Efficient Parallel Algorithms for a Class of Graph Theoretic Problems

Abstract

In this paper, we present efficient parallel algorithms for the following graph problems: finding the lowest common ancestors for vertex pairs of a directed tree; finding all fundamental cycles, a directed spanning forest, all bridges, all bridge-connected components, all separation vertices, all biconnected components, and testing the biconnectivity of an undirected graph. All these algorithms achieve the O(lg n) time bound, with the first two algorithms using n[n/lg n] processors and the remaining algorithms using n[n/lg n] processors. In all cases, our algorithms are better than the previously known algorithms and in most cases reduce the number of processors used by a factor of n lg n. Moreover, our algorithms are optimal with respect to the time-processor product for dense graphs, with the exception of the first two algorithms.The machine model we use is the PRAM which is a SIMD model allowing simultaneous reads but not simultaneous writes to the same memory location.

Efficient Multicore Algorithms For Identifying Biconnected Components

“…The other is for graphs which are sparse in nature and it requires O(log 2 n log k) time with O(mn + log 2 n) processors where k is the number of biconnected components in the given graph. Tsin and Chin [20] developed an optimal algorithm for dense graphs that runs in O(log 2 n) time using O(n 2 / log 2 n) processors. Tarjan and Vishkin [19] provided a O(log n) time algorithm that uses O(n + m) processors.…”

Updating Properties of Directed Acyclic Graphs on a Parallel Random Access Machine.

“…It has been shown in [18] that several graph properties of an undirected graph can be computed by first constructing a spanning subtree for the graph. Consequently, update algorithms for these properties involve updating a spanning tree for the new graph (see [11,131).…”

“…In the case of undirected graphs, the processor complexities of computing a minimum spanning tree, bridges, bridge-connected components, cut points, and biconnected components are same as that of computing connected components (see [18]). …”

Linear Time Algorithms for Finding Articulation and Hinge Vertices of Circular Permutation Graphs

“…An O(n + m) time algorithm exists for solving the articulation vertex problem in simple graphs by using the traditional depth-first spanning tree method [1]. Moreover, efficient parallel algorithms for finding articulation vertices, bridges, and biconnected components in general graphs are given in [2], [3].…”