is as set forth in the following equation (5):





  
    
      
        
          
            ∑
            
              i
              ∈
              N
            
          
          ⁢
          
            
              α
              k
            
            ⁢
            
              t
              ij
            
          
        
        =
        
          
            α
            k
          
          ⁢
          
            
              C
              j
            
            .
          
        
      
    
    
      
        (
        5
        )
      
    
  




Accordingly, since in phase 1, k is essentially the same as j, and in phase 2, k is essentially i, the total demand from node i to node j as a result of routing in phases 1 and 2 is (αjRi+αiCj), which can be derived without knowledge of the matrix TεT(R,C). The following three properties characterize this two-phase routing scheme:

    
    
        (i) Routing is oblivious to traffic variations. The demands that need to be routed during phases 1 and 2 do not depend on the specific traffic matrix TεT(R,C), but only on the row and column sum bounds that constrain T (i.e., the set T(R,C)).
        (ii) Routed demands are traffic matrix-independent. The total demand between nodes i and j as a result of routing in phases 1 and 2 is tij′=αjRi+αiCj and does not depend on the specific matrix TεT(R,C)
        (iii) Provisioned capacity is completely used. For each matrix TεT(R,C), the routing scheme completely utilizes the associated point-to-point demands on phases 1 and 2.
    
    


Property (ii) implies that the scheme handles variability in traffic matrix TεT(R,C) by effectively routing a transformed matrix T′=<tij′> that depends only on the row and column sum bounds and the distribution ratios α1, α2, . . . , αn, and not on the specific matrix TεT(R,C), thereby rendering the routing scheme oblivious to changes in the traffic distribution.
Ensuring that the traffic distribution obeys the row or column sum bounds can be accomplished by making the row or column sum bounds equal to the sum of the line card capacities that connect to external interfaces at a node, thereby enforcing the constraint in a hard manner at the physical layer. Alternatively, a differentiated services (DiffServ)-type policing scheme, whereby traffic entering a network is classified and possibly conditioned at the boundaries of the network and assigned to different behavior aggregates, can rate-limit the total traffic that enters the network at each ingress node and guarantee that each node is not over-subscribed.
Thus, in a routing method consistent with one embodiment of the present invention, routing decisions at each source node during phase 1 do not require any network-wide state information (e.g., how the traffic at other peering points is varying), and routing decisions during phase 2 are based on the packet destination only. Additionally, the network can meet any traffic distribution as long as the ingress/egress points are not oversubscribed, and congestion can be avoided by either hard rate guarantees of line cards connecting to other carriers, or by implementing a differentiated services-type policing scheme for rate-limiting the traffic entering the network at a node. Moreover, the routing scheme is oblivious and robust to any changes in the traffic distribution, and providing end-to-end bandwidth guarantees does not require any reconfiguration of the network in real-time.
As illustrated in the flowchart of FIG. 5, a routing architecture consistent with one embodiment of the present invention may be implemented in the following exemplary method: At step 501, the method begins with computing row (or column) bounds Ri (or Ci) using inter-autonomous system peering agreements and/or rates of line cards at each node connecting to other carriers. Next, at step 502, traffic distribution ratios α1, α2, . . . , αn are computed (using an exemplary algorithm that optimizes the required network bandwidth, as will be described in further detail below). Next, at step 503, for each node pair i, j, two sets of connections (e.g., MPLS LSPs, IP tunnels, or optical layer circuits) are provisioned: one set for phase 1 of bandwidth αjRi from node i to one or more intermediate nodes and the other set for phase 2 of bandwidth αiCj from the one or more intermediate nodes to node j. Next, at step 504, traffic is routed in accordance with phases 1 and 2 (as has been described in further detail above, which requires only local operations at source and intermediate nodes. Next, at step 505, a differentiated services-type policing mechanism is used to rate-limit the total traffic that enters the network at each node. Next, at step 506, a determination is made whether the row (or column) bounds Ri (or Ci) have changed, e.g., as a result of new peering agreements or modifications to existing ones. If the bounds have not changed, then the method returns to step 504 to continue the routing operation. If the bounds have changed, then at step 507, the αi distribution ratios are re-optimized, and at step 508, the bandwidth of the LSPs (or optical layer circuits, or IP tunnels) for routing during phases 1 and 2 may be adjusted accordingly before returning to step 504.
In the foregoing method, packets belonging to the same end-to-end connection may arrive out of order at the egress node if traffic is split within the same connection. This situation may be avoided by using per-flow splitting in phase 1 of the scheme. Additionally and/or additionally, the traffic split ratios αi can be generalized to depend on source and/or destination nodes of the traffic, as will be explained in further detail below.
In a network having link capacities and constraints Ri, Ci on the egress/ingress traffic, it is desirable to route so as to minimize the maximum utilization of any link in the network. The utilization of a link may be defined as the traffic on the link divided by its capacity. If λ·T(R,C) denotes the set of all traffic matrices in T(R,C) with their entries multiplied by λ, then a linear program may be used to find the maximum multiplier α (throughput) such that all matrices in λ·T(R,C) may be routed.
For the case of equal split ratios, i.e., as αi=1/n ∀iεN, the demand between nodes i and j is (Ri+Cj)/n, and the problem reduces to the maximum concurrent flow problem, as explained in F. Shahrokhi and D. Matula, “The Maximum Concurrent Flow Problem”, Journal of ACM, 37 (2):318-334, 1990, the teachings of which are incorporated herein by reference.
An exemplary link flow-based linear programming formulation in one embodiment of the present invention will now be described, wherein flows are augmented in the solution of a primal problem, and weights are updated in a multiplicative fashion in the solution to the corresponding dual problem. The primal and dual problems and solutions may be characterized as follows:
1. If the primal problem has n variables and m resource constraints, then the dual problem will have m variables and n resource constraints. Therefore, the constraint matrix of the dual problem is the transpose of the constraint matrix of the primal problem.
2. There is a one-to-one correspondence between the primal constraints and the dual variables, i.e., a variable in the dual problem is paired with an inequality in the primal problem, and similarly for the primal variables and the dual constraints.
3. The objective function of the dual is determined by the right-hand side of the primal constraints, and similarly for the objective function of the primal and the right-hand side of the dual constraints.
In the following exemplary linear programming formulation, given a commodity index k, wherein the term “commodity” refers to the flow between a source and a destination, the source node for commodity k is denoted by s(k), the destination node by d(k), and the amount of flow corresponding to commodity k by f(k). The vector xk(e) denotes the amount of flow of commodity k on link e in the network, and δ−(i) and δ+(i) denote the sets of incoming and outgoing edges at node i, respectively. The exemplary link flow-based linear programming formulation, with the constraints of equations (6-7) and inequality (8), is set forth as follows:




  
    
      
        
          maximize
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                i
                ∈
                N
              
              n
            
            ⁢
            
              α
              i
            
          
        
        ,
      
    
    
      
        
      
    
  
  
    
      
        subject
        ⁢
        
          
        
        ⁢
        to
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  ∑
                  
                    e
                    ∈
                    
                      δ
                      -
                      
                        (
                        i
                        )
                      
                    
                  
                
                ⁢
                
                  
                    x
                    k
                  
                  ⁡
                  
                    (
                    e
                    )
                  
                
              
              =
              
                
                  ∑
                  
                    e
                    ∈
                    
                      δ
                      +
                      
                        (
                        i
                        )
                      
                    
                  
                
                ⁢
                
                  
                    x
                    k
                  
                  ⁡
                  
                    (
                    e
                    )
                  
                
              
            
          
          
            
              
                ∀
                
                  i
                  ≠
                  
                    s
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                
              
              ,
              
                d
                ⁡
                
                  (
                  k
                  )
                
              
              ,
              
                ∀
                k
              
              ,
            
          
        
      
    
    
      
        (
        6
        )
      
    
  
  
    
      
        
          
            
              
                
                  ∑
                  
                    e
                    ∈
                    
                      δ
                      +
                      
                        (
                        i
                        )
                      
                    
                  
                
                ⁢
                
                  
                    x
                    k
                  
                  ⁡
                  
                    (
                    e
                    )
                  
                
              
              =
              
                
                  
                    a
                    
                      s
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    C
                    
                      d
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                
                +
                
                  
                    a
                    
                      d
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                  ⁢
                  
                    R
                    
                      s
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                
              
            
          
          
            
              
                i
                =
                
                  s
                  ⁡
                  
                    (
                    k
                    )
                  
                
              
