NC algorithms for the single most vital edge problem with respect to all pairs shortest paths

Abstract

For a weighted, undirected graph G = (V, E) where |V| = n and |E| = m, we examine the single most vital edge with respect to two measurements related to all-pairs shortest paths (APSP). The first measurement considers only the impact of the removal of a single edge from the APSP on the shortest distance between each vertex pair. The second considers the total weight of all the edges which make up the APSP, that is, calculate the sum of the distance between each vertex pair after the deletion of a tree edge. We give a sequential algorithm for this problem, and show how to obtain an NC algorithm running in O(log n) time using mn ² processors and O(mn ²) space on the MINIMUM CROW PRAM. Given the shortest distance between each pair of vertices u and v, the diameter of the graph is defined as the longest of these distances. The Most vital edge with respect to the diameter is the edge lying on such a u - v shortest path which when removed causes the greatest increase in the diameter. We show how to modify the above algorithm to solve this problem using the same time and number of processors. Both algorithms compare favourably with the straightforward solution which simply recalculates the all pairs shortest path information.