An improvement of Goldberg, Plotkin and Vaidya's maximal node-disjoint paths algorithm

Goldberg, Plotkin, and Vaidya recently developed a sublinear-time parallel algorithm for finding maximal node-disjoint paths [3] with the concurrent-read concurrent-write random access machine model (CRCW PRAM) [2] by balancing two approaches to the problem appropriately. We improve their results by finding a better balance factor. Our results are as follows: we can find maximal node-disjoint paths for undirected graphs in O(√ nlog 2 n) time with O( n + m) processors improved from O(√ nlog 3 n) time with the same number of processors; for directed graphs in O(√ nlog 5/2 n) time with BFS( n, m)processors improved from O(√ nlog 3 n) time with the same number of processors. Here BFS( n, m) denotes the maximum of n + m and the number of processors required to find a breadth-first search tree in O(log 2 n) time for a directed graph with n vertices and m edges. As a consequence of our result, we show that a depth-first search tree in an undirected graph can be found in O(√ nlog 5 n) time with O( n + m) processors.