              ,
              
                ∀
                k
              
              ,
            
          
        
      
    
    
      
        (
        7
        )
      
    
  
  
    
      
        
          
            
              
                
                  ∑
                  k
                
                ⁢
                
                  
                    x
                    k
                  
                  ⁡
                  
                    (
                    e
                    )
                  
                
              
              ≤
              
                u
                e
              
            
          
          
            
              ∀
              
                e
                ∈
                
                  E
                  .
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




The above linear program includes two sets of decision variables: the traffic split ratios as and the flows on link e for commodity k, denoted by xk(e). It is noted that the demand for commodity k will be given by αs(k)Cd(k)+αd(k)Rs(k). The αi values in an optimal solution of the above linear program are represented by αi*, and the optimum objective function value is represented by λ*, wherein λ*=Σiαi*. If λ*≧1, then the problem is feasible, i.e., the given demands can be routed on the network. The αi*values can be reduced by a factor of λ* to get the actual split ratios, and the explicit paths along which demands are routed can be determined from the solution of the above problem. If the value λ*<1, then the problem is infeasible. In this case, the egress (or ingress) constraints Ri (Ci) may be scaled down by division by a factor of 1/λ*, and the problem will then be feasible for routing under the given link capacities. Alternatively, the link capacities may be scaled up by multiplication by a factor of 1/λ* to accommodate routing of all the given demands.

An exemplary path flow-based linear programming formulation in one embodiment of the present invention, which may be used to develop a fast combinatorial Fully Polynomial Time Approximation Scheme (FPTAS) algorithm, will now be described. In the following exemplary formulation, Pij denotes the set of all paths from node i to node j, and x(P) denotes the traffic on path P. The exemplary path flow-based linear programming formulation, with the constraints of equation (9) and inequality (10), is set forth as follows:




  
    
      
        
          maximize
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                i
                ∈
                N
              
            
            ⁢
            
              a
              i
            
          
        
        ,
      
    
    
      
        
      
    
  
  
    
      
        subject
        ⁢
        
          
        
        ⁢
        to
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  ∑
                  
                    P
                    ∈
                    
                      P
                      ij
                    
                  
                
                ⁢
                
                  x
                  ⁡
                  
                    (
                    P
                    )
                  
                
              
              =
              
                
                  
                    α
                    j
                  
                  ⁢
                  
                    R
                    i
                  
                
                +
                
                  
                    α
                    i
                  
                  ⁢
                  
                    C
                    j
                  
                
              
            
          
          
            
              
                ∀
                i
              
              ,
              
                j
                ∈
                N
              
              ,
              
                i
                ≠
                j
              
              ,
            
          
        
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        
          
            
              
                
                  ∑
                  
                    P
                    :
                    
                      
                    
                    ⁢
                    
                      e
                      ∈
                      P
                    
                  
                
                ⁢
                
                  x
                  ⁡
                  
                    (
                    P
                    )
                  
                
              
              ≤
              
                u
                e
              
            
          
          
            
              ∀
              
                e
                ∈
                
                  E
                  .
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




Since a network generally can have an exponential number of paths (in the size of the network), the foregoing (primal) linear program can possibly have an exponential number of variables, and its dual (which will be provided in detail below) can have an exponential number of constraints. Thus, these programs might not be well suited for running on medium- to large-sized networks. Nevertheless, such a primal/dual formulation is useful in designing a fast, polynomial-time combinatorial algorithm for the problem, as will be explained below.
A fast combinatorial approximation algorithm may be used to compute the split ratio up to (1+ε)-factor of the optimal objective function value for any ε>0. The value of ε may be selected to provide the desired degree of optimality for the solution. This algorithm is preferably an FPTAS scheme and runs in time that is polynomial in the input size and 1/ε. Since the algorithm maintains primal and dual solutions at each step, the optimality gap can be estimated by computing the ratio of the primal and dual objective function values.
A dual formulation of the linear program set forth in equation (9) and inequality (10) above associates a variable w(e) with each link-capacity constraint in inequality (10) and a variable πij with each demand constraint in equation (9). SP (i,j) denotes the shortest path PεPij under weights w(e), as set forth in the following equation (11):




  
    
      
        
          SP
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          
            min
            
              P
              ∈
              
                P
                ij
              
            
          
          ⁢
          
            
              ∑
              
                e
                ∈
                P
              
            
            ⁢
            
              
                w
                ⁡
                
                  (
                  e
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




After simplification and removal of the dual variable πij, the dual linear programming formulation may be written as follows, with the constraints of inequalities (12-13):





  
    
      
        
          minimize
          ⁢
          
            
              
            
            ⁢
            
              
            
          
          ⁢
          
            
              ∑
              
                e
                ∈
                E
              
            
            ⁢
            
              
                u
                e
              
              ⁢
              
                
              
              ⁢
              
                w
                ⁡
                
                  (
                  e
                  )
                
              
            
          
        
        ,
      
    
    
      
        
      
    
  
  
    
      
        subject
        ⁢
        
          
        
        ⁢
        to
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  
                    ∑
                    
                      
                        i
                        ⁢
                        
                          :
                        
                        ⁢
                        
                          
                        
                        ⁢
                        i
                      
                      ≠
                      k
                    
                  
                  ⁢
                  
                    
                      R
                      i
                    
                    ⁢
                    
                      SP
                      ⁡
                      
                        (
                        
                          i
                          ,
                          k
                        
                        )
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      
                        j
                        ⁢
                        
                          :
                        
                        ⁢
                        
                          
                        
                        ⁢
                        j
                      
                      ≠
                      k
                    
                  
                  ⁢
                  
                    
                      C
                      j
                    
                    ⁢
                    
                      SP
                      ⁡
                      
                        (
                        
                          k
                          ,
                          j
                        
                        )
                      
                    
                  
                
              
              ≥
              1
            
          
          
            
              
                ∀
                
                  k
                  ∈
                  N
                
              
              ,
            
          
        
      
    
    
      
        (
        12
        )
      
    
  
  
    
      
        
          
            
              
                w
                ⁡
                
                  (
                  e
                  )
                
              
              ≥
              0
            
          
          
            
              ∀
              
                e
                ∈
                
                  E
                  .
                
              
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




For a given node k, V(k) denotes the left-hand side of the constraint in inequality (12). Given the weights w(e), it is noted that V(k) can be computed in polynomial time by two shortest-path computations, one for the shortest-path tree with node k as root and reaching all destinations, and the other for the reverse shortest path tree reaching node k from all other nodes.
Given a set of weights w(e), a feasible solution for the dual program exists if and only if the following inequality (14) is satisfied:




  
    
      
        
          
            min
            
              k
              ∈
              N
            
          
          ⁢
          
            V
            ⁡
            
              (
              k
              )
            
          
        
        ≥
        1.
      
    
    
      
        (
        14
        )
      
    
  




The algorithm begins with equal initial weights w(e)=δ (the quantity δ depends on ε and is derived later). Then, the following steps (1-5) are repeated until the dual feasibility constraints are satisfied:
(1) As illustrated in FIG. 6, the node k is computed for which V(k) is minimum, thereby identifying a link k as well as paths Pi from node i to node k for all i and paths Qj from node k to node j for all j.
(2) For each eεE, Np(e) is defined as the set of nodes i for which Pi contains link e, and NQ(e) is defined as the set of nodes j for which Qj contains link e. The fraction α is then computed using the following equation (15):




  
    
      
        a
        =
        
          
            min
            
              e
              ∈
              E
            
          
          ⁢
          
            
              
                u
                e
              
              
                
                  
                    ∑
                    
                      i
                      ∈
                      
                        
                          N
                          P
                        
                        ⁡
                        
                          (
                          e
                          )
                        
                      
                    
                  
                  ⁢
                  
                    R
                    i
                  
                
                +
                
                  
                    ∑
                    
                      j
                      ∈
                      
                        
                          N
                          Q
                        
                        ⁡
                        
                          (
                          e
                          )
                        
                      
                    
                  
                  ⁢
                  
                    C
                    j
                  
                
              
            
            .
          
