Abstract:
A method is described for one-to-all route selection in Communications Networks with multiple QoS metrics. This method takes a first metric (say, delay) as a constraint and a second metric (say, cost) as an optimization target. A potential objective is to find a path between a source node and each node in a communications network such that the delay of the path does not exceed a path delay constraint and the cost of the path is minimized. The method selects a first path which is a shortest path from a source node to each node in terms of the first metric using Dijkstra&#39;s algorithm. A reachability graph is then constructed based on the first metric path constraint. Within the reachability graph, another path is found, which is a shortest path from a source node to each node in terms of the second metric, using Dijkstra&#39;s algorithm. Any path to a particular node selected within the reachability graph replaces the first path to said particular node. This method can guarantee to find a nearly optimal path with the given constraint satisfied as long as there exists such a path.

Description:
FIELD OF THE INVENTION 
     The present invention relates to communication networks and is particularly concerned with QoS-oriented route selection. 
     BACKGROUND OF THE INVENTION 
     In traditional data communication networks, routing is primarily concerned with connectivity. To achieve efficient utilization of network resources, the cheapest routes are usually selected by finding the shortest paths using algorithms such as Dijkstra&#39;s. 
     Dijkstra&#39;s algorithm, which was developed by Edsger Dijkstra in 1959, is a well known method described in, for example, C. Papadimitriou, K. Steiglitz, (1982),  Combinatorial Optimization: Algorithms and Complexity , Prentice-Hall, the contents of which are incorporated herein by reference. Dijkstra&#39;s algorithm takes a graph with weighted links and a given root vertex as its input and returns, as its output, a label for each vertex on the graph. In the case where the weights represent the length of the links, each vertex label represents the length of the shortest path from the root vertex to the particular vertex. Thus the algorithm allows finding the shortest path for travelling from a given vertex on a graph to every other vertex. 
     With the arrival of multimedia applications such as video and audio, it is realized that connectivity alone is far from adequate for such applications to be successful. Unlike traditional data communication applications such as e-mail and remote file transfers, live video and audio cannot tolerate excessive delay and need guaranteed maximum delay and bandwidth. To deliver such Quality of Service (QoS) guarantees, a network must make resource reservations and exercise network control. In “Private Network-Network Interface Specification Version 1.0 (PNNI 1.0),” ATM Forum, 1996, resource reservation has been incorporated in the PNNI protocol for Asynchronous Transfer Mode (ATM) communication. As well, “RSVP” has been developed as a resource reservation protocol for the Internet. Multiple QoS metrics such as cost, delay, delay variation, loss probability and bandwidth are accommodated by the PNNI protocol. As a result, the routing problem, that of achieving efficient utilization of network resources, is further complicated by these QoS metrics. The route selection problem is now to find a path from the source to the destination that satisfies all QoS constraints and has the lowest cost. 
     The most studied QoS metrics fall into two categories. The first category is concave, in which the aggregate metric over a path is the minimum of the values of this metric for all sections of the path. The second category is additive, in which the aggregate metric over a path is the sum of the values of this metric for all sections of the path. 
     Zheng Wang and Jon Crowcroft, “Quality of Service Routing for Supporting Multimedia Applications,” (1996),  IEEE Jour. Sel. Area. Comm ., Vol. 14, No. 17, pp. 1228-1234, and R. Guerin et al., “QoS Routing Mechanism and OSPF Extensions,” Internet Draft, Mar. 25, 1997, demonstrated that concave metrics can be easily handled by simply ignoring those links between nodes that do not satisfy the constraints since a decision can be made locally at a link. For example, all links that do not satisfy the bandwidth constraints can be removed before selecting a route. 
     Selecting a route is not simple, however, for additive metrics. This is because a decision cannot be made until the value of the metric for each link on a path has been examined. For example, to determine if a path satisfies a given delay constraint it is necessary to compare the sum of the delays of all links on the path to the constraint. Finding an algorithm that can handle two or more additive parameters has been challenging. No efficient, that is to say “polynomial-time”, algorithms have been found so far to solve the problem completely, even if not considering any other QoS metrics. Worst of all, it is unlikely that any polynomial-time algorithm can ever be found. The intrinsic difficulty lies in that this problem is in the category of NP-HARD problems. An NP-HARD problem is defined as a problem with n variables for which the computation time is higher than a n , for some a&gt;1. This problem was identified as NP-HARD in J. M. Jaffe, “Algorithms for Finding Paths with Multiple Constraints,” (1984),  Networks , Vol. 14, pp. 95-116. For practical purposes, no polynomial time algorithms can be found for any NP-HARD problems. 
