Super critical tree numbering and optimal tree ranking are in NC

Abstract

This paper places the optimal tree ranking problem in NC. A ranking is a labeling of the nodes with natural numbers such that if nodes u and v have the same label then there exists another node with a greater label on the path between them. An optimal ranking is a ranking in which the largest label assigned to any node is as small as possible among all rankings. An O(n) sequential algorithm is known. Researchers have speculated that the problem is P-complete. The authors show that for an n node tree, one can compute an optimal ranking in O(log/sup 2/n) time using n/sup 2//log n EREW PRAM processors. In fact, their ranking is super critical in that the label assigned to each node is absolutely as small as possible. They achieve their results by introducing and showing that a more general problem, which they call the super critical numbering problem, is in NC. No NC algorithm for the super critical tree ranking problem, approximate or otherwise, was previously known; the only known NC algorithm for optimal tree ranking was an approximate one. >