        
      
    
    
      
        (
        15
        )
      
    
  




(3) An amount of flow αRi is sent on path Pi for all i, and an amount of flow αCj is sent on path Qj for all j, and the total flow Δ(e) that is sent on link e is computed for all eεE. The flow on link e is incremented by Δ(e).
(4) The weights w(e) are updated for all eεE as w(e)→w(e)(1+εΔ(e)/ue).
(5) The split ratio α k associated with node k is incremented by α.
When the foregoing procedure terminates, dual feasibility constraints will be satisfied. However, primal capacity constraints on each link may be violated, since the original (and not residual) link capacity at each stage is employed in the procedure. To remedy this, the split ratios may be scaled down uniformly so that capacity constraints are obeyed.
Pseudo-code for an exemplary algorithm that may be employed to implement the foregoing described exemplary method is provided below. In the pseudo-code, array flow(e) keeps track of the traffic on link e. The variable G is initialized to 0 and remains less than 1 as long as the dual constraints remain unsatisfied. After the while loop terminates, the factor by which the capacity constraint on each link e gets violated is computed into array scale(e). Finally, the αi values are divided by the maximum capacity violation factor, and the resulting values are output as the optimum.
Two theorems related to the exemplary algorithm are provided, as follows.
Theorem 1: If L=(n−1)(ΣiεNRi+ΣjεNCj), and L′ is the minimum non-zero value of the Ri's and Cj's, and considering that the values of ε and δ are related to the approximation factor guarantee of the algorithm set forth below, then for any given ε′>0, the algorithm computes a solution with an objective function value within (1+ε′)-factor of the optimum for the following equations (16-17):




  
    
      
        
          δ
          =
          
            
              1
              +
              ɛ
            
            
              
                
                  L
                  ′
                
                ⁡
                
                  [
                  
                    
                      (
                      
                        1
                        +
                        ɛ
                      
                      )
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      L
                      
                        L
                        ′
                      
                    
                  
                  ]
                
              
              
                1
                /
                ɛ
              
            
          
        
        ,
      
    
    
      
        (
        16
        )
      
    
  
  
    
      
        ɛ
        =
        
          1
          -
          
            
              1
              
                
                  1
                  +
                  
                    ɛ
                    ′
                  
                
              
            
            .
          
        
      
    
    
      
        (
        17
        )
      
    
  




Theorem 2: For any given ε>0 chosen to provide the desired approximation factor guarantee in accordance with Theorem 1, the algorithm is polynomial in the input size and 1/ε, i.e.,




  O
  ⁢
  
    
  
  ⁢
  
    
      (
      
        
          
            n
            ⁢
            
              
            
            ⁢
            m
          
          ɛ
        
        ⁢
        
          (
          
            m
            +
            
              n
              ⁢
              
                
              
              ⁢
              log
              ⁢
              
                
              
              ⁢
              n
            
          
          )
        
        ⁢
        
          
        
        ⁢
        
          log
          
            1
            +
            ɛ
          
        
        ⁢
        
          L
          
            L
            ′
          
        
      
      )
    
    .
  




The following exemplary pseudo-code may be used to implement the exemplary algorithm set forth above:
αk→0∀kεN;
w(e)→δ∀eεE;
flow(e)→0∀eεE;
G→0;
while G<1 do
Compute shortest path of cost SP(i,j) from i to j under link costs w(e) ∀i, j εN;

    
    
        V(k)→Σi≠kRiSP(i,k)+Σj≠kCjSP(k,j);
        G→minkεNV(k);
        if G≧1 break;
        Let k be the node for which g(k) is minimum;
        Let Pi be the shortest path from i to k for all i;
        Let Qj be the shortest path from k to j for all j;
        NP(e)→{i: Pi contains e} for all e;
        NQ(e)→{j: Qj contains e} for all e;
    
    






  
    α
    ←
    
      
        min
        
          e
          ∈
          E
        
      
      ⁢
      
        
          u
          e
        
        
          
            
              ∑
              
                i
                ∈
                
                  
                    N
                    P
                  
                  ⁡
                  
                    (
                    e
                    )
                  
                
              
            
            ⁢
            
              R
              i
            
          
          +
          
            
              ∑
              
                j
                ∈
                
                  
                    N
                    Q
                  
                  ⁡
                  
                    (
                    e
                    )
                  
                
              
            
            ⁢
            
              C
              j
            
          
        
      
    
  
  ;




    
    
        Send αRi flow on path Pi for all i and αCi flow on path Qj for all j and compute resulting capacity usageΔ(e) on link e for all e;
        flow(e)→flow(e)+Δ(e) for all e;
        w(e)→w(e)(1+εΔ(e)/ue) for all e;
        α k→α k−+α;

end while

scale(e)→flow(e)/ue for all eεE;

scale_max→maxeεEscale(e);

αk→αk/scale_max for all kεN;

Output αk as the optimalt raffic split ratios;

    
    


The proofs and underlying lemmas for Theorems 1 and 2 are as follows:
Given a set of dual weights w(e), wherein D(w) represents the dual objective function value, and Γ(e) denotes the minimum value of the left-hand side of the dual program constraint set forth in inequality (12) over all nodes kεN, solving the dual program is equivalent to finding a set of weights w(e) such that D(w)/Γ(w) is minimized. The optimal objective function value of D(w)/Γ(w) is denoted by θ, i.e., θ=minwD(w)/Γ(w). The weight function at the beginning of iteration t of the while loop is denoted by wt−1, and ft−1 is the value of ΣjεNαj (primal objective function) up to the end of iteration t−1. As defined above, L=(n−1)(ΣiεNRi+ΣjεNCj), and L′ is the minimum non-zero value of the Ri's and Cj's. The algorithm terminates after iteration N.
Lemma 1: At the end of every iteration t of the algorithm ∀1≦t≦K, the following inequality (17.1) is satisfied:




  
    
      
        
          Γ
          ⁡
          
            (
            
              w
              t
            
            )
          
        
        ≤
        
          δ
          ⁢
          
            
          
          ⁢
          L
          ⁢
          
            
          
          ⁢
          
            
              ∏
              
                j
                =
                1
              
              t
            
            ⁢
            
              
                [
                
                  1
                  +
                  
                    
                      ɛ
                      θ
                    
                    ⁢
                    
                      (
                      
                        
                          f
                          j
                        
                        -
                        
                          f
                          
                            j
                            -
                            1
                          
                        
                      
                      )
                    
                  
                
                ]
              
              .
            
          
        
      
    
    
      
        (
        17.1
        )
      
    
  




Proof: The node for which V(k) is minimum is k= k, and the corresponding paths along which flow is augmented during iteration t are represented by Pi, Qj, as defined above. The weights are updated as wt(e)=wt−1(e)(1+εΔ(e)/ue) ∀eεE, where Δ(e) is the total flow sent on link e during iteration t. Using this, D(wt) may be derived as set forth in the following equation (17.2):




  
    
      
        
          
            
              
                D
                ⁡
                
                  (
                  
                    w
                    t
                  
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  
                    e
                    ∈
                    E
                  
                
                ⁢
                
                  
                    u
                    e
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      w
                      t
                    
                    ⁡
                    
                      (
                      e
                      )
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      e
                      ∈
                      E
                    
                  
                  ⁢
                  
                    
                      u
                      e
                    
                    ⁢
                    
                      
                        w
                        
                          t
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        e
                        )
                      
                    
                  
                
                +
                
                  ɛ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        e
                        ∈
                        E
                      
                    
                    ⁢
                    
                      
                        Δ
                        ⁡
                        
                          (
                          e
                          )
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          w
                          
                            t
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          e
                          )
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  D
                  ⁡
                  
                    (
                    
                      w
                      
                        t
                        -
                        1
                      
                    
                    )
                  
                
                +
                
                  ɛ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        e
                        ∈
                        E
                      
                    
                    ⁢
                    
                      
                        
                          w
                          
                            t
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          e
                          )
                        
                      
                      ⁢
                      
                        
                      
                      [
                      
                        
                          
                            ∑
                            
                              i
                              ∈
                              
                                
                                  N
                                  P
                                
                                ⁡
                                
                                  (
                                  e
                                  )
                                
                              
                            
                          
                          ⁢
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              R
                              i
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              ∈
                              
                                
                                  N
                                  Q
                                
                                ⁡
                                
                                  (
                                  e
                                  )
                                
                              
                            
                          
                          ⁢
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              C
                              j
                            
                          
                        
                      
                      ]
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  D
                  ⁡
                  
                    (
                    
                      w
                      
                        t
                        -
                        1
                      
                    
                    )
                  
                
                +
                
                  ɛ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    α
                    [
                    
                      
                        
                          ∑
                          i
                        
                        ⁢
                        
                          
                            R
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                e
                                ∈
                                
                                  P
                                  i
                                
                              
                            
                            ⁢
                            
                              
                                w
                                
                                  t
                                  -
                                  1
                                
                              
                              ⁡
                              
                                (
                                e
                                )
                              
