Typically-Correct Derandomization for Small Time and Space

TL;DR

This paper introduces a derandomization technique that reduces randomness in algorithms running in small space and time, leading to nearly deterministic algorithms with low failure probabilities for a broad class of problems.

Contribution

It presents a new derandomization method that converts randomized algorithms into nearly deterministic ones with minimal randomness and failure on most inputs.

Findings

Reduces randomness usage to O(S) bits in space-bounded algorithms.

Creates algorithms with negligible failure probability on most inputs.

Provides complexity-theoretic applications of the derandomization technique.

Abstract

Suppose a language $L$ can be decided by a bounded-error randomized algorithm that runs in space $S$ and time $n \cdot \text{poly}(S)$. We give a randomized algorithm for $L$ that still runs in space $O(S)$ and time $n \cdot \text{poly}(S)$ that uses only $O(S)$ random bits; our algorithm has a low failure probability on all but a negligible fraction of inputs of each length. An immediate corollary is a deterministic algorithm for $L$ that runs in space $O(S)$ and succeeds on all but a negligible fraction of inputs of each length. We also give several other complexity-theoretic applications of our technique.