Shortest‐path algorithms: Taxonomy and annotation

Abstract

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...

Average-case complexity of single-source shortest-paths algorithms: lower and upper bounds

“…Naturally, due to the importance of shortest-paths problems and the intensive research on them, our list cannot (and is not intended to) provide a survey of the whole field. Appropriate overview papers for classical and recent sequential shortest-paths results are, e.g., [6,11,52,58,65,71].…”

“…Shortest-paths computations are used as subroutines in solution procedures for computational biology (DNA sequence alignment [68]), VLSI design [8], knapsack packing problems [25], and traveling salesman problems [43] and for many other problems. A diverse set of shortest-paths models and algorithms have been developed to accommodate these various applications [11].…”

An effective cost distance calculation based on raster data model improved algorithm

“…2. For this model, many scholars give solving algorithm for calculation the cost from source point to each target point [1][2][3]. To visual representation the algorithm follow-up, we can spend the consuming grid as a spread medium, then the best path between two points that can be seen as waves emitted from a source (such as light, sound or other waves) to the target point which path is the fastest in the transmission medium.…”

Performance analysis of fast distributed link restoration algorithms

“…The survey article of Deo and Pang [11] classified the shortest path algorithms into four classes: 1) one (node) to all (nodes) [12], 2) one to one [13], 3) all to all [14], and 4) k-shortest path algorithms [15,16,17]. The k-shortest path algorithms try to find k paths between the source node and the sink node.…”