                            
                          
                        
                      
                      +
                      
                        
                          ∑
                          j
                        
                        ⁢
                        
                          
                            C
                            j
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                e
                                ∈
                                
                                  Q
                                  j
                                
                              
                            
                            ⁢
                            
                              
                                w
                                
                                  t
                                  -
                                  1
                                
                              
                              ⁡
                              
                                (
                                e
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  D
                  ⁡
                  
                    (
                    
                      w
                      
                        t
                        -
                        1
                      
                    
                    )
                  
                
                +
                
                  ɛ
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  Γ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      (
                      
                        w
                        
                          t
                          -
                          1
                        
                      
                      )
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        17.2
        )
      
    
  




Using the foregoing derived equation for each iteration down to the first iteration, D(wt) may be defined as in the following equation (17.3):





  
    
      
        
          D
          ⁡
          
            (
            
              w
              t
            
            )
          
        
        =
        
          
            D
            ⁡
            
              (
              
                w
                0
              
              )
            
          
          +
          
            ɛ
            ⁢
            
              
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  1
                
                t
              
              ⁢
              
                
                  (
                  
                    
                      f
                      j
                    
                    -
                    
                      f
                      
                        j
                        -
                        1
                      
                    
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                
                  
                    Γ
                    ⁡
                    
                      (
                      
                        w
                        
                          j
                          -
                          1
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        17.3
        )
      
    
  




Considering now the weight function wt−w0, it is known that D(wt−w0)=D(wt)−D(w0), and also that Γ(w0)≦Σi(n−1)δRi+Σj(n−1)δCj=δL, since any of the paths Pi, Qj are at most n−1 hops in length. Hence, Γ(wt−w0)≧Γ(wt)−δL. Since θ is the optimal dual objective function value, the following inequalities (17.4-17.5) hold true:





  
    
      
        
          0
          ≤
          
            
              D
              ⁡
              
                (
                
                  
                    w
                    t
                  
                  -
                  
                    w
                    0
                  
                
                )
              
            
            
              Γ
              ⁡
              
                (
                
                  
                    w
                    t
                  
                  -
                  
                    w
                    0
                  
                
                )
              
            
          
          ≤
          
            
              
                D
                ⁡
                
                  (
                  
                    w
                    t
                  
                  )
                
              
              -
              
                D
                ⁡
                
                  (
                  
                    w
                    0
                  
                  )
                
              
            
            
              
                Γ
                ⁡
                
                  (
                  
                    w
                    t
                  
                  )
                
              
              -
              
                δ
                ⁢
                
                  
                
                ⁢
                L
              
            
          
        
        ,
      
    
    
      
        (
        17.4
        )
      
    
  
  
    
      
        
          
            D
            ⁡
            
              (
              
                w
                t
              
              )
            
          
          -
          
            D
            ⁡
            
              (
              
                w
                0
              
              )
            
          
        
        ≥
        
          
            θ
            ⁡
            
              (
              
                
                  Γ
                  ⁡
                  
                    (
                    
                      w
                      t
                    
                    )
                  
                
                -
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  L
                
              
              )
            
          
          .
        
      
    
    
      
        (
        17.5
        )
      
    
  




By combining inequality (17.5) with equation (17.3), the following inequality (17.6) may be derived:





  
    
      
        
          Γ
          ⁡
          
            (
            
              w
              t
            
            )
          
        
        ≤
        
          
            δ
            ⁢
            
              
            
            ⁢
            L
          
          +
          
            
              ɛ
              θ
            
            ⁢
            
              
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  1
                
                t
              
              ⁢
              
                
                  (
                  
                    
                      f
                      j
                    
                    -
                    
                      f
                      
                        j
                        -
                        1
                      
                    
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                
                  
                    Γ
                    ⁡
                    
                      (
                      
                        w
                        
                          j
                          -
                          1
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        17.6
        )
      
    
  




The property in Lemma 1 may now be proved using inequality (17.6) and mathematical induction on the iteration number. It is noted that the induction basis case (iteration t=1) holds true since w0(e)=δ∀eεE and Γ(w0)≦δL. Now, an estimate can be made of the factor by which the objective function value fK in the primal solution at the time the algorithm terminates needs to be scaled, in order to ensure that link capacity constraints are not violated.
Lemma 2: When the algorithm terminates, to ensure primal feasibility, the primal solution should be scaled by a factor of, at most, the following value:




  
    log
    
      1
      +
      ɛ
    
  
  ⁢
  
    
      
        1
        +
        ɛ
      
      
        δ
        ⁢
        
          
        
        ⁢
        
          L
          ′
        
      
    
    .
  




Proof: Considering any link e and its associated weight w(e), the value of w(e) is updated when flow is augmented on edge e. The sequence of flow augmentations (per iteration) on link e is Δ1, Δ2, . . . , Δr, where r≦K. The total flow routed on link e exceeds its capacity by a factor of κ, i.e., Σt=1rΔt=κue. Since the algorithm terminates when Γ(w)≧1, and since dual weights are updated by a factor of at most 1+ε after each iteration, then Γ(wK)≦1+ε. It is noted that, immediately prior to each augmentation mentioned above, the weight w(e), with coefficient at least L′, is one of the summing components of Γ(w). Hence, L′wK (e,f)≦1+ε, and the value of wK (e,f) may be given by the following equation (17.7):




  
    
      
        
          
            w
            K
          
          ⁡
          
            (
            
              e
              ,
              f
            
            )
          
        
        =
        
          δ
          ⁢
          
            
          
          ⁢
          
            
              ∏
              
                t
                =
                1
              
              r
            
            ⁢
            
              
                (
                
                  1
                  +
                  
                    
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      
                        u
                        e
                      
                    
                    ⁢
                    ɛ
                  
                
                )
              
              .
            
          
        
      
    
    
      
        (
        17.7
        )
      
    
  




Using the fact that (1+βx)≧(1+x)β∀x≧0 and any 0≦β≦1, and setting x=ε and β=(Δt/ue)≦1, the following inequalities (17.8-17.9) hold true:





  
    
      
        
          
            
              1
              +
              ɛ
            
            
              L
              ′
            
          
          ≥
          
            
              w
              K
            
            ⁡
            
              (
              
                e
                ,
                f
              
              )
            
          
          ≥
          
            δ
            ⁢
            
              
                ∏
                
                  t
                  =
                  1
                
                r
              
              ⁢
              
                
                  (
                  
                    1
                    +
                    ɛ
                  
                  )
                
                
                  
                    Δ
                    t
                  
                  /
                  
                    u
                    e
                  
                
              
            
          
          ≥
          
            
              δ
              ⁡
              
                (
                
                  1
                  +
                  ɛ
                
                )
              
            
            ⁢
            
              
                ∑
                
                  t
                  =
                  1
                
                r
              
              ⁢
              
                
                  Δ
                  t
                
                /
                
                  u
                  e
                
              
            
          
          ≥
          
            
              δ
              ⁡
              
                (
                
                  1
                  +
                  ɛ
                
                )
              
            
            κ
          
        
        ,
      
    
    
      
        (
        17.8
        )
      
    
  
  
    
      
        κ
        ≤
        
          
            log
            
              1
              +
              ɛ
            
          
          ⁢
          
            
              
                1
                +
                ɛ
              
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  L
                  ′
                
              
            
            .
          
        
      
    
    
      
        (
        17.9
        )
      
    
  




Proof of Theorem 1: Using Lemma 1 and the fact that 1+x≦ex∀x>0, the following inequality (17.10) may be derived:





  
    
      
        
          Γ
          ⁡
          
            (
            
              w
              t
            
            )
          
        
        ≤
        
          δ
          ⁢
          
            
          
          ⁢
          L
          ⁢
          
            
              ∏
              
                j
                =
                1
              
              t
            
            ⁢
            
              e
              
                
                  ɛ
                  θ
                
                ⁢
                
                  (
                  
                    
                      f
                      j
                    
                    -
                    
                      f
                      
                        j
                        -
                        1
                      
                    
                  
                  )
                
              
            
          
        
        ≤
        
          δ
          ⁢
          
            
          
          ⁢
          L
          ⁢
          
            
          
          ⁢
          
            
              e
              
                ɛ
                ⁢
                
                  
                
                ⁢
                
                  
                    f
                    t
                  
                  /
                  θ
                
              
            
            .
          