     A few heuristic approaches have been attempted to solve the QoS-oriented route selection problem. A common heuristic algorithm used by several ATM vendors uses Dijkstra&#39;s algorithm to find a minimum cost route and check the validity of this route against other constraints. (See, for example, Atshushi Iwata, et al., “PNNI Routing Algorithms for Multimedia ATM Internet,” (1997),  NEC Res. and Develop ., Vol. 38, No. 1; Data Connection, (1997),  DC-PNNI Specification .) In the case of failure to find a route, the Dijkstra algorithm is used to find a shortest path route in terms of another additive metric, say delay, and then the validity of this route is checked against other metrics. The problem with this approach is that the algorithm gives up looking for minimum cost routes too easily and is likely to miss existing routes that satisfy all constraints and have minimum cost. 
     U.S. Pat. No. 5,467,343 issued Nov. 14, of 1995 to Lee proposes to combine all additive metrics into one and then solve the problem in polynomial time. (See also W. C. Lee, et al., “Rule based Call-by-Call Source Routing for Integrated Communication Networks”, Infocom &#39;93, pp.987-993, 1993 and W. C. Lee, et al., “Multi-Criteria Routing Subject to Resource and Performance Constraints,” ATM Forum, 94-0280, March 1994.) Unfortunately, a function can not generally be formulated for this conversion. This is especially so where the metrics are independent. Consider, for example, delay and cost. At first glance, the delay and cost might be considered related and proportional. However, a carrier may want to consider any route going through a third party&#39;s network to be more expensive than his own. Thus, in general, these metrics must be considered to be independent. 
     SUMMARY OF THE INVENTION 
     In accordance with an aspect of the present invention there is provided a method for selecting paths from a source node to other nodes in a communications network having a primary link metric to be limited to a primary link metric path constraint and a secondary link metric to be optimized. Each metric is additive. The method includes constructing a reachability graph, the reachability graph including (i) links and (ii) nodes which terminate the links, for which a sum of the primary link metrics along every path in the reachability graph starting at the source node and passing through the link is no greater than the primary link metric path constraint and selecting a path from the source node to each node in the reachability graph, using only the links in the reachability graph, corresponding to a path wherein the secondary link metric is minimized. 
     In accordance with another aspect of the present invention there is provided a method for selecting paths from a source node to other nodes in a communications network having a primary link metric to be limited to a primary link metric path constraint and a secondary link metric to be optimized. Each the metric is additive. The method includes adding to a reachability graph from the source node any link from a node in the reachability graph to a first node where the link is associated with a primary link metric which, when summed with a primary link metric assigned to the node in the reachability graph from which the link extends, is less than the primary link metric path constraint. The method also includes, if at least one link to the first node is added to the reachability graph, adding the first node to the reachability graph and assigning, as a primary link metric for the first node, a maximum sum utilizing only links in the reachability graph, selecting paths from the source node through the reachability graph links to nodes in the reachability graph which optimize the secondary link metric. In accordance with other aspects of the present invention there is provided a computer readable medium for providing program control for a processor to carry out this method, a communications system including a controller with means for carrying out this method and a router including means for carrying out this method. 
     In accordance with a further aspect of the present invention there is provided, in a communication system including nodes and links between the nodes, a method for selecting a route from a source node to a plurality of nodes in the system. The method is based on two quality of service link metrics, a primary link metric to be limited to a primary link metric path constraint and a secondary link metric to be optimized. Each of the metrics is additive. The method includes selecting a first route from the source node to each node corresponding to a path wherein the primary link metric is minimized, eliminating from consideration those nodes for which a sum of the primary link metrics along the path wherein the primary link metric is minimized is greater than the primary link metric path constraint, the eliminating resulting in a plurality of nodes still under consideration. The method also includes constructing a reachability graph, the reachability graph including (i) links and (ii) nodes which terminate the links, for which a sum of the primary link metrics along every path in the reachability graph starting at the source node and passing through the link is no greater than the primary link metric path constraint. The method further includes determining a second route from the source node to each node in the reachability graph, using only the links in the reachability graph, corresponding to a path wherein the secondary link metric is minimized and replacing each selected first route to each node still under consideration with the second route if the second route differs from the first route. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In the figures which illustrate an example embodiment of this invention: 
     FIG. 1 is a schematic network of nodes representing a communications network. 
     FIG. 2 schematically illustrates an ATM PNNI network suitable for use with this invention. 
     FIG. 3 is a flow diagram illustrating the steps of an algorithm in accordance with an embodiment of the invention. 
     FIG. 4 is a flow diagram illustrating the steps of Dijkstra&#39;s algorithm. 
     FIG. 5 illustrates, in a flow diagram, the steps of a reachability graph constructing algorithm of the process of FIG.  3 . 
     FIG. 6 illustrates, in a flow diagram, the steps of a reachability graph stage adding procedure of the algorithm of FIG.  5 . 
     FIG. 7 illustrates, in a flow diagram, the steps of a node judging process of the procedure of FIG.  6 . 
     FIG. 8 illustrates, in a flow diagram, the steps of a constraint reassigning method of the process of FIG.  7 . 
