This paper places the optimal tree ranking problem in [Formula: see text]. 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 this problem is P-complete. We show that for an n-node tree, one can compute an optimal ranking in O( log n) time using n2/ log n CREW PRAM processors. In fact, our ranking is super critical in that the label assigned to each node is absolutely as small as possible. We achieve these results by showing that a more general problem, which we call the super critical numbering problem, is in [Formula: see text]. No [Formula: see text] algorithm for the super critical tree ranking problem, approximate or otherwise, was previously known; the only known [Formula: see text] algorithm for optimal tree ranking was an approximate one.
On Finding the Optimal Tree of a Complete Weighted Graph