        
      
    
    
      
        (
        17.10
        )
      
    
  




The simplification in the foregoing step uses telescopic cancellation of the sum (fj−fj−1) over j. Since the algorithm terminates after iteration K, then Γ(w)≧1. Thus, the following inequalities (17.11-17.12) hold true:





  
    
      
        
          1
          ≤
          
            Γ
            ⁡
            
              (
              
                w
                K
              
              )
            
          
          ≤
          
            δ
            ⁢
            
              
            
            ⁢
            L
            ⁢
            
              
            
            ⁢
            
              e
              
                ɛ
                ⁢
                
                  
                
                ⁢
                
                  
                    f
                    t
                  
                  /
                  θ
                
              
            
          
        
        ,
      
    
    
      
        (
        17.11
        )
      
    
  
  
    
      
        
          θ
          
            f
            K
          
        
        ≤
        
          
            ɛ
            
              ln
              ⁡
              
                (
                
                  
                    1
                    /
                    δ
                  
                  ⁢
                  
                    
                  
                  ⁢
                  L
                
                )
              
            
          
          .
        
      
    
    
      
        (
        17.12
        )
      
    
  




From Lemma 2, the objective function value of the feasible primal solution after scaling is at least the following value:





  
    
      f
      K
    
    
      
        log
        
          1
          +
          ɛ
        
      
      ⁢
      
        
          1
          +
          ɛ
        
        
          δ
          ⁢
          
            
          
          ⁢
          
            L
            ′
          
        
      
    
  
  .




The approximation factor for the primal solution is at most the (ratio) gap between the primal and dual solutions. Using inequality (17.12), this gap may be given by the following inequality (17.13):




  
    
      
        
          θ
          
            f
            K
          
        
        ≤
        
          
            ɛ
            ⁢
            
              
            
            ⁢
            
              log
              
                1
                +
                ɛ
              
            
            ⁢
            
              
                1
                +
                ɛ
              
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  L
                  ′
                
              
            
          
          
            ln
            ⁡
            
              (
              
                
                  1
                  /
                  δ
                
                ⁢
                
                  
                
                ⁢
                L
              
              )
            
          
        
        ≤
        
          
            ɛ
            
              ln
              ⁡
              
                (
                
                  1
                  +
                  ɛ
                
                )
              
            
          
          ⁢
          
            
              
                ln
                ⁢
                
                  
                
                ⁢
                
                  
                    1
                    +
                    ɛ
                  
                  
                    δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      L
                      ′
                    
                  
                
              
              
                ln
                ⁡
                
                  (
                  
                    
                      1
                      /
                      δ
                    
                    ⁢
                    
                      
                    
                    ⁢
                    L
                  
                  )
                
              
            
            .
          
        
      
    
    
      
        (
        17.13
        )
      
    
  




The quantity ln





  
    
      1
      +
      ɛ
    
    
      δ
      ⁢
      
        
      
      ⁢
      
        L
        ′
      
    
  
  /
  
    ln
    ⁡
    
      (
      
        
          1
          /
          δ
        
        ⁢
        
          
        
        ⁢
        L
      
      )
    
  




equals 1/(1−ε) for





  δ
  =
  
    
      
        1
        +
        ɛ
      
      
        L
        ′
      
    
    /
    
      
        
          [
          
            
              (
              
                1
                +
                ɛ
              
              )
            
            ⁢
            
              L
              
                L
                ′
              
            
          
          ]
        
        
          1
          /
          ɛ
        
      
      .
    
  




Using this value of δ, the approximation factor is upper-bounded by the following inequality (17.14):





  
    
      
        
          
            ɛ
            
              ln
              ⁡
              
                (
                
                  1
                  +
                  ɛ
                
                )
              
            
          
          ⁢
          
            1
            
              (
              
                1
                -
                ɛ
              
              )
            
          
        
        ≤
        
          ɛ
          
            
              (
              
                ɛ
                -
                
                  
                    ɛ
                    2
                  
                  /
                  2
                
              
              )
            
            ⁢
            
              (
              
                1
                -
                ɛ
              
              )
            
          
        
        ≤
        
          
            1
            
              
                (
                
                  1
                  -
                  ɛ
                
                )
              
              2
            
          
          .
        
      
    
    
      
        (
        17.14
        )
      
    
  




Setting 1+ε′=1/(1−ε)2 and solving for ε, the value of ε stated in Theorem 1 is obtained.

Proof of Theorem 2: First, the running time of each iteration of the algorithm is considered during which a node k and its associated paths Pi, Qj are chosen to augment flow. Selection of this node and the paths involves an all-pairs shortest-path computation which may be implemented in O(nm+n2 log n) time using Dijkstra's shortest-path algorithm with Fibonacci heaps, as is explained in R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows: Theory, Algorithms, and Applications, Prentice Hall, February 1993, the teachings of which are incorporated herein by reference. All other operations within an iteration are absorbed (up to a constant factor) by the time taken for this all-pairs shortest-path computation, leading to a total of O(n(m+n log n)) time per iteration.
Next, the number of iterations before the algorithm terminates is estimated, with reference to the fact that in each iteration, flow is augmented along paths Pi, Qj, the value being such that the total flow Δ(e) sent on link e during that iteration is at most ue. Thus, for at least one link e, Δ(e)=ue, and w(e) increases by a factor of 1+ε.
The weight w(e) for fixed eεE is now considered. Since w0(e)=δ and wK(e)≦(1+ε)/L′, the maximum number of times that this weight can be associated with any iteration may be defined by the following equation (18):




  
    
      
        
          
            log
            
              1
              +
              ɛ
            
          
          ⁢
          
            
              1
              +
              ɛ
            
            
              δ
              ⁢
              
                
              
              ⁢
              
                L
                ′
              
            
          
        
        =
        
          
            
              1
              ɛ
            
            ⁢
            
              (
              
                1
                +
                
                  
                    log
                    
                      1
                      +
                      ɛ
                    
                  
                  ⁢
                  
                    L
                    
                      L
                      ′
                    
                  
                
              
              )
            
          
          =
          
            
              O
              ⁡
              
                (
                
                  
                    1
                    ɛ
                  
                  ⁢
                  
                    log
                    
                      1
                      +
                      ɛ
                    
                  
                  ⁢
                  
                    L
                    
                      L
                      ′
                    
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        18
        )
      
    
  




Since there are a total of m weights w(e), the total number of iterations is upper-bounded by




  
    O
    ⁡
    
      (
      
        
          m
          ɛ
        
        ⁢
        
          log
          
            1
            +
            ɛ
          
        
        ⁢
        
          L
          
            L
            ′
          
        
      
      )
    
  
  .




Multiplying this by the running time per iteration, the overall algorithm running time may be obtained as





  
    O
    ⁡
    
      (
      
        
          
            n
            ⁢
            
              
            
            ⁢
            m
          
          ɛ
        
        ⁢
        
          (
          
            m
            +
            
              n
              ⁢
              
                
              
              ⁢
              log
              ⁢
              
                
              
              ⁢
              n
            
          
          )
        
        ⁢
        
          log
          
            1
            +
            ɛ
          
        
        ⁢
        
          L
          
            L
            ′
          
        
      
      )
    
  
  .




It is noted that log (L/L′) is polynomial in log n and the number of bits used to represent the Ri and Cj values.

Considering now the capacity performance of the routing scheme set forth above, the “routing guarantee” of a routing scheme should first be defined and then compared with that of the best possible scheme in the class of all schemes that route all matrices in T(R,C). Given a network with link capacities and bounds Ri, Ci on the traffic matrix, the optimum objective function value is represented by λ*, which is an output of the linear problem formulation set forth above that provides a guarantee that all matrices in λ*−T(R,C) can be routed by a routing scheme consistent with one embodiment of the present invention. As illustrated in FIG. 7, a schematic of a routing guarantee for row sum {circumflex over (R)} and column sum Ĉ, supposing that the highest possible value of λ admitted by any routing scheme is {circumflex over (λ)}, then λ*≦{circumflex over (λ)}, and the efficiency of the routing scheme can be measured by the quantity λ*/{circumflex over (λ)} (≦1).
The value {circumflex over (λ)} may be difficult to compute. Nevertheless, supposing that a single matrix TεT(R,C) exists, and computing the maximum multiplier λ(T), e.g., using a maximum concurrent flow formulation, such that λ(T)·T can be feasibly routed in the network with the given link capacities, then {circumflex over (λ)}≦λ(T), and hence, the following inequality (19) is satisfied:




  
    
      
        
          
            λ
            *
          
          
            λ
            ⁡
            
              (
              T
              )
            
          
        
        ≤
        
          
            λ
            *
          
          
            λ
            ^
          
        
        ≤
        1.
      
