Cost-optimal parallel algorithms for the tree bisector and related problems

Abstract

An edge is a bisector of a simple path if it contains the middle point of the path. Let T=(V,E) be a tree. Given a source vertex s /spl isin/ V, the single-source tree bisector problem is to find, for every vertex /spl upsi/ /spl isin/ V, a bisector of the simple path from s to /spl upsi/. The all-pairs tree bisector problem is to find for, every pair of vertices u, /spl upsi/ /spl isin/ V, a bisector of the simple path from u to /spl upsi/. In this paper, it is first shown that solving the single-source tree bisector problem of a weighted tree has a time lower bound /spl Omega/(n log n) in the sequential case. Then, efficient parallel algorithms are proposed on the EREW PRAM for the single-source and all-pairs tree bisector problems. Two O(log n) time single-source algorithms are proposed. One uses O(n) work and is for unweighted trees. The other uses O(n log n) work and is for weighted trees. Previous algorithms for the single-source problem could achieve the same time O(log n) and the same optimal work, O(n) for unweighted trees and O(n log n) for weighted trees, on the CRCW PRAM. The contribution of our single-source algorithms is the improvement from CRCW to EREW. One all-pairs parallel algorithm is proposed. It requires O(log n) time using O(n/sup 2/) work. All the proposed algorithms are cost-optimal. Efficient tree bisector algorithms have practical applications to several location problems on trees. Using the proposed algorithms, efficient parallel solutions for those problems are also presented.