🤖 AI Summary
Existing simplex partitioning algorithms rely on restrictive geometric assumptions, limiting their applicability to diverse set systems and hindering generality in data structures such as range searching. This paper introduces the first generalization of simplex partitioning to arbitrary abstract set systems, proposing a geometry-agnostic greedy heuristic that minimizes the crossing number to construct low-complexity partitions. The method provides theoretical guarantees on partition quality while demonstrating empirical robustness across both geometric and non-geometric set systems—including hypergraphs, relational databases, and combinatorial families. An open-source implementation ensures reproducibility and facilitates practical deployment. Our core contributions are threefold: (i) eliminating geometric prerequisites for simplex partitioning; (ii) establishing a unified, model-agnostic partitioning framework; and (iii) providing an efficient, analyzable, and implementable construction paradigm with provable approximation bounds.
📝 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.