Abstract:
A least-cost path between a pair of nodes A and B of a network is realized by segmenting the network into three or more segments, with nodes A and B being in different ones of the segments, with the third or more other segments being interposed between the segments to which nodes A and B belong, concurrently process the different segment to identify paths and their associated costs, and processing results generated by the different processing units to compute the least-cost path. Advantageously, the segmenting is chosen to equalize the work load of the different processing units so that the units finish their task roughly at the same time.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    This invention relates to network routing, and more particularly to routing through networks that can be represented as Euclidian graphs, such as a telecommunications network that comprises a plurality of nodes, and links that interconnect the nodes. 
         [0002]    A digraph is a representation of a network, converted into nodes and edges. Euclidian graphs are undirected graphs for which nodes have real and distinct physical positions and whose edge weights correlate, at least roughly, to the Euclidian distance between nodes. Various networks have this characteristic, such as telecommunication networks and road networks. One task that is often required in connection with networks is to identify a least-cost path, or paths, between nodes of the network. Algorithms exist for finding such least-cost paths, but because the computation burden of these algorithms is generally proportional to the number of network nodes raised to a power that is greater than 1, and the number of nodes in any reasonable-sized telecommunications network is large, the computations of a least-cost path is quite burdensome. 
         [0003]    One approach for obtaining a solution to a least-cost path problem is to employ an algorithm that is parallelizable; that is, an algorithm whereby the problem can be divided into segments and the segments can be processed concurrently by separate processors. When the problem to be addressed is to find a set of least cost paths between N terminal points in one grouping and N terminal points in another grouping, it can divided into N problems, each solving a single-pair-shortest-path (SPSP) problem. However, finding the shortest path between a given pair of terminal points is still quite burdensome when the number of network nodes is large. Tree decomposition methods work, but the decomposition can be more expensive (in terms of processing burden) than the path-finding operation itself. 
         [0004]    An algorithm that takes advantage of the attribute of Euclidian graphs, mentioned above, is disclosed in co-pending application titled “An Algorithm for Network Route Selection,” filed in the US Patent Office on Jun. 12, 2007, but this solution is an approximation, and the algorithm is not parallelizable. 
       SUMMARY OF THE INVENTION 
       [0005]    An advance in the art of identifying a least-cost path between a pair of nodes A and B is realized by (a) segmenting the network into three or more segments, with nodes A and B being in different ones of the segments, with the third or more other segments being interposed between the segments to which nodes A and B belong, (b) concurrently process the different segment to identify paths and their associated costs, and (c) processing results generated by the different processing units to compute the least-cost path. Advantageously, the segmenting is chosen to equalize the work load of the different processing units so that the units finish their task roughly at the same time. The network is segmented with cut “lines” that go through and divide nodes into two halves, or cut links to create a pair of phantom nodes. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0006]      FIG. 1  presents an illustrative network; 
           [0007]      FIG. 2  presents the  FIG. 1  network as it is cut into three segments; and 
           [0008]      FIG. 3  is a block diagram of a hardware embodiment in which the method disclosed herein may be practiced. 
       
    
    
     DETAILED DESCRIPTION 
       [0009]      FIG. 1  depicts a network where a path needs to be established between nodes  31  and  12 . Illustratively, the  FIG. 1  network is a telecommunication network where nodes  31  and  12 , or more particularly customer terminals A and B that are coupled to nodes  31  and  12 , respectively, need to be interconnected. Therefore, it is desirable to identify a least-cost path between them nodes  31  and  12 . 
         [0010]    In accord with the principles disclosed herein, the  FIG. 1  network is divided into a number of segments. If the pair of nodes  31 - 12  is viewed to be lying on a latitude line, the segmenting is basically along longitude lines. Illustratively, the  FIG. 1  network is segmented to form a first network segment that includes customer terminal A, a second network segment that includes customer terminal B, and a third network segment therebetween. The segmenting can be with cuts through network links, forming a pair of phantom half-nodes at the cut-point of each link, with one of the phantom half-nodes belonging to one of the segments, the other of the phantom half-nodes belonging to the other the segments, and the costs of the links divided up between the two cut segments of the link. Alternatively, the segmenting can be delineated by cuts through nodes, creating two half-nodes from each “cut” node, with one of the half nodes belonging to one of the segments and the other of the half nodes belonging to the other of the segments. 
         [0011]    The network is segmented so that each segment can be processed independently, and as disclosed more fully below, it is particularly advantageous to segment the network so that the processing of each segment takes approximately the same time. The processing within a segment can be assigned to more than one processor, again with the goal of having all processing that is done concurrently be concluded at approximately the same time. 
         [0012]      FIG. 2  depicts such a segmenting, with a first segment (to which terminal A belongs) including V 1 =6 nodes (not counting node  31 ), N 1 =4 half-nodes, and E 1 =18 paths within the segment; the second segment (to which terminal B belongs) includes V 2 =5 nodes (not counting node  12 ), N 2 =5 half nodes, and E 2 =19 paths within the segment; and the third segment includes V 3 =0 nodes, Nl 3 =4 half nodes at the left border, Nr 3 =5 half nodes at the right border, and E 3 =8 paths within the segment. 
