Parallel Breadth-first and Breadth-depth traversals of general trees

Abstract

Two adaptive, level-order tree traversal algorithms are proposed for an exclusive-read and exclusive-write (EREW), parallel random access machine (PRAM) model of computation. Our breadth-first traversal algorithm for a general tree with n nodes achieves O((n/p)*log n /log(n/p)) time complexity using p processors on the EREW model, and hence it attains optimal speedup for p ≤ n^{1 − e}, where 0 < e ≤ 1. This algorithm performs better (in terms of processor-time product) than an existing algorithm [12] which has O(k log n) time complexity using O(n^{1 + 1/k}) processors on a concurrent-read and exclusive-write (CREW), PRAM model. The proposed breadth-depth algorithm for traversing a general tree requires O(n/p + log n) time on the EREW model, and thus it achieves optimal speedup for p ≤ n/log n. This algorithm provides a significant improvement over an existing parallel breadth-depth algorithm [4] which requires O(log n) time with O(n²) processors on CREW model. Our breadth-first traversal algorithm uses an Euler tour technique [20], and a list construction technique which is similar to the one used in [18] for solving the adjacency list construction problem for graphs. The breadth-depth traversal algorithm, on the other hand, is based on a special characterization which enables the reduction of this problem into a variety of list ranking problems.

This work was in part supported by a faculty research grant from the University of North Texas.