High Probability Work Efficient Parallel Algorithms

TL;DR

This paper presents the first parallel semisort algorithm achieving $O(n) work and $O(\text{polylog } n)$ depth whp, improving upon the expected work bound of Gu et al.

Abstract

Randomized parallel algorithms for many fundamental problems achieve optimal linear work in expectation, but upgrading this guarantee to hold with high probability (whp) remains a recurring theoretical challenge. In this paper, we address this gap for several core parallel primitives. First, we present the first parallel semisort algorithm achieving $O(n)$ work and $O(\text{polylog } n)$ depth whp, improving upon the $O(n)$ expected work bound of Gu et al. [SPAA 2015]. Our analysis introduces new concentration arguments based on simple tabulation hashing and tail bounds for weighted sums of geometric random variables. As a corollary, we obtain an integer sorting algorithm for keys in $[n]$ matching the same bounds. Second, we introduce a framework for boosting randomized parallel graph algorithms from expected to high probability linear work. The framework applies to \emph{locally extendable} problems -- those admitting a deterministic procedure that extends a solution across a graph cut in work proportional to the cut size. We combine this with a \emph{culled balanced partition} scheme: an iterative culling phase removes a polylogarithmic number of high-degree vertices, after which the remaining graph admits a balanced random vertex whp via a bounded-differences argument. Applying work-inefficient whp subroutines to the small pieces and deterministic extension across cuts yields overall linear work whp. We instantiate this framework to obtain $O(m)$ work and polylogarithmic depth whp algorithms for $(Δ+1)$-vertex coloring and maximal independent set.

Theorems & Definitions (21)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.1
• Theorem 2.4: McDiarmid mcdiarmid89
• Theorem 2.5: Pǎtraşcu--Thorup tab-hasing
• Lemma 2.1
• Corollary 2.1
• Theorem 3.1
• Lemma 3.1