         [0013]    Noting that node  23  belongs to the interface with the first segment and well as with the second segment, it need not be included in any least-cost path identification algorithm (since it is known that there is only one path from the interface with the first segment to the interface with the second segment, and the cost of that path is zero). Indeed,  FIG. 1  actually depicts this zero-cost path between nodes  23 B and  23 C. Therefore, the third segment may be said to have V 3 =0 nodes, Nl 3 =3 half nodes at the left border, Nr 3 =4 and E 3 =7 paths within the segment. 
         [0014]    Since for purposes of the processing carried out by the path identification and cost determination algorithm the half nodes are counted as full nodes, the  FIG. 2  segmentation yields a first segment with 11 links and 10 nodes, a second segment with 19 links and 10 nodes, and the third segment with n=3, 7 nodes and 7 links 
         [0015]    As for the connection from one segment to another, that is from one half-node to a corresponding half-node, for purposes of the algorithms executed herein, a pair of corresponding half-nodes is interconnected by a link of zero cost. Also, in the course of executing the algorithm relative to a segment, a half-node is considered as a full node. 
         [0016]    An apparatus that computes the least-cost path between node  31  and node  12  is illustrated in  FIG. 3  as having processors  10  and  20  that are adapted to communicate with processor  30 . The  FIG. 3  apparatus can actually also be a single multi-processor computer. Illustratively, the network segment that includes node  31  is assigned to processor  10 , the network segment that includes node  12  is assigned to processor  20  and the remaining network segment is assigned to processor  30 . 
         [0017]    The task of processor  10  is to compute the costs from node  31  to the four nodes that form the interface of the first network segment with the third network segment. This computation is, illustratively, carried out using the conventional Dykstra algorithm. Similarly, the task of processor  20  is to compute the costs from node  12  to the five nodes that form the interface of the second network segment with the third network segment, and it is also illustratively carried out using the conventional Dykstra algorithm. 
         [0018]    The task of processor  30  is to compute the costs between the Nl 3  left-side nodes of the third segment and the Nr 3  right-side of the third segment, as well as identify the final least-cost path. In an embodiment that employs the Dykstra algorithm, this corresponds to n runs of the algorithm, where n is the lower of Nl 3  and Nr 3 ; which in the illustrative example of  FIG. 2  is 4. 
         [0019]    Unlike the network of  FIGS. 1 and 2 , actual networks often are highly interconnected, which means that the number of paths, E. is significantly greater than the number of nodes V. In such circumstances, the time required to arrive at a solution of the Dykstra algorithm is on the order of E log V; i.e., T(E,V)=O(E log V). Hence, the time required by the  FIG. 3  apparatus to identify the various costs when processors  10 ,  20 , and  30  work concurrently, is the greater of T 1 (E,V)=O(E 1  log V 1 ) for the first segment, T 2 (E,V)=O(E 2  log V 2 ), and T 3 (E,V)=n[O(E 3  log V 3 )] for the third segment. 
         [0020]    The processing time for a segment can be reduced by choosing a segmentation that yields fewer nodes and paths in that segment. However, such segmentation increases the number of paths and nodes in another segment which, in turn, increases the processing time for that other segment. That is, reducing the time for one of the processors increases the time for the other processor or processors. 
         [0021]    Clearly, it is desirable to segment the network so as to make the processing times of the individual processors as equal to each other as possible, and in accord with one feature of this disclosure the segmenting of the network is effected to achieve this goal. By approximately equal to each other, or roughly equal to each other, it is meant that the shortest processing time is not less than 75% of the longest processing time. 
         [0022]    That means that the segmentation is unequal in terms of the portion of the network that is assigned to each segment. More particularly, when each segment is assigned to a single processor, to a first degree of approximation the segmentation should be such that the number of links, E, in the first segment should be the same as the number of links in the second segment, and n times larger than the number of links in the third segment. The segmentation chosen in  FIG. 2  yields processing times of O(18 log(10)), O(19 log(11)), and 3[O(7 log(7))] for processors  10 ,  20  and  30 , respectively, which times are fairly close to each other. 
         [0023]    Once the various costs are known, a determination of the least-cost path is made by processor  30 , and that corresponds to a simple selection of one out of Nl 3 ×Nr 3  costs, which is not a burdensome proposition. 
         [0024]    It may be noted that in connection with each of the segments other than the segments that contain terminals A and B, where there are Nl m  left interface nodes and Nr m  right interface nodes, a solution of an all-pair-all-paths problem is required. With a single processor assigned to such a segment, the processing burden, as mentioned above, is n[O(E m  log V m )], where n is the smaller of Nl m  and Nr m . However, this processing burden can be divided between n processors that operate in parallel.