The Probabilistic Method Yields Deterministic Parallel Algorithms

We present a technique for converting RNC algorithms into NC algorithms. Our approach is based on a parallel implementation of the method of conditional probabilities. This method was used to convert probabilistic proofs of existence of combinatorial structures into polynomial time deterministic algorithms. It has the apparent drawback of being extremely sequential in nature. We show certain general conditions under which it is possible to use this technique for devising deterministic parallel algorithms. We use our technique to devise an NC algorithm for the set balancing problem. This problem turns out to be a useful tool for parallel algorithms. Using our de-randomization method and the set balancing algorithm, we provide an NC algorithm for the lattice approximation problem. We also use the lattice approximation problem to bootstrap the set balancing algorithm, and the result is a more processor eecient algorithm. The set balancing algorithm also yields an NC algorithm for near-optimal edge coloring of simple graphs. Our methods also extend to the parallelization of various algorithms in computational geometry that rely upon the random sampling technique of Clarkson. Finally, our methods apply to constructing certain combinatorial structures, e.g., Ramsey graphs and independent sets and covers of hypergraphs.