Typically-Correct Derandomization for Small Time and Space

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. \nNext, suppose a language $L$ can be decided by a nondeterministic algorithm that runs in space $S$ and time $n \\cdot \\text{poly}(S)$. We give an unambiguous algorithm for $L$ that runs in space $O(S \\sqrt{\\log S})$ and time $2^{O(S)}$ that succeeds on all but a negligible fraction of inputs of each length. \nFinally, we prove that $\\mathbf{BPL} \\subseteq \\mathbf{L}/(n + O(\\log^2 n))$ and $\\mathbf{NL} \\subseteq \\mathbf{UL}/(n + O(\\log^2 n))$, improving results by Fortnow and Klivans (STACS 06) and Reinhardt and Allender (SICOMP 00), respectively. If the original randomized/nondeterministic algorithm runs in quasilinear time, we show that fewer than $n$ bits of advice suffice (for disambiguation, this involves increasing the space complexity to $O(\\log n \\sqrt{\\log \\log n})$).