Splitters and Near-Optimal Derandomization

We present a fairly general method for nding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)-universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2 k con gurations appear) and families of perfect hash functions. The near-optimal constructions of these objects imply the very e cient derandomization of algorithms in learning, of xed-subgraph nding algorithms, and of near optimal threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a local-coloring protocol, and for exhaustive testing of circuits.