Near-Linear Time Computation of Welzl Orders on Graphs with Linear Neighborhood Complexity

TL;DR

This paper introduces a fast randomized algorithm for computing Welzl orders on set systems with linear shatter functions, enabling efficient graph processing and model checking in near-linear time.

Contribution

It presents a near-linear time randomized algorithm for computing Welzl orders on set systems with linear neighborhood complexity, improving previous runtimes significantly.

Findings

Computes Welzl orders in b5(S \, ext{log} \, S) time.

Enables (n \, ext{log} \, n) time neighborhood cover computations.

Reduces first-order model checking time from (n^{5+}) to (n^{3+}).

Abstract

Orders with low crossing number, introduced by Welzl, are a fundamental tool in range searching and computational geometry. Recently, they have found important applications in structural graph theory: set systems with linear shatter functions correspond to graph classes with linear neighborhood complexity. For such systems, Welzl's theorem guarantees the existence of orders with only $\mathcal{O}(\log^2 n)$ crossings. A series of works has progressively improved the runtime for computing such orders, from Chazelle and Welzl's original $\mathcal{O}(|U|^3 |\mathcal{F}|)$ bound, through Har-Peled's $\mathcal{O}(|U|^2|\mathcal{F}|)$, to the recent sampling-based methods of Csik\'os and Mustafa. We present a randomized algorithm that computes Welzl orders for set systems with linear primal and dual shatter functions in time $\mathcal{O}(\|S\| \log \|S\|)$, where $\|S\| = |U| + \sum_{X \in \mathcal{F}} |X|$ is the size of the canonical input representation. As an application, we compute compact neighborhood covers in graph classes with (near-)linear neighborhood complexity in time \(\mathcal{O}(n \log n)\) and improve the runtime of first-order model checking on monadically stable graph classes from $\mathcal{O}(n^{5+\varepsilon})$ to $\mathcal{O}(n^{3+\varepsilon})$.