Showing papers on "Longest path problem published in 1977"
TL;DR: Taking advantage of the concept of direction present in grid graphs, an algorithm is developed which is 0(n) in the worst case and 0(√n)In the best case.
Abstract: Grid graphs are a simple class of planar graphs for which the vertices can be assigned integer coordinates so that neighbors agree in one coordinate and differ by one in the other coordinate. Grid graphs arise in applications from the layout design of integrated circuits to idealized models of city street networks. In many applications, a shortest path between two given vertices is needed. The best known algorithms for the shortest path in a general graph of n vertices are of complexity 0(n2). However, if edge lengths are of uniform length, the shortest path can be determined in time 0(n). In this paper, taking advantage of the concept of direction present in grid graphs, an algorithm is developed which is 0(n) in the worst case and 0(√n) in the best case.
136 citations
TL;DR: A heuristic given here for finding a near optimal path cover for the general case is based upon applying the maximum-matching algorithm to the subgraphs of an interval decomposition.
Abstract: A point-disjoint path cover of a directed graph is a collection of point-disjoint paths (some paths possibly having zero length) which covers all the points. A path cover which minimizes the number of paths corresponds to an optimal sequence of the steps of a computer program for efficient coding and documentation. The minimization problem for the general directed graph is hard in the sense of being NP-complete. In the case of cycle-free digraphs, however, the problem is polynomial, for it is shown that it can be reduced to the maximum-matching problem. A heuristic given here for finding a near optimal path cover for the general case is based upon applying the maximum-matching algorithm to the subgraphs of an interval decomposition.
86 citations
TL;DR: In this article, the problem of finding paths from a fixed node to all other nodes of a directed graph which minimise a function defined on the paths is considered, and two algorithms which are generalisations of standard shortest path methods are given.
Abstract: This paper considers the problem of finding paths from a fixed node to all other nodes of a directed graph which minimise a function defined on the paths. Under certain assumptions a characterisation of optimal paths is derived. Two algorithms which are generalisations of standard shortest path methods are then given.
14 citations
TL;DR: In this paper, a situation is considered where a constraint is such that no two consecutive arcs in the path are traversed by the same agency, and functional equations using Bellman's principle of optimality are developed and solved.