The minimum spanning tree problem consists of determining for a connected and undirected graph a spanning tree of minimal total edge cost. The sensitivity analysis problem involves a determination of the extent to wh...
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The minimum spanning tree problem consists of determining for a connected and undirected graph a spanning tree of minimal total edge cost. The sensitivity analysis problem involves a determination of the extent to which each edge cost can be perturbed individually without upsetting the minimality of the subgraph. Sensitivity analysis also can be applied to the shortest path tree problem. This problem requires computing for a given directed graph and a given vertex a spanning tree rooted at the vertex and containing a minimum-cost path from the vertex to every other vertex. algorithms are presented for performing sensitivity analysis of minimum spanning trees and shortest path trees. The algorithms involve the use of an auxiliary graph called a transmuter. Figures.
The problem of sink-finding in a directed graph represented by an adjacency matrix, was first introduced as a counterexample to an early version of the Anderaa-Rosenberg (1973) conjecture, which stated that, for P a &...
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The problem of sink-finding in a directed graph represented by an adjacency matrix, was first introduced as a counterexample to an early version of the Anderaa-Rosenberg (1973) conjecture, which stated that, for P a ''natural'' property of graphs, the problem of ascertaining whether P holds for a graph with n vertices, represented by its adjacency matrix, necessitates the examination of 0(n2) matrix entries in the worst case. Anderaa disproved this conjecture by demonstrating that sink-finding can be accomplished using only about 3n matrix accesses. The problem of ascertaining whether a graph has a sink utilizing only 0(n) matrix accesses has since been introduced as an exercise in 2 texts on algorithms. Although it is not difficult to construct an algorithm that performs at most 3n-4 matrix accesses, this algorithm is not optimal. It is demonstrated that the optimal number of matrix accesses is actually 3n-(log n)-3.
In a strongly connected digraph whose edges have distinct non-negative weights, the hierarchy of vertex partitions may be represented by means of a rooted tree. This tree is called the strong component decomposition ...
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In a strongly connected digraph whose edges have distinct non-negative weights, the hierarchy of vertex partitions may be represented by means of a rooted tree. This tree is called the strong component decomposition tree. The strong component decomposition tree has been proposed as a clustering method suitable for asymmetric similarity matrices. An especially fast algorithm has been devised to find strong component decomposition trees. The algorithm, which is recursive, consists of several steps. It can be adapted to handle non-distinct edge weights. A different adaptation will cause it to return the edges defining the strong components as well as the vertex sets. Figures.
Matrix multiplication algorithms for cube connected and perfect shuffle computers are presented. It is shown that in both these models two n×nn×nn \times n matrices can be multiplied in <span class="...
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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...
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A general method is described for solving path problems on directed graphs. Such path problems include finding shortest paths, solving sparse systems of hnear equaUons, and carrying out global flow analysis of compute...
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Let G be a directed graph such that for each vertex v in G, the successors of v are ordered Let C be any equivalence relation on the vertices of G. The congruence closure C* of C is the finest equivalence relation con...
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A fast algorithm for finding dominators in a flowgraph is presented. The algorithm uses depth-first search and an efficient method of computing functions defined on paths in trees. A simple implementation of the algor...
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