The Lovász Local Lemma is a tool that enables one to show that certain events hold with pos-itive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.

A ray–nonsingular matrix is a square complex matrix, A, such that each complex matrix whose entries have the same arguments as the corresponding entries of A is nonsingular. Extremal properties of ray– nonsingular matrices are studied in this paper. Combinatorial and probabilistic arguments are used to prove that if the order of a ray– nonsingular matrix is at least 6, then it must contain a zero entry, and that if each of its rows and columns have an equal number, k, of nonzeros, then k ≤ 13. ∗ This paper was written while Professor Lee was visiting the University of Wyoming and was supported by a 1996 Post-doctoral Fellowship from KOSEF.