Showing papers on "Longest path problem published in 1981"

Fast Algorithms for Solving Path Problems

Abstract: Let G = (V,E) be a directed graph with a distinguished source vertex s. The single-source path expression problem is to find, for each vertex v, a regular expression P(s,v) which represents the set of all paths in G from s to v. A solution to this problem can be used to solve shortest path problems, solve sparse systems of linear equations, and carry out global flow analysis. We describe a method to compute path expressions by dividing G into components, computing path expressions on the components by Gaussian elimination, and combining the solutions. This method requires O(m $\alpha$(m,n)) time on a reducible flow graph, where n is the number of vertices in G, m is the number of edges in G, and $\alpha$ is a functional inverse of Ackermann''s function. The method makes use of an algorithm for evaluating functions defined on paths in trees. A simplified version of the algorithm, which runs in O(m log n) time on reducible flow graphs, is quite easy to implement and efficient in practice.

A Unified Approach to Path Problems

Abstract: We describe a general method for solving path problems on directed graphs. Such path problems include finding shortest paths, solving sparse systems of linear equations, and carrying out global flow analysis of computer programs. Our method consists of two steps. First, we construct a collection of regular expressions representing sets of paths in the graph. This can be done by using any standard algorithm, such as Gaussian or Gauss-Jordan elimination. Next, we apply a natural mapping from regular expressions into the given problem domain. We exhibit the mappings required to find shortest paths, solve sparse systems of linear equations, and carry out global flow analysis. Our results provide a general-purpose algorithm for solving any path problem, and show that the problem of constructing path expressions is in some sense the most general path problem.

An $O(n\log ^2 n)$ Algorithm for the kth Longest Path in a Tree with Applications to Location Problems

Abstract: Many known algorithms are based on selection in a set whose cardinality is superlinear in terms of the input length. It is desirable in these cases to have selection algorithms that run in sublinear time in terms of the cardinality of the set. This paper presents a successful development in this direction. The methods developed here are applied to improve the previously known upper bounds for the time complexity of various location problems.

Nonpreemptive LP-Scheduling on Homogeneous Multiprocessor Systems

Abstract: Unequal execution time task systems are nonpreemptively scheduled on $m \geqq 2$ identical processors without additional resource constraints. Worst-case bounds for the ratio of the length of an LP-schedule (longest path) and an optimal schedule are given for two classes of dependency structures—chains and trees. Moreover, the asymptotic bounds, which are independent of the number of processors, are given for these classes and for anti-tree-systems.

A note on finding shortest path trees

Abstract: Two shortest path algorithms are compared and it is shown that, while one outperforms the other in practice, the former's running time is exponential in the worst case while the latter's is polynomial. A procedure which constructs such worst case examples is given.

Longest paths in digraphs

Abstract: In this paper, we give a sufficient condition on the degrees of the vertices of a digraph to insure the existence of a path of given length, and we characterize the extremal graphs.

A note on path and cycle decompositions of graphs

Abstract: In the study of decompositions of graphs into paths and cycles, the following questions have arisen: Is it true that every graph G has a smallest path (resp. path-cycle) decomposition P such that every odd vertex of G is the endpoint of exactly one path of P? This note gives a negative answer to both questions.