     FIG. 9 illustrates, a result of a first application of the algorithm of FIG. 3 applied to the network of FIG.  1 . 
     FIGS. 10 to  13  illustrate an example of the application of FIGS. 6 through 8 to the network of FIG.  1 . 
     FIG. 14 illustrates an example of the result of the application of FIGS. 6 through 8 to the network of FIG.  1 . 
     FIG. 15 illustrates an example of the algorithm of FIG. 3 applied to the network of FIG.  14 . 
     FIG. 16 illustrates, in a flow diagram, an adaptation of FIG. 3 to perform an alternative embodiment of the invention. 
     FIG. 17 illustrates, in a flow diagram, the steps of a reachability graph constructing algorithm of the process of FIG.  16 . 
     FIG. 18 illustrates, in a flow diagram, the steps of a reachability graph stage adding procedure of the algorithm of FIG.  17 . 
     FIG. 19 illustrates, in a flow diagram, the steps of a node judging process of the procedure of FIG.  18 . 
     FIG. 20 illustrates, in a flow diagram, the steps of a constraint reassigning method of the process of FIG.  19 . 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     A communication network may be modelled by a connection graph G(N,L,d(L),c(L)) , where: N is a set of nodes in the network; L is a set of links connecting the set of nodes in the network; d(L) is a mapping from L to positive reals; and c(L) is a mapping from L to positive reals. The link (u,v) is in L if and only if u and v are nodes in the network and there exists a physical link from u to v. 
     The mapping d((u,v)) may be termed the primary link metric while the mapping c((u,v)) is termed the secondary link metric. By way of example, the primary link metric, d((u,v)), may be the delay of the link (u,v) and the secondary link metric c((u,v)) the cost of the link (u,v). The objective is to find a path between a source node, s, and each node in graph G such that the delay of the path (which is the sum of link delays over the path) does not exceed a path delay constraint D and the cost of the path (which is the sum of link costs over the path) is minimized. 
     Referring to FIG. 1, which models a communication system as a graph of nodes  100 , the link (u,v) between each pair of nodes is shown to have an associated delay and cost. 
     FIG. 2 illustrates an ATM PNNI network suitable for use with this invention. PNNI network  200  comprises a number of ATM backbone switch nodes  210  which are nodes interconnected by PNNI links  208 . A number of ATM devices (work stations, servers or private ATM switches)  206  are connected to the backbone switch nodes  210  by UNI (User-to-Network Interface)  204  links. The backbone switch comprises a processor  212  and a memory  214  loaded with route selection software for executing the method of this invention from software medium  216  which could be a disk, a tape, a chip or a random access memory containing a file downloaded from a remote source. 
     In operation, if a backbone switch  210 A receives a destination ATM address from an ATM device  206 A indicating a request to connect to ATM device  206 B, the switch, under control of software in memory  214 , undertakes the method of this invention to determine an appropriate path to device  206 B. In a conventional manner, the call is then set up over this path. 
     In overview, with reference to FIG. 3, a one-to-all route selection algorithm in accordance with an embodiment of this invention begins with selecting a shortest path from a source s to each node in terms of delay (step  302 ). This step may be accomplished using Dijkstra&#39;s Algorithm, where the weight of each link is representative of delay. Those nodes for which the delay of the associated shortest delay path exceeds the path delay constraint D are disqualified from further consideration (step  304 ). A “reachability graph” is constructed for source node s (step  306 ). The path to each node with minimum cost is determined using Dijkstra&#39;s Algorithm, using only links in the reachability graph, where the weight of each link is representative of cost (step  308 ). Due to the criteria used to qualify links for addition to the reachability graph, the path determined in step  308  to a particular node may have a path delay which is not minimum, but is acceptable with regard to the path delay constraint D. For each node, the path determined in step  308  is selected if it differs from the path selected in step  302  (step  310 ). 
     The steps involved in Dijkstra&#39;s Algorithm are outlined in FIG.  4 . It is assumed for purposes of illustration that w((u,v)) represents the weight of the link (u,v), where weight is a generic term which may represent an additive metric such as delay or cost. A constraint W(v) is associated with every node v (step  402 ). W(v) is initially set to w((s,v)), the weight of the link from node s to v or to where a link does not exist between s and v (step  404 ). 
     A set L is initialized with one element, that being the source node, s (step  406 ) and Dijkstra&#39;s Algorithm proceeds as follows. A node, u, is chosen from the nodes, v, that are not already in set L such that the weight associated with node u, W(u), is the minimum of all W(v)&#39;s (step  408 ). The node u is then added to set L (step  410 ). If set L contains all nodes, Dijkstra&#39;s Algorithm is finished (step  412 ). If some nodes are not in L, the W(v)&#39;s of such nodes are reevaluated as follows. For each node v not in set L, a new weight W(v) is assigned which is the lesser of either the current weight W(v) or the sum of the weight associated with the most recently added node W(u) and the weight w((u,v)) of the link from u to v (step  414 ). When the new W(v)&#39;s have been assigned, the process returns to step  408  to select another node to add to set L. After the termination of Dijkstra&#39;s Algorithm, the resulting updated weights, W(v), associated with each node represent the weight of the minimum weight path from the source node s to each node v. That is, a path from the source node s to each node v exists with that minimum weight. 
