Fast Local Computation Algorithms

TL;DR

This paper introduces local computation algorithms that efficiently answer queries about large outputs without reading entire inputs, applicable to problems like coloring, scheduling, and SAT, using probabilistic techniques.

Contribution

It develops a novel technique based on small sample spaces and the Lovász Local Lemma to construct polylogarithmic time and space local algorithms for various problems.

Findings

Algorithms run in polylogarithmic time and space.

Applicable to maximal independent set, hypergraph coloring, and SAT.

Provides a unified framework for local computation methods.

Abstract

For input $x$, let $F(x)$ denote the set of outputs that are the "legal" answers for a computational problem $F$. Suppose $x$ and members of $F(x)$ are so large that there is not time to read them in their entirety. We propose a model of {\em local computation algorithms} which for a given input $x$, support queries by a user to values of specified locations $y_i$ in a legal output $y \in F(x)$. When more than one legal output $y$ exists for a given $x$, the local computation algorithm should output in a way that is consistent with at least one such $y$. Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on known constructions of small sample spaces of $k$-wise independent random variables and Beck's analysis in his algorithmic approach to the Lov{\'{a}}sz Local Lemma, which under certain conditions can be applied to construct local computation algorithms that run in {\em polylogarithmic} time and space. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying $k$-SAT formulas.