Improved parallel algorithms for finding the most vital edge of a graph with respect to minimum spanning tree

Let G be a connected, undirected and weighted graph with n vertices and n edges. A most vital edge of G with respect to minimum spanning tree is an edge whose removal from G results in the greatest weight-increase in the minimum spanning tree of the remaining graph. This paper presents fast parallel algorithms that compute the most vital edge of G in O(log n) time using O(max{n, m log log log n/log n}) processors on a CRCW PRAM with O(m) space, and in O(log n, log log n) time using O(max{m, n(2)/(log n log log n)}) processors on both a CREW PRAM with O(m + n log n) space and an EREW PRAM with O(n(2)) space, respectively. These significantly improve the known results of O(log n) time and O(m) processors on the CRCW PRAM [10, 13], and of O(log(2) n) time and O(n(2)/ log(2) n) processors on the CREW PRAM [13] and O(n(1+epsilon)) time using n(1-epsilon) processors and O(m log m/N/N + n alpha (m,n) log m/n) time using N less than or equal to m log m/(n alpha (m, n) log m/n) processors on the EREW PRAM, respectively.