Large-scale semi-discrete optimal transport with distributed Voronoi diagrams

TL;DR

This paper introduces a scalable, parallel algorithm for semi-discrete optimal transport problems involving extremely large point sets, enabling applications in cosmology, fluid simulation, and data science.

Contribution

It presents the Distributed Voronoi Diagram (DVD) algorithm that computes Voronoi diagrams in parallel for giga-scale point sets, advancing the state-of-the-art in large-scale optimal transport solutions.

Findings

The DVD algorithm efficiently computes Voronoi diagrams on distributed clusters.

Experimental results show potential for giga-scale applications in cosmology.

The method significantly extends the limits of existing optimal transport computations.

Abstract

In this article, we propose a numerical method to solve semi-discrete optimal transport problems for gigantic pointsets (108 points and more). By pushing the limits by several orders of magnitude, it opens the path to new applications in cosmology, fluid simulation and data science to name but a few. The method is based on a new algorithm that computes (generalized) Voronoi diagrams in parallel and in a distributed way. First we make the simple observation that the cells defined by a subgraph of the Delaunay graph contain the Voronoi cells, and that one can deduce the missing edges from the intersections between those cells. Based on this observation, we introduce the Distributed Voronoi Diagram algorithm (DVD) that can be used on a cluster and that exchanges vertices between the nodes as need be. We also report early experimental results, demonstrating that the DVD algorithm has the potential to solve some giga-scale semi-discrete optimal transport problems encountered in computational cosmology.