Showing papers on "Longest path problem published in 1975"

A Preliminary Evaluation of the Critical Path Method for Scheduling Tasks on Multiprocessor Systems

Abstract: The problem of scheduling tasks on a system of independent identical processors is discussed and the performance of a suboptimal method is evaluated. The computation is modeled by an acyclic directed graph G(T,<), where node set T represents the set of tasks to be completed and edge set < defines the precedence between tasks. The objective is to minimize the finishing time of the computation graph. Known theoretical results are reviewed and a general branch-and-bound algorithm for finding optimal solutions is presented. The schedules produced by a simple critical path priority method are shown to be near optimal for randomly generated computation graphs.

Solving path problems on directed graphs.

Abstract: This paper considers path problems on directed graphs which are solvable by a method similar to Gaussian elimination. The paper gives an axiom system for such problems which is a weakening of Salomaa''s axioms for a regular algebra. The paper presents a general solution method which requires O($n^3$) time for dense graphs with n vertices and considerably less time for sparse graphs. The paper also presents a decomposition method which solves a path problem by breaking it into subproblems, solving each subproblem by elimination, and combining the solutions. This method is a generalization of the "reducibility" notion of data flow analysis, and is a kind of single-element "tearing". Efficiently implemented, the method requires O(m $\alpha$(m,n)) time plus time to solve the subproblems, for problem graphs with n vertices and m edges. Here $\alpha$(m,n) is a very slowly growing function which is a functional inverse of Ackermann''s function. The paper considers variants of the axiom system for which the solution methods still work, and presents several applications including solving simultaneous linear equations and analyzing control flow in computer programs.

On the decision tree complexity of the shortest path problems

Abstract: In this paper we present two new bounds on the decision tree complexity of the all-pairs shortest path problem, one for the worst case complexity and one for the average complexity. Let G be a graph on N vertices with N(N-I) directed edges joining each ordered pair of distinct vertices, and assume that each edge is assigned a nonnegative weight. Between any two vertices V.,V., i I j, there exists a directed path from v. to 1 .1 1 v. of minimum weighted length, the value of which we .1 denote by L... The computation of the N(N-I) values 1.1 L .. , 1 ~ i I j ~ N, is referred to as the all-pairs 1.1 shortest path problem. This problem and some variations of it have been investigated by a number of authors.