    
    
      
        (
        19
        )
      
    
  




Accordingly, for any traffic matrix TεT(R,C), the quantity λ*/λ(T) is a lower bound on the efficiency of the routing scheme. To obtain a tighter lower bound on the routing efficiency, a matrix TεT(R,C) should be identified for which λ(T) is minimum, which may be difficult to compute, since such a matrix will take significant bandwidth to route. The following exemplary heuristic approach may be used to approximate the matrix that takes the most bandwidth to route, wherein C(T) denotes the least bandwidth for routing the matrix TεT(R,C). The matrix TεT(R,C) that maximizes C(T) is computable in polynomial time using a linear programming formulation. Since the problem has no capacity constraints, it can be assumed that the traffic from node i to node j is routed unsplittably along a single shortest path. If dij denotes the hop count of a shortest path from node i to node j, then the problem of determining the traffic matrix TεT(R,C) that takes the maximum bandwidth to route may be formulated as the following exemplary linear program, with the constraints of equations (20-21) and inequality (22):




  
    
      
        
          maximize
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                i
                ,
                
                  j
                  ∈
                  N
                
              
            
            ⁢
            
              
                d
                ij
              
              ⁢
              
                t
                ij
              
            
          
        
        ,
        
          

        
        ⁢
        
          subject
          ⁢
          
            
          
          ⁢
          to
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              ∑
              
                
                  j
                  ∈
                  N
                
                ,
                
                  j
                  ≠
                  i
                
              
            
            ⁢
            
              t
              ij
            
          
          =
          
            
              R
              i
            
            ⁢
            
              ∀
              
                i
                ∈
                N
              
            
          
        
        ,
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          
            
              ∑
              
                
                  j
                  ∈
                  N
                
                ,
                
                  j
                  ≠
                  i
                
              
            
            ⁢
            
              t
              ji
            
          
          =
          
            
              C
              i
            
            ⁢
            
              ∀
              
                i
                ∈
                N
              
            
          
        
        ,
      
    
    
      
        (
        21
        )
      
    
  
  
    
      
        
          
            t
            ij
          
          ≥
          
            0
            ⁢
            
              ∀
              i
            
          
        
        ,
        
          j
          ∈
          
            N
            .
          
        
      
    
    
      
        (
        22
        )
      
    
  




The required bandwidth is the objective function of the linear program, and the row and column sum bounds (that define T(R,C)) form the constraints. This formulation may be used to compute a matrix T and the value λ(T) so as to provide a lower bound to the efficiency of a routing scheme consistent with one embodiment of the present invention.

Two variations on the split ratios of a two-phase routing scheme consistent with one embodiment of the present invention provide generalizations of the routing scheme, as follows:

    
    
        (I) Split Ratio Dependent on Both the Source and Destination: In this scheme, it is assumed that a fraction αkij of the traffic that originates at node i whose destination is node j is routed to node k in the intermediate stage. The capacity that is needed between nodes i and j in the second phase is then calculated. In the first phase, the capacity needed between nodes i and j is defined by the following inequality (23):
    
    






  
    
      
        
          
            ∑
            k
          
          ⁢
          
            
              α
              j
              ik
            
            ⁢
            
              t
              ik
            
          
        
        ≤
        
          
            max
            k
          
          ⁢
          
            
              α
              j
              ik
            
            ⁢
            
              
                ∑
                k
              
              ⁢
              
                t
                ik
              
            
          
        
        ≤
        
          
            max
            k
          
          ⁢
          
            
              α
              j
              ik
            
            ⁢
            
              
                R
                i
              
              .
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




    
    
        For the second phase, the capacity needed between nodes i and j is given by the following inequality (24):
    
    






  
    
      
        
          
            ∑
            k
          
          ⁢
          
            
              α
              i
              kj
            
            ⁢
            
              t
              kj
            
          
        
        ≤
        
          
            max
            k
          
          ⁢
          
            
              α
              i
              kj
            
            ⁢
            
              
                ∑
                k
              
              ⁢
              
                t
                kj
              
            
          
        
        ≤
        
          
            max
            k
          
          ⁢
          
            
              α
              i
              kj
            
            ⁢
            
              
                C
                j
              
              .
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




    
    
        Therefore, the total capacity needed between nodes i and j in the two phases together may be defined by the following inequality (25):

Cij≧αjikRi+αimjCj∀k∀m.  (25)

        (II) Split Ratio Dependent only on the Source: In this scheme, a represents a fraction of the traffic entering node i to node j. The amount of traffic that flows from node i to node j in phase 1 is given by αjiRi, which is the required capacity between nodes i and j in phase 1. The required capacity between nodes i and j may be defined by the following inequality (26):
    
    






  
    
      
        
          
            ∑
            k
          
          ⁢
          
            
              α
              k
              i
            
            ⁢
            
              t
              kj
            
          
        
        ≤
        
          
            max
            k
          
          ⁢
          
            
              α
              k
              i
            
            ⁢
            
              
                ∑
                k
              
              ⁢
              
                t
                kj
              
            
          
        
        ≤
        
          
            max
            k
          
          ⁢
          
            
              α
              k
              i
            
            ⁢
            
              
                C
                j
              
              .
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




    
    
        Therefore, the total required capacity Cij between nodes i and j is given by the following inequality (27):

Cij≧αjiRi+αikCj∀k.  (27)

        It is noted that in both cases, the constraints are linear and independent of the individual entries in the traffic matrix, being dependent only on the row and column sums.
    
    


Simulations using the implementation set forth above were performed on two network topologies representative of carrier backbone networks in their size ranges. As illustrated in FIG. 8, the first network was a 15-node network, including nodes n1-n15, with 28 bi-directional links. The second network was a 20-node network with 33 bi-directional links (not shown in the figures). For different runs, the capacity of each network link was chosen from the set {OC-3, OC-12, OC-48, OC-192}. For the results, the Ri's and Ci's were assumed to be equal and normalized to 1, i.e., Ri=Ci=1∀i. The matrix T was computed for each of the two topologies above: (I) Split Ratio Dependent on Both the Source and Destination and (II) Split Ratio Dependent only on the Source. The quantity λ(T) is an upper bound on the λ value for the cases of both equal (denoted by λequal) and unequal (denoted by λunequal) traffic split ratios, and λ(T)≧λunequal≧λequal.
FIGS. 9 and 10 illustrate plots for the above three lambda values for five different runs, wherein the relative ordering of the λ values is as expected. As can be seen, the routing efficiency of a method consistent with one embodiment of the present invention is very close to 1.0 in all of the cases plotted. For the plots of both networks, the routing efficiency varies from 0.9 to 0.99, thus indicating that a method consistent with one embodiment of the present invention may perform close to optimum. The results also illustrate the increase in network throughput when the split ratios αi are allowed to be unequal. For the 15-node topology runs, the percentage increase in throughput (λunequal−λequal)/λequal varies from 10% to as high as 85%. For the 20-node topology, the percentage increase varies from 2% to as high as 53%. From these results, the following two conclusions may be drawn: (1) a routing scheme consistent with one embodiment of the present invention may be able to route efficiently with traffic uncertainty (under the defined traffic variation model) with network throughput not significantly lower than that for a single matrix chosen from the traffic distribution; and (2) by allowing the traffic split ratios to be unequal, network throughput may be increased significantly over the case of equal split ratios.
Thus, it has been shown that a routing strategy consistent with one embodiment of the present invention may address several known issues related to handling extreme traffic variability in a network without requiring dynamic routing changes and without requiring high capacity overprovisioning. The ability to handle traffic variation without routing adaptation can lead to more stable and robust Internet behavior. Using a routing scheme consistent with one embodiment of the present invention may permit a service provider to route all traffic distributions (under the defined model) with (i) network throughput substantially close to that for routing a single matrix, and (ii) no additional necessity either to detect traffic variations in real time or to reconfigure the network. The present invention thus provides a simple network routing scheme that is not much more complex to implement than shortest-path routing, and the scheme has the following additional advantageous properties: (i) the scheme may effectively handle all traffic patterns permissible within the capacity constraints of ingress-egress links, (ii) the scheme may avoid network congestion under high traffic variability without requiring dynamic reconfiguration of routing parameters (such as link weights or routing policies), and (iii) the scheme may be bandwidth efficient and the capacity requirements of the scheme may be close to those needed to accommodate a single bad traffic pattern, even though the scheme can handle all possible traffic patterns subject to ingress-egress capacity constraints.
A method of routing consistent with one embodiment of the present invention may provide one or more of the advantages of more efficient utilization of network service-level capacity, reducing congestion of routers at network nodes, and higher packet throughput of the network. The method may be implemented for a requested LSP by either a centralized network management system or by each node of the network, or both. An implementation employing a centralized network management system that distributes results to network nodes may be preferred for coordination of provisioning of new paths. A distributed implementation in each node of the network may be preferred when no centralized network management system exists and/or if the requested LSP is a distributed request implemented with control packets routed through a network.
The various functions of a method of routing consistent with one embodiment of the present invention may be implemented with circuit elements or may also be implemented in the digital domain as processing steps in a software program. Such software may be employed in, for example, a digital signal processor, micro-controller, or general-purpose computer.
It should be understood that the term “router,” as used herein, may refer to a single hardware device or a plurality of interconnected hardware devices, such as a switch fabric, a combination of software and hardware elements, or a software program.
The present invention can be embodied in the form of methods and apparatuses for practicing those methods. The present invention can also be embodied in the form of program code embodied in tangible media, such as floppy diskettes, CD-ROMs, hard drives, or any other machine-readable storage medium, wherein, when the program code is loaded into and executed by a machine, such as a computer, the machine becomes an apparatus for practicing the invention. The present invention can also be embodied in the form of program code, for example, whether stored in a storage medium, loaded into and/or executed by a machine, or transmitted over some transmission medium, such as over electrical wiring or cabling, through fiber optics, or via electromagnetic radiation, wherein, when the program code is loaded into and executed by a machine, such as a computer, the machine becomes an apparatus for practicing the invention. When implemented on a general-purpose processor, the program code segments combine with the processor to provide a unique device that operates analogously to specific logic circuits.
It should be understood that the steps of the exemplary methods of routing set forth herein are not necessarily required to be performed in the order described, and the order of the steps of such methods should be understood to be merely exemplary. Likewise, additional steps may be included in such methods, and certain steps may be omitted or combined, in routing methods consistent with various embodiments of the present invention.
It will be further understood that various changes in the details, materials, and arrangements of the parts which have been described and illustrated in order to explain the nature of this invention may be made by those skilled in the art without departing from the principle and scope of the invention as expressed in the following claims.

