Trading Determinism for Time in Space Bounded Computations

TL;DR

This paper demonstrates that for every language in NL, there exists an unambiguous nondeterministic algorithm that operates within logarithmic squared space and runs in polynomial time, partially addressing a long-standing open problem.

Contribution

It shows that NL problems can be solved by unambiguous algorithms with polylogarithmic space and polynomial time, advancing understanding of space-time trade-offs in complexity theory.

Findings

Existence of unambiguous NL algorithms with O(log^2 n) space

Algorithms run in polynomial time

Partial solution to a long-standing open problem in complexity theory

Abstract

Savitch showed in $1970$ that nondeterministic logspace (NL) is contained in deterministic $\mathcal{O}(\log^2 n)$ space but his algorithm requires quasipolynomial time. The question whether we can have a deterministic algorithm for every problem in NL that requires polylogarithmic space and simultaneously runs in polynomial time was left open. In this paper we give a partial solution to this problem and show that for every language in NL there exists an unambiguous nondeterministic algorithm that requires $\mathcal{O}(\log^2 n)$ space and simultaneously runs in polynomial time.