     To begin the discussion of a “reachability graph,” some terminology is introduced. A “neighbor set” of node u, {right arrow over (N)}(u) , may be defined as the set of nodes to which there are links coming from node u. Similarly, the neighbor set of a subset N′ of N, {right arrow over (N)}(N′), is the set of nodes to which there are links coming from nodes in N′ but which are not members of N′ themselves. 
     With the connection graph denoted as G(N,L,d(L),c(L)) , a “reachability” graph may be denoted as G′(N′,L′,d(L′),c(L′),s,D) starting from a source node s. The reachability graph is a subgraph of the corresponding connection graph. A link (u,v) is eligible to be in the reachability graph only if there is one path starting at the source node, passing through this link and terminating at v that satisfies a path delay constraint D. 
     A reachability graph is constructed by initiating a sub set of nodes, N 0 ′, with only the source node s. The source node delay constraint d(s) is set to 0. In stage 1, each node, v, with a link from s is then evaluated for addition to the reachability graph. Node v is deemed reachable and selected if and only if d(s)+d((s,v))≦D. After all candidate nodes have been evaluated, N 1 ′ results which includes all selected nodes v along with the source node s. The delay constraint d(v) of a given selected node is set to d(s)+d((s,v)). In stage 2, each node, v, in the neighbour set {right arrow over (N)}(N 1 ′) to N 1 ′ is evaluated and selected if there is a link to v from a node u in N 1 ′ such that d(u)+d((u,v))≦D. For N 1 ′, it is possible that there is more than one node u with a link to a given v. In such case, the maximum of the various calculated values for d(v) is chosen as d(v) for node v: this is known as the “max link qualification rule”. After all candidates have been evaluated, N 2 ′ results. For stage 3, the same procedure is repeated. This continues until all nodes in graph G have been evaluated; the nodes selected at each stage are included in the reachability graph. In addition, every link to a given node in the reachability graph is itself included in the reachability graph if, for any such link, there is a path through such link to the given node which does not exceed D. Thus, there may be more than one path to a given node in the reachability graph. 
     More generally, the procedure in constructing the reachability graph may be described as follows. At each step of the construction of the reachability graph from the connection graph, nodes in the neighbor set—{right arrow over (N)}(N′)—of the set of nodes—N′—that have already been reached are considered. Each node u in N′ has a delay constraint d(u) associated with it with respect to paths reaching node u from source node s. A node v in {right arrow over (N)}(N′) is deemed reachable and selected for inclusion in the reachability graph if and only if there is at least one node u in N′ such that d(u)+d((u, v))≦D. For a given node v in {right arrow over (N)}(N′) and u in N′, link (u,v) is selected for inclusion in the reachability graph as long as d(u)+d((u,v))≦D. The delay constraint d(v) is assigned as the maximum of d(u)+d((u,v)) (the “max link qualification rule”). Note that all links in the reachability graph are in the forward direction from one neighbor set to the next neighbor set (i.e., progressively farther from the source node s, from u to v; links from v to u are ignored). 
     An exemplary procedure for constructing a reachability graph is next presented in conjunction with FIGS. 5 to  8 . 
     Referencing FIG. 5, the set of nodes in the reachability graph, N′, is initialized with the start node s (step  502 ). The value of d(s) is set to zero (step  504 ). A stage (i.e., nodes from the neighbour set of nodes {right arrow over (N)}(N′)) is added to the reachability graph (step  506 ) using a procedure described hereafter in conjunction with FIG.  6  and stages are added until all nodes have been evaluated for inclusion in the reachability graph (step  508 ). A reachability graph can be constructed by recursively calling step  506  on the current node set N′ and using the max link qualification rule. 
     Referencing FIG. 6, the steps in adding a stage to a reachability graph are as follows. Given an unfinished construction of reachability graph G′ (covering a subset of nodes N′), the procedure continues the construction of the reachability graph considering the neighbor set of subset N′. A node u is selected from N′ which has an outgoing link to {right arrow over (N)}(N′) (step  602 ). For each node v, for which there exists a link (u,v), a judgement is made, using criteria described in conjunction with FIG. 7, regarding the addition of v to N′ (step  604 ). If some nodes u in N′ have not yet been considered, the process reverts to step  602  (step  606 ). If no nodes v were marked for addition to N′ in step  604 , the procedure for the current stage is complete (step  608 ). Those nodes v marked for addition to N′ in step  604  are added to N′ (step  610 ). The delay constraints, d(v), associated with nodes v marked for addition to N′ in step  604 , are assigned the value of a corresponding alternate delay constraint d″(v) (step  612 ). The valuation of d″(v) is described hereafter in conjunction with FIGS. 7 and 8. 