## Claims

1. A computer-implemented method of routing data through a network of nodes interconnected by links and having at least one ingress point and at least one egress point, comprising the steps of: (a) a computer receiving a request for routing data between the ingress point and the egress point; (b) the computer selecting at least first and second intermediate nodes between the ingress point and the egress point; (c) the computer determining, after completion of step (b) and based on a bandwidth of said network, a first, non-zero fraction of the data to send from the ingress point to the first selected intermediate node; (d) the computer determining, after completion of step (b) and based on the bandwidth of said network, a second, non-zero fraction of the data to send from the ingress point to the second selected intermediate node; (e) the computer routing, from the ingress point to the first intermediate node, a first subset of the data corresponding to the determined first, non-zero fraction; (f) the computer routing, from the ingress point to the second intermediate node, a second subset of the data corresponding to the determined second, non-zero fraction; (g) the computer routing the first subset of the data from the first selected intermediate node to the egress point; and (h) the computer routing the second subset of the data from the second selected intermediate node to the egress point; wherein the determinations of the first and second fractions in steps (c) and (d) are made by solving a linear program having primal and dual solutions, wherein flows along said links are augmented in the primal solution, and weights of the links are updated in a multiplicative fashion in the dual solution.

2. The invention of claim 1, wherein: the primal solution is represented by a linear programming formulation as follows:



  
    
      maximize
      ⁢
      
        
      
      ⁢
      
        
          ∑
          
            i
            ∈
            N
          
        
        ⁢
        
          a
          i
        
      
    
    ,
    
      

    
    ⁢
    
      subject
      ⁢
      
        
      
      ⁢
      to
    
  




  
    
      
        
          ∑
          
            P
            ∈
            
              P
              ij
            
          
        
        ⁢
        
          x
          ⁡
          
            (
            P
            )
          
        
      
      =
      
        
          
            α
            j
          
          ⁢
          
            R
            i
          
        
        +
        
          
            α
            i
          
          ⁢
          
            C
            j
          
          ⁢
          
            ∀
            i
          
        
      
    
    ,
    
      j
      ∈
      N
    
    ,
    
      i
      ≠
      j
    
    ,
    
      

    
    ⁢
    
      
        
          
            ∑
            
              P
              :
              
                e
                ∈
                P
              
            
          
          ⁢
          
            x
            ⁡
            
              (
              P
              )
            
          
        
        ≤
        
          
            u
            e
          
          ⁢
          
            ∀
            
              e
              ∈
              E
            
          
        
      
      ;
      and
    
  



the dual solution is represented by a linear programming formulation as follows:



  
    
      minimize
      ⁢
      
        
      
      ⁢
      
        
          ∑
          
            e
            ∈
            E
          
        
        ⁢
        
          
            u
            e
          
          ⁢
          
            w
            ⁡
            
              (
              e
              )
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      subject
      ⁢
      
        
      
      ⁢
      to
    
  




  
    
      
        
          
            ∑
            
              i
              :
              
                i
                ≠
                k
              
            
          
          ⁢
          
            
              R
              i
            
            ⁢
            
              SP
              ⁡
              
                (
                
                  i
                  ,
                  k
                
                )
              
            
          
        
        +
        
          
            ∑
            
              j
              :
              
                j
                ≠
                k
              
            
          
          ⁢
          
            
              C
              j
            
            ⁢
            
              SP
              ⁡
              
                (
                
                  k
                  ,
                  j
                
                )
              
            
          
        
      
      ≥
      
        1
        ⁢
        
          ∀
          
            k
            ∈
            N
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        
          w
          ⁡
          
            (
            e
            )
          
        
        ≥
        
          0
          ⁢
          
            ∀
            
              e
              ∈
              E
            
          
        
      
      ;
    
  



wherein:
N represents the set of all nodes in the network, which includes source node i, destination node j, and intermediate node k;
E represents the set of all links e in the network;
P represents a given path from node i to node j;
x(P) represents traffic on path P;
αi represents a distribution ratio for traffic sent to node i;
αj represents a distribution ratio for traffic sent to node j;
R1 represents a maximum total bandwidth of traffic that node i sends into the network at any time;
Cj represents a maximum total bandwidth of traffic that node j receives from the network at any time;
ue represents available capacity for link e;
w(e) represents a set of weights for link e;
SP(i,k) represents a shortest path from node i to node k under weights w(e); and
SP(k,j) represents a shortest path from node k to node j under weights w(e).

3. The invention of claim 1, wherein the determinations of the first and second fractions in steps (c) and (d) result in respective fractions that are unequal.

4. The invention of claim 1, wherein the determinations of the first and second fractions in steps (c) and (d) are made without taking into account the identity of the egress point.

5. The invention of claim 1, wherein the determinations of the first and second fractions in steps (c) and (d) are made so as to optimize bandwidth.

6. The invention of claim 1, wherein the determinations of the first and second fractions in steps (c) and (d) are made without taking into account a traffic matrix corresponding to the ingress and egress points.

7. The invention of claim 1, wherein a traffic matrix corresponding to the ingress and egress points has row and column sum bounds, and wherein the determinations of the first and second fractions in steps (c) and (d) are made using solely at least one of the row and column sum bounds of the traffic matrix.

8. The invention of claim 1, wherein the routing of the data from the first and second intermediate nodes to the egress point in steps (g) and (h) is performed based on the identity of the egress point, without knowledge of any other information regarding a traffic matrix corresponding to the ingress and egress points.

9. The invention of claim 1, wherein data routing is adapted to be performed by a centralized decision-maker or by a plurality of distributed routing agents.

