Parallelism in Randomized Incremental Algorithms

In this paper we show that most sequential randomized incremental algorithms are in fact parallel. We consider several random incremental algorithms including algorithms for comparison sorting and Delaunay triangulation; linear programming, closest pair, and smallest enclosing disk in constant dimensions; as well as least-element lists and strongly connected components on graphs. We analyze the dependence between iterations in an algorithm, and show that the dependence structure is shallow for all of the algorithms, implying high parallelism. We identify three types of dependences found in the algorithms studied and present a framework for analyzing each type of algorithm. Using the framework gives work-efficient polylogarithmic-depth parallel algorithms for most of the problems that we study. Some of these algorithms are straightforward (e.g., sorting and linear programming), while others are more novel and require more effort to obtain the desired bounds (e.g., Delaunay triangulation and strongly connected components). The most surprising of these results is for planar Delaunay triangulation for which the incremental approach is by far the most commonly used in practice, but for which it was not previously known whether it is theoretically efficient in parallel.