Parallel Tree Contraction, Part 2: Further Applications

This paper applies the parallel tree contraction techniques developed in Miller and Reif’s paper [Randomness and Computation, Vol. 5, S. Micali, ed., JAI Press, 1989, pp. 47’72] to a number of fundamental graph problems. The paper presents an $O(\log n)$ time and $n / \log n$ processor, a 0-sided randomized algorithm for testing the isomorphism of trees, and an $O(\log n)$ time, n algorithm for maximal subtree isomorphism and for common subexpression elimination. An O(log n) time, n-processor algorithm for computing the canonical forms of trees and subtrees is given. An Olog n time algorithm for computing the tree of 3-connected components of a graph, an $O(\log ^2 n)$ time algorithm for computing an explicit planar embedding of a planar graph, and an $O(\log ^3 n)$ time algorithm for computing a canonical form for a planar graph are also given. All these latter algorithms use only $n^{O(1)} $ processors on a Parallel Random Access Machine (PRAM) model with concurrent writes and concurrent reads.