A Greedy Algorithm for Low-Crossing Partitions for General Set Systems

TL;DR

This paper introduces a simple greedy heuristic for constructing simplicial partitions applicable to any set system, demonstrating good empirical performance across diverse geometric and non-geometric instances.

Contribution

It presents a novel, general-purpose greedy algorithm for low-crossing partitions in set systems, expanding applicability beyond specific geometric cases.

Findings

Performs well on most tested instances

Applicable to both geometric and non-geometric set systems

Implementation available on Github

Abstract

Simplicial partitions are a fundamental structure in computational geometry, as they form the basis of optimal data structures for range searching and several related problems. Current algorithms are built on very specific spatial partitioning tools tailored for certain geometric cases. This severely limits their applicability to general set systems. In this work, we propose a simple greedy heuristic for constructing simplicial partitions of any set system. We present a thorough empirical evaluation of its behavior on a variety of geometric and non-geometric set systems, showing that it performs well on most instances. Implementation of these algorithms is available on Github.