     Turning to FIG. 7 which illustrates the process for the application of criteria relating to making a determination regarding the addition of v to N′ in step  604 , a node v is selected from {right arrow over (N)}(N′), the neighbor set to the set of nodes in the unfinished reachability graph (step  702 ). The sum of d(u) (the delay of the path from the source s to node u) and d((u,v)) (the delay of link (u,v)) is compared to the path delay constraint D (step  704 ). If the sum exceeds the path delay constraint D, no further consideration is given to the selected v and the selected v is not included in the reachability graph. (That is, based on the link of v with u, v does not qualify for inclusion in the graph. It is still possible, however, that v will qualify based on a link between v and another node in N′). If the sum does not exceed the path delay constraint D, the selected v is marked for addition to N′ (step  706 ). If the marking of the selected v for addition to N′ is occurring for the first time (determined in step  708 ), the alternate delay constraint, d″(v), is assigned an initial value equivalent to d(u)+d((u,v)) (step  710 ). The alternate delay constraint is then reevaluated (step  712 ), using a method disclosed hereafter in conjunction with FIG.  8 . It is then ascertained whether all nodes in the neighbour set to N′ have been considered (step  714 ). If there exists a node within {right arrow over (N)}(N′) which has not yet been considered, the process begins again (step  702 ). If all nodes with a link to u have been considered, the process returns to FIG. 6 at step  606 . 
     Steps in a method for reevaluating the alternate delay constraint are illustrated in FIG.  8 . Link (u,v) is added to the reachability graph (step  802 ). If it is determined that the sum of d(u) and d((u,v)) is greater than the current value of the alternate delay constraint d″(v) (step  804 ), the alternate delay constraint d″(v) is assigned the value of the sum of d(u) and d((u,v)) (step  806 ). If the sum is less than or equal to the alternate delay constraint, the method is complete (i.e., d″(v) remains unchanged). It will be apparent that these steps implement the max link qualification rule. 
     Since step  802  adds a link to v to the reachability graph whenever step  704  finds d(u)+d((u,v)) is less than D, it will be apparent that if more than one u has a link to v, more than one link to v may be included in the reachability graph. 
     The procedure for adding a stage to a reachability graph has the advantage that every single path in the reachability graph sourced at the source node s satisfies the path delay constraint (step  704 , FIG. 7) and there are no cyclic paths in the reachability graph. Note that once a given node v is placed in the node set N′, no more links leading to this given node will be added to the reachability graph in later stages. 
     By way of example, consider the algorithm of FIG. 4 as it applies to network G of FIG. 1 with source node, s, being  102  and a path delay constraint D of 40 time units. FIG. 9 illustrates the result of the application of Dijkstra&#39;s Algorithm (FIG. 4) to the network G of FIG. 1 using the delay of each link as weights (step  302 ). The heavy lines represent links in the shortest delay path to each node. Each node is assigned a delay constraint d(v) with a value equivalent to the sum of link delays over the shortest delay path from the source node s to the particular node. For each node in network G a path does exist which satisfies the path delay constraint D, thus no nodes are disqualified (step  304 ). A reachability graph is constructed (step  306 ) for network G, where the start node, s, is node  102 . Construction is accomplished in three stages. 
     Stage 1—FIG. 10 
     N′ is initialized with node s (FIG. 5, step  502 ). As N′ comprises only node s, it is selected as node u (FIG. 6, step  602 ). Node  104  is selected as node v (FIG. 7, step  702 ). Since the sum d(s)+d((s, 104 )) is less than the path delay constraint D, node  104  is marked for addition to N′ (FIG. 7, step  706 ). As it is the first time that node  104  has been marked, the alternate delay constraint d″( 104 ) is initialized with the value of the sum d(s)+d((s, 104 )) (FIG. 7, step  710 ). Link (s, 104 ) is added to the unfinished reachability graph N′ (FIG. 8, step  802 ). Since the alternate delay constraint d″( 104 ) is equivalent to the sum d(s)+d((s, 104 )), the alternate delay constraint is not reassigned (FIG. 8, step  804 ). 
