A Note on Finding All Shortest Paths

Glover, Fred; Klingman, Darwin; Napier, A

doi:10.1287/trsc.8.1.3

Cited by 6 publications

(2 citation statements)

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“…Special decomposition algorithms have been proposed for shortest-path problems on large-scale sparse networks [2,[8][9][10]12,14,17,19,22]. In most of these algorithms shortest paths are obtained between a source node and all the other nodes or between every pair of nodes.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, he indicated that star-shaped networks and linear networks are two subclasses of tree-structured networks. Glover, Klingman, and Napier [8] suggested a decomposition procedure which also takes advantage of the linear structure of a network. However, the procedure may sometimes produce very few subnetworks or even no decomposition at all.…”

Section: Introductionmentioning

confidence: 99%

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Decomposition algorithms for finding the shortest path between a source node and a sink node of a network

Tufekci

1983

Naval Research Logistics

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Decomposition algorithms for finding a shortest path between a source node and a sink node of an arbitrary distance network are developed. Different decomposition algorithms are proposed for different network topologies. Since Shier's algorithm compares very favorably with other decomposition algorithms in all the network topologies, we compare our algorithms against Shier's algorithm. It is shown that the efficiency of the proposed algorithms compares very favorably with Shier's algorithm. For special types of networks the computational requirements of the proposed algorithm is a polynomial of O(nZ).

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Decomposition algorithms for finding the shortest path between a source node and a sink node of a network

Tufekci

1983

Naval Research Logistics

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A decomposition algorithm for locating a shortest path between two nodes in a network

Jarvis

Tufekci

1982

Networks

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A decomposition algorithm for locating a shortest path between two nodes of a network with circuits (directed cycles) and arbitrary arc distances (with the customary assumption of no negative circuits) is introduced. Unlike similar decomposition algorithms by Hu [ 12, 13 1, Yen [ 23 1, and Glover, Klingman, and Napier [ lo], this algorithm finds only the shortest distance between two specified nodes. Computational complexity of this algorithm is shown to be better than the previous decomposition algorithms.

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Shortest‐path algorithms: Taxonomy and annotation

1984

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We have evolved a classification scheme to characterize algorithms for solving shortestpath problems. The algorithms are classified according t o (A) the problem type, i.e., the question being asked about the given network; (B) the input type, i.e., the salient features of the given network which impact on the design of the algorithm and selection of data structures; and (C) the type of underlying technique employed t o solve the problem. An annotated bibliography of 79 selected references on shortest-path algorithms is included. We have also provided a more complete listing of 222 references carefully culled out of a larger body of literature on shortest-path algorithms through the year 1979. DefinitionsA directed graph G = ( V , E ) consists of a finite set V of nodes and a finite set E of arcs. We will use n and e to denote the number of elements in V and E , respectively.The node set is assumed to have elements { 1, 2, . . . , n } . An arc a in E is an ordered pair (u, u ) of nodes. We say that a = (u, u) is directed from u t o u, and we also say a begins at u and ends at u. The indegree of a node u is the number of arcs in E that end at u . The outdegree of a node u is the number of arcs in E that begin at u. A directed network (or simply network) is a directed graph G together with a real-valued function l on the set of arcs. For any arc a, l(a) is called the arc length (or weight) of a . A positive arc is an arc with a positive arc length, and a negative arc is an arc with SHORTEST-PATH ALGORITHMS 277 SHORTEST PATH PROBLEMS . \ ALGEBRAICALLY RELATED PROBLEMS \ WITH CYCLES FIG. 1. Classification of shortest-path problems.a negative arc length. The weight matrix or arc length matrix (or adjacency matrix, if all edges have unit weight) of a network is an n X n matrix whose (u, u)th entry equals Z(u, u) if the arc (u, u) exists, and equals infinity otherwise. An undirected network is a network that has a symmetric arc length matrix. A path is a finite sequence of arcs P = ( a l , a?, . . . , ak), such that ai starts where ai-l ends, for i = 2, . . . , k. The individual arcs, the q ' s , are called the tracks of P. The path length (or simply length) of P is defined to be d(P) = l(al ) t . * t l(ak). If ai = ui) for i = 1 , . . . , k , the path P is also denoted by the node sequence (uo, u l , . . . , uk); uo is called the source (or origin) of P, uk is called the destination of P, and P is called a path from uo to uk. P is simple if uf # ul for all i # j . P is a cycle if uo = uk. A cycle with negative path length is a negative cycle. Pis a shortest path from u to u if d(P) is the minimum length of any path from u to u; in this case, d(P) is the shortest distance from u to u. The shortest distance matrix is an n X n matrix whose (u, u)th entry is the shortest distance from u to u. Note that there may be more than one shortest path from u to u. Therefore one talks of a shortest path but the shortest distance from x toy. When track sequence of a shortest path is emphasized some authors use the term shortest route...

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A Note on Finding All Shortest Paths

Cited by 6 publications

References 10 publications

Decomposition algorithms for finding the shortest path between a source node and a sink node of a network

Decomposition algorithms for finding the shortest path between a source node and a sink node of a network

A decomposition algorithm for locating a shortest path between two nodes in a network

Shortest‐path algorithms: Taxonomy and annotation

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