Randomized minimum spanning tree algorithms using exponentially fewer random bits

Abstract

For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best-known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel minimum spanning tree problems, and the local sorting and set maxima problems. (For the first two problems there are provably optimal deterministic algorithms with unknown, and possibly superlinear, running times.) One downside of the randomized methods for solving these problems is that they use a number of random bits linear in the size of input. In this article we develop some general methods for reducing exponentially the consumption of random bits in comparison-based algorithms. In some cases we are able to reduce the number of random bits from linear to nearly constant, without affecting the expected running time. Most of our results are obtained by adjusting or reorganizing existing randomized algorithms to work well with a pairwise or O (1)-wise independent sampler. The prominent exception, and the main focus of this article, is a linear-time randomized minimum spanning tree algorithm that is not derived from the well-known Karger-Klein-Tarjan algorithm. In many ways it resembles more closely the deterministic minimum spanning tree algorithms based on soft heaps. Further, using our algorithm as a guide, we present a unified view of the existing “nongreedy” minimum spanning tree algorithms. Concepts from the Karger-Klein-Tarjan algorithm, such as F -lightness, MST verification, and sampled graphs, are related to the concepts of edge corruption, subgraph contractibility, and soft heaps, which are the basis of the deterministic MST algorithms of Chazelle and Pettie-Ramachandran.