     Node  106  is then selected as node v. Since the sum d(s)+d((s, 106 )) is less than the path delay constraint D, node  106  is marked for addition to N′. As it is the first time that node  106  has been marked, the alternate delay constraint d″( 106 ) is initialized with the value of the sum d(s)+d((s, 106 )). Link (s, 106 ) is added to the unfinished reachability graph N′ (FIG. 8, step  802 ). Since the alternate delay constraint d″( 106 ) is equivalent to the sum d(s)+d((s, 106 )), the alternate delay constraint is not reassigned. Since all neighbour nodes v have been considered (FIG. 7, step  714 ), all nodes u in N′ have been selected (FIG. 6, step  606 ) and there are nodes marked for addition to N′ (FIG. 6, step  608 ), nodes  104  and  106  are added to N′ (FIG. 6, step  610 ). The delay constraint s d( 104 ) and d( 106 ) are assigned the values of d″( 104 ) and d″( 106 ) respectively (step  612 ). FIG. 11 shows the new subset N′ which includes nodes  104 ,  106  as well as the values for d( 104 ), d( 106 ) and, in heavy lines, links ( 102 ,  104 ) and ( 102 ,  106 ) which are part of the reachability graph. 
     Stage 2—FIG. 11 
     Node  104  may be selected as node u and node  108  may be selected as node v. Since the sum d( 104 )+d(( 104 ,  108 )) is less than the path delay constrainst D, node  108  is marked for addition to N′. As it is the first time that node  108  has been marked, the alternate delay constrainst d″ ( 104 ,  108 ) is initialized with the value of the sum d( 104 )+d(( 104 ,  108 )). Link ( 104 ,  108 ) is added to the unfinished reachability graph N′. Since the alternate delay constrainst d″ ( 108 ) is equivalent to the sum d( 104 )+d(( 104 ,  108 )), the alternate delay constrainst is not reassigned. 
     Node  110  is next selected as node v. Since the sum d( 104 )+d(( 104 , 110 )) is less than the path delay constraint D, node  110  is marked for addition to N′. As it is the first time that node  110  has been marked, the alternate delay constraint d″( 110 ) is initialized with the value of the sum d( 104 )+d(( 104 , 110 )). Link ( 104 , 110 ) is added to the unfinished reachability graph N′. Since the alternate delay constraint d″( 110 ) is equivalent to the sum d( 104 )+d(( 104 , 110 )), the alternate delay constraint is not reassigned. 
     Since all nodes v with a link to node  104  have been considered, node  106  is next selected as node u. Node  110  may be selected as node v. Since the sum d( 106 )+d(( 106 , 110 )) is less than the path delay constraint D, node  110  is marked for addition to N′ and link ( 106 , 110 ) is added to the unfinished reachability graph N′. As it is not the first time that node  110  has been marked, the reevaluation of the alternate delay constraints is the next step. Since the alternate delay constraint d″( 110 ) is greater than the sum d( 106 )+d(( 106 , 110 )), the alternate delay constraint is reassigned to the latter value. 
     Node  112  is then selected as node v. Since the sum d( 106 )+d(( 106 , 112 )) is greater than the path delay constraint D, node  112  is not marked for addition to N′. 
     At this point, all nodes v with a link to node  106  have been considered. Also, all nodes u in N′ with a link to a neighbour node have been selected. There are nodes marked for addition to N′, nodes  108  and  110 , which are now added to N′ and the delay constraints d( 108 ) and d( 110 ) are assigned the values of d″( 108 ) and d″( 110 ), respectively. Earlier in stage 2, links ( 104 , 108 ) and ( 104 , 110 ) were added to the unfinished reachability graph N′. 
     Stage 3—FIG. 12 
     Note that node  110  and node  106  are the only nodes which have outgoing links to {right arrow over (N)}(N′) . Also note that the link ( 106 ,  112 ) was considered and invalidated in the last stage and does not need re-evaluation. 
     Node  110  may then be selected as node u and node  112  (which was not added to the reachability graph in stage 2) may be selected as node v. Since the sum d( 110 )+d(( 110 , 112 )) is less than the path delay constraint D, node  112  is marked for addition to N′. As it is the first time that node  112  has been marked, the alternate delay constraint d″( 112 ) is initialized with the value of the sum d( 110 )+d(( 110 , 112 )). Link ( 110 , 112 ) is added to the unfinished reachability graph N′. Since the alternate delay constraint d″( 112 ) is equivalent to the sum d( 110 )+d(( 110 , 112 )), the alternate delay constraint is not reassigned. Node  114  may then be selected as node v. Since the sum d( 110 )+d(( 110 , 114 )) is less than the path delay constraint D, node  114  is marked for addition to N′. As it is the first time that node  114  has been marked, the alternate delay constraint d″( 114 ) is initialized with the value of the sum d( 110 )+d(( 110 , 114 )) . Link ( 110 , 114 ) is added to the unfinished reachability graph N′. Since the alternate delay constraint d″( 114 ) is equivalent to the sum d( 110 )+d(( 110 , 114 )), the alternate delay constraint is not reassigned. Node  116  is next selected as node v. 
     Since the sum d( 110 )+d(( 110 , 116 )) is less than the path delay constraint D, node  116  is marked for addition to N′. As it is the first time that node  116  has been marked, the alternate delay constraint d″( 116 ) is initialized with the value of the sum d( 110 )+d(( 110 , 116 )). Link ( 110 , 116 ) is added to the unfinished reachability graph N′. Since the alternate delay constraint d″( 116 ) is equivalent to the sum d( 110 )+d(( 110 , 116 )), the alternate delay constraint is not reassigned. 
