Parallelism in Randomized Incremental Algorithms

TL;DR

This paper demonstrates that many sequential randomized incremental algorithms are inherently parallelizable, providing a framework for efficient parallel algorithms with optimal work and polylogarithmic depth for various computational geometry problems.

Contribution

It introduces a framework for analyzing dependence structures in randomized incremental algorithms, enabling the design of work-efficient parallel algorithms for multiple problems, including the first optimal parallel Delaunay triangulation.

Findings

Most algorithms have shallow dependence structures with high probability.

The framework yields polylogarithmic-depth parallel algorithms for several problems.

First incremental Delaunay triangulation algorithm with optimal work and polylogarithmic depth.

Abstract

In this paper we show that many sequential randomized incremental algorithms are in fact parallel. We consider algorithms for several problems including Delaunay triangulation, linear programming, closest pair, smallest enclosing disk, least-element lists, and strongly connected components. We analyze the dependences between iterations in an algorithm, and show that the dependence structure is shallow with high probability, or that by violating some dependences the structure is shallow and the work is not increased significantly. We identify three types of algorithms based on their dependences and present a framework for analyzing each type. Using the framework gives work-efficient polylogarithmic-depth parallel algorithms for most of the problems that we study. This paper shows the first incremental Delaunay triangulation algorithm with optimal work and polylogarithmic depth, which is an open problem for over 30 years. This result is important since most implementations of parallel Delaunay triangulation use the incremental approach. Our results also improve bounds on strongly connected components and least-elements lists, and significantly simplify parallel algorithms for several problems.