10. Apparatus for routing data through a network of nodes interconnected by links and having at least one ingress point and one egress point, comprising: an input module adapted to receive (i) a request for routing data between the ingress point and the egress point and (ii) the data associated with the request; a processing module adapted to process the request by: (a) selecting at least first and second intermediate nodes between the ingress point and the egress point; (b) determining, after completion of step (a) and based on a bandwidth of said network, a first, non-zero fraction of the data to send from the ingress point to the first selected intermediate node; and (c) determining, after completion of step (a) and based on the bandwidth of said network, a second, non-zero fraction of the data to send from the ingress point to the second selected intermediate node; and a router adapted to transfer the packets from the input module to an output module of the router in accordance with the request, wherein the router is adapted to: (d) route, from the ingress point to the first intermediate node, a first subset of the data corresponding to the determined first, non-zero fraction; (e) route, from the ingress point to the second intermediate node, a second subset of the data corresponding to the determined, second non-zero fraction; (f) route the first subset of the data from the first selected intermediate node to the egress point; and (g) route the second subset of the data from the second selected intermediate node to the egress point; wherein the processing module is adapted to determine the respective fractions by solving a linear program having primal and dual solutions, wherein flows along said links are augmented in the primal solution, and weights of the links are updated in a multiplicative fashion in the dual solution.

11. The invention of claim 10, wherein:
the primal solution is represented by a linear programming formulation as follows:




  
    
      maximize
      ⁢
      
        
      
      ⁢
      
        
          ∑
          
            i
            ∈
            N
          
        
        ⁢
        
          a
          i
        
      
    
    ,
    
      

    
    ⁢
    
      subject
      ⁢
      
        
      
      ⁢
      to
    
  




  
    
      
        
          ∑
          
            P
            ∈
            
              P
              ij
            
          
        
        ⁢
        
          x
          ⁡
          
            (
            P
            )
          
        
      
      =
      
        
          
            α
            j
          
          ⁢
          
            R
            i
          
        
        +
        
          
            α
            i
          
          ⁢
          
            C
            j
          
          ⁢
          
            ∀
            i
          
        
      
    
    ,
    
      j
      ∈
      N
    
    ,
    
      i
      ≠
      j
    
    ,
    
      

    
    ⁢
    
      
        
          
            ∑
            
              P
              :
              
                e
                ∈
                P
              
            
          
          ⁢
          
            x
            ⁡
            
              (
              P
              )
            
          
        
        ≤
        
          
            u
            e
          
          ⁢
          
            ∀
            
              e
              ∈
              E
            
          
        
      
      ;
      and
    
  






  
    
      maximize
      ⁢
      
        
      
      ⁢
      
        
          ∑
          
            i
            ∈
            N
          
        
        ⁢
        
          a
          i
        
      
    
    ,
    
      

    
    ⁢
    
      subject
      ⁢
      
        
      
      ⁢
      to
    
  




  
    
      
        
          ∑
          
            P
            ∈
            
              P
              ij
            
          
        
        ⁢
        
          x
          ⁡
          
            (
            P
            )
          
        
      
      =
      
        
          
            α
            j
          
          ⁢
          
            R
            i
          
        
        +
        
          
            α
            i
          
          ⁢
          
            C
            j
          
          ⁢
          
            ∀
            i
          
        
      
    
    ,
    
      j
      ∈
      N
    
    ,
    
      i
      ≠
      j
    
    ,
    
      

    
    ⁢
    
      
        
          
            ∑
            
              P
              :
              
                e
                ∈
                P
              
            
          
          ⁢
          
            x
            ⁡
            
              (
              P
              )
            
          
        
        ≤
        
          
            u
            e
          
          ⁢
          
            ∀
            
              e
              ∈
              E
            
          
        
      
      ;
      and
    
  



the dual solution is represented by a linear programming formulation as follows:



  
    
      minimize
      ⁢
      
        
      
      ⁢
      
        
          ∑
          
            e
            ∈
            E
          
        
        ⁢
        
          
            u
            e
          
          ⁢
          
            w
            ⁡
            
              (
              e
              )
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      subject
      ⁢
      
        
      
      ⁢
      to
    
  




  
    
      
        
          
            ∑
            
              i
              :
              
                i
                ≠
                k
              
            
          
          ⁢
          
            
              R
              i
            
            ⁢
            
              SP
              ⁡
              
                (
                
                  i
                  ,
                  k
                
                )
              
            
          
        
        +
        
          
            ∑
            
              j
              :
              
                j
                ≠
                k
              
            
          
          ⁢
          
            
              C
              j
            
            ⁢
            
              SP
              ⁡
              
                (
                
                  k
                  ,
                  j
                
                )
              
            
          
        
      
      ≥
      
        1
        ⁢
        
          ∀
          
            k
            ∈
            N
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        
          w
          ⁡
          
            (
            e
            )
          
        
        ≥
        
          0
          ⁢
          
            ∀
            
              e
              ∈
              E
            
          
        
      
      ;
    
  



wherein:
N represents the set of all nodes in the network, which includes source node i, destination node j, and intermediate node k;
E represents the set of all links e in the network;
P represents a given path from node i to node j;
x(P) represents traffic on path P;
αi represents a distribution ratio for traffic sent to node i;
αj represents a distribution ratio for traffic sent to node j;
Ri represents a maximum total bandwidth of traffic that node i sends into the network at any time;
Cj represents a maximum total bandwidth of traffic that node j receives from the network at any time;
ue represents available capacity for link e;
w(e) represents a set of weights for link e;
SP(i,k) represents a shortest path from node i to node k under weights w(e); and
SP(k,j) represents a shortest path from node k to node j under weights w(e).

12. The invention of claim 10, wherein the first and second fractions determined by the processing module are unequal.

13. The invention of claim 10, wherein the processing module is adapted to determine the first and second fractions without taking into account the identity of the egress point.

14. The invention of claim 10, wherein the processing module is adapted to determine the first and second fractions so as to optimize bandwidth.

15. The invention of claim 10, wherein the processing module is adapted to determine the first and second fractions without taking into account a traffic matrix corresponding to the ingress and egress points.

16. The invention of claim 10, wherein a traffic matrix corresponding to the ingress and egress points has row and column sum bounds, and wherein the processing module is adapted to determine the first and second fractions by using solely at least one of the row and column sum bounds of the traffic matrix.

17. The invention of claim 10, wherein the router is adapted to route the data from the first and second intermediate nodes to the egress point in steps (f) and (g) based on the identity of the egress point, without any other information regarding a traffic matrix corresponding to the ingress and egress points.

18. The invention of claim 10, wherein data routing is adapted to be performed by a centralized decision-maker or by a plurality of distributed routing agents.

19. A non-transitory computer-readable medium having stored thereon a plurality of instructions, the plurality of instructions including instructions which, when executed by a processor, cause the processor to implement a method for routing data through a network of nodes interconnected by links and having at least one ingress point and one egress point, the method comprising the steps of: (a) receiving a request for routing data between the ingress point and the egress point; (b) selecting at least first and second intermediate nodes between the ingress point and the egress point; (c) determining, after completion of step (b) and based on a bandwidth of said network, a first, non-zero fraction of the data to send from the ingress point to the first selected intermediate node; (d) determining, after completion of step (b) and based on the bandwidth of said network, a second, non-zero fraction of the data to send from the ingress point to the second selected intermediate node; (e) routing, from the ingress point to the first intermediate node, a first subset of the data corresponding to the determined first, non-zero fraction; (f) routing, from the ingress point to the second intermediate node, a second subset of the data corresponding to the determined second, non-zero fraction; (g) routing the first subset of the data from the first selected intermediate node to the egress point; and (h) routing the second subset of the data from the second selected intermediate node to the egress point; wherein the determinations of the first and second fractions in steps (c) and (d) are made by solving a linear program having primal and dual solutions, wherein flows along said links are augmented in the primal solution, and weights of the links are updated in a multiplicative fashion in the dual solution.

20. A network of nodes interconnected by links and having at least one ingress point and one egress point, the network adapted to: (a) receive a request for routing data between the ingress point and the egress point; (b) select at least first and second intermediate nodes between the ingress point and the egress point; (c) determine, after completion of step (b) and based on a bandwidth of said network, a first, non-zero fraction of the data to send from the ingress point to the first selected intermediate node; (d) determine, after completion of step (b) and based on the bandwidth of said network, a second, non-zero fraction of the data to send from the ingress point to the second selected intermediate node; (e) route, from the ingress point to the first intermediate node, a first subset of the data corresponding to the determined first, non-zero fraction; (f) route, from the ingress point to the second intermediate node, a second subset of the data corresponding to the determined second, non-zero fraction; (g) route the first subset of the data from the first selected intermediate node to the egress point; and (h) route the second subset of the data from the second selected intermediate node to the egress point; wherein the determinations of the first and second fractions in steps (c) and (d) are made by solving a linear program having primal and dual solutions, wherein flows along said links are augmented in the primal solution, and weights of the links are updated in a multiplicative fashion in the dual solution.