     Since all nodes v have been considered for node  110 , all nodes u have been selected and there are nodes marked for addition to N′, nodes  112 ,  114  and  116  are added to N′. Further, constraints d( 112 ), d( 114 ) and d( 116 ) are assigned the values of d″( 112 ), d″( 114 ) and d″( 116 ), respectively (FIG.  13 ). Since all nodes have been evaluated for inclusion in the reachability graph, it is complete (FIG. 5, step  506 ). 
     There is now a complete reachability graph from start node s (node  102 ) to every node in the graph that does not exceed the path delay constraint (FIG.  14 ). 
     FIG. 15 illustrates the result of the application of Dijkstra&#39;s Algorithm (FIG. 3) to the reachability graph of FIG. 14 using the cost of each link as weights (FIG. 4, step  406 ). Each node is assigned a cost constraint with a value equivalent to the sum of link costs over a minimum cost path from the source node s to the particular node. For each node, the minimum cost path from s to the particular node, the heavy line links in the graph of FIG. 15, is selected (FIG. 3, step  310 ). This minimum cost path may differ from the minimum delay path of FIG.  9 . Where a call is destined from the source node to a given other node on the graph, it is routed along the selected path. 
     In the foregoing, the max link qualification rule was used in constructing the reachability graph. More specifically, the step of adding a stage to the reachability graph (step  506 , detailed in FIGS. 6,  7  and  8 ) relies on the conditions of the “max link qualification rule”. A drawback with the foregoing is that application of the max link qualification rule may exclude some nodes from the reachability graph which may be reached without exceeding the path delay constraint D. In such instance, the path to such nodes will not be replaced from that established by application of Dijkstra&#39;s algorithm in step  302 . Consequently, a cost optimization will not be performed for such nodes. 
     To increase the number of nodes in respect of which a cost optimization is performed, a modified algorithm may be employed which makes use of a “min link qualification rule”. The “min link qualification rule” is as follows: For a given node V G  in {right arrow over (N)}(N′) and U G  in N′, link (U G , V G ) is selected for inclusion in the reachability graph if and only if d(U G )+d((U G , V G ))≦D and d(U G )+d((U G , V G )) has a value which is the minimum value as compared with the d(u)+d((u,V G )) for all other nodes u in N′ with a link to V G  which satisfy the condition d(u)+d((u, V G ))&lt;D. (In contrast to the situation where the max link qualification rule is employed, this usually—but not necessarily—results in only one link to any given v being included in the reachability graph. This increases the possibility that paths can further extend from V G  so that more paths are eligible for cost optimization.) The delay constraint d(V G ) at node V G  is set to the minimum of d(u)+d ((u, V G )) . Use of the “min link qualification rule” allows for only the link with lower delay to be selected when a link from two different u&#39;s in N′ to a given v have identical cost. 
     FIG. 16 outlines an iterative process that replaces the process of FIG. 3 for the alternative embodiment. A shortest path from a source s to each node in terms of delay is first selected using Dijkstra&#39;s Algorithm (step  1602 ). Those nodes for which the delay of the associated shortest delay path exceeds the path delay constraint D are disqualified from further consideration (step  1604 ). A counter, m, of the number of applications of the max link qualification rule, is initialized at zero (step  1605 ). In the first iteration, a reachability graph is constructed from source node s, using the min link qualification rule at every stage (step  1606 ). The path to each node with minimum cost is determined using Dijkstra&#39;s Algorithm, using only links in the reachability graph (step  1608 ). In subsequent iterations, a previously selected path to a particular node is replaced by the path determined in step  1608  if the cost of the latter path is no more than the cost of the former path (step  1610 ). Note that the old path is replaced by the new path even when the costs are identical. This results in improved load balancing. One application of the above process (constructing a reachability graph, determining the minimum cost paths within the reachability graph and replacing previously selected paths) may be called an iteration. The process is repeated and, for each iteration, the number of stages of the reachability graph construction using the max link qualification rule is increased by one (step  1616 ). The process is complete when the most recent iteration has used only the max link qualification rule (step  1614 ). 
     Only the min link qualification rule is used in the first iteration. In the second iteration, the max link qualification rule is used in the first stage and the min link qualification rule in the rest of the stages of construction. In general, the max link qualification rule is used in the first 0, 1, 2, . . . , stages and the min link qualification rule in the rest of the stages. In particular, the first iteration uses the min link qualification rule only and the last iteration uses the max link qualification rule only. 
     To implement this procedure, the reachability graph constructed in step  1606  is constructed according to the procedure set out in FIG.  17 . Turning to FIG. 17, it will be apparent that, except for steps  1706 , the procedure is identical to that of FIG. 5 in the first embodiment, and like steps have been given like reference numerals. Step  1706  calls the procedure of FIG. 6 when the next stage is be added to the reachability graph using the max link qualification rule (step  1706 B) and FIG. 18 when the next stage is to be added using the min link qualification rule (step  1706 C). 
     Turning to FIG. 18, the procedure to add a stage to the reachability graph under the min link qualification rule is similar to that of the procedure of FIG. 6 used in conjunction with the max link qualification rule and like steps have been given like reference numerals. However, where step  604  of FIG. 6 calls the procedure of FIG. 7 to judge v&#39;s for addition to N′, step  1804  of FIG. 18 calls the procedure of FIG.  19 . Turning to FIG. 19, the procedure is identical to that of FIG. 7, and like steps are given like reference numerals, except that step  1910  initialises d′(v)—in contrast to step  710  of FIG. 7 which initialises d″(v). Also, step  1912  reassigns d′(v)—in contrast to step  712  of FIG. 7 which reassigns d″(v). This reassignment is accomplished with the steps of FIG.  20 . Turning to FIG. 20, if it is determined in step  2004  that the sum of d(u) and d((u,v)) is less than the current value of the alternate delay constraint d′(v), the alternate delay constraint d′(v) is assigned the value of the sum of d(u) and d((u,v)) (step  2006 ). If the sum is greater than the alternate delay constraint, the method is complete (i.e., d′(v) remains unchanged). It will be apparent that these steps implement the min link qualification rule. 
     Returning to FIG. 18, when adding a stage in accordance with the min link qualification rule, steps  1811  and  1812  illustrated in FIG. 18 replace step  612  of FIG.  6 . In consequence, after all marked v&#39;s are added to N′ (step  610 ), all links that satisfy d′(v)=d(u)+d((u,v)) are added to the reachability graph (step  1811 ). Then d(v) for each added v is set equal to d′(v) (step  1812 ). 
     In consequence, a link from a given node u to a particular node v will only be added to the reachability graph if its delay, when added to the delay associated with given node u, is at least as small as the delay associated with each other link to particular node v summed with the delay associated with the node u from which each other link is sourced. In the typical case, one of these sums is smaller than all others and so typically only one link to particular node v is added to the reachability graph. It is, however, possible that more than one link to node v is added to the reachability graph. The delay constraints, d(v), associated with nodes v marked for addition to N′ in step  604 , are assigned the value of a corresponding alternate delay constraint, which is d′(v) for the min link qualification rule. 
     While a particular node, x, may be reachable using the min link qualification rule, it may not be reachable using the max link qualification rule. The min link qualification rule therefore tends to allow reachability graph construction to extend further forward and possibly cover more nodes. This is because node x is usually assigned a smaller delay value d(x) with the min link qualification rule than with the max qualification rule. As a result, it is possible that some nodes may be reachable with respect to the delay constraint when the min link qualification rule is applied but not reachable when the max qualification rule is used at node x. 
     In general, once a reachability graph has been constructed, a greater number of links normally exist between the source node s and an arbitrary node y if the max link qualification rule was applied in constructing the graph than if the min link qualification rule was applied. Conversely, a greater likelihood exists that a path from source node s through arbitrary node y to a further node x will exist if the min link qualification rule was applied in constructing the graph than than if the max link qualification rule was applied. 
     Note also that load balancing (i.e., use of different links in paths to different nodes) is promoted by this algorithm. Paths selected by one application of Dijkstra&#39;s Algorithm in step  1608  tend to have heavy link sharing, especially towards the source node. The progressive increasing use of the max link qualification rule in many iterations of reachability graph construction allows alternative links to be used not only for the purpose of cost Optimization but also for the purpose of load balancing. 
     To simplify the route selection algorithm, the first two steps, namely selecting a shortest path from a source s to each node in terms of delay and disqualifying those nodes for which the delay of the associated shortest delay path exceeds the path delay constraint D, may be eliminated. In this simplified embodiment, a reachability graph is constructed which includes links, and nodes terminating the links, whose sum of delays along every path in the reachability graph starting at the source node s and passing through the link is no greater than the path delay constraint D. Then, using only the links in the reachability graph, a minimum cost route is selected to each node in the reachability graph. 
     The max link qualification rule may be used in the reachability graph construction step of the above simplified algorithm. However, a reachability graph so constructed might not include all nodes to which paths exist whose path delay satisfies the path delay constraint D. 
     Another simplified embodiment may employ only two iterations, one using the max link qualification rule throughout while the other uses the min link qualification rule throughout. 
     Since concave metrics can be handled simply by ignoring those links where the path constraints are not satisfied, discussion of concave metrics has been omitted in the preceding to simplify the description. Any actual implementation of the algorithm may take into consideration any concave metrics. 
     Other modifications will be apparent to those skilled in the art and, therefore, the invention is defined in the claims.