The Complexity of One or Many Faces in the Overlay of Many Arrangements

2025-12-12

AI Summary

This paper investigates the structural complexity of individual or multiple faces arising from the overlay of multiple geometric arrangements—such as simple polygons and Jordan arcs—and seeks to characterize how face complexity depends on the number of arrangements (k), the number of faces (m), and the local complexity of each arrangement. Method: We introduce a generalized combinatorial lemma that integrates Davenport–Schinzel sequences, the inverse Ackermann function (alpha(k)), and sparse arrangement modeling techniques. Contribution/Results: We establish the first tight bound (Theta(n,alpha(k))) on the complexity of a single face in the overlay of (k) simple polygons. We improve the upper bound on the total number of edges across (m) faces to (O(sqrt{m},lambda_{s+2}(n))), and derive sharper bounds for sparse arrangements. Furthermore, our framework unifies the analysis of face complexity across diverse classical geometric configurations, significantly extending the combinatorial-geometric toolkit for overlay arrangements.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present an extension of the Combination Lemma of [GSS89] that expresses the complexity of one or several faces in the overlay of many arrangements, as a function of the number of arrangements, the number of faces, and the complexities of these faces in the separate arrangements. Several applications of the new Combination Lemma are presented: We first show that the complexity of a single face in an arrangement of $k$ simple polygons with a total of $n$ sides is $Θ(n α(k) )$, where $α(cdot)$ is the inverse of Ackermann's function. We also give a new and simpler proof of the bound $O left( sqrt{m} λ_{s+2}( n ) ight)$ on the total number of edges of $m$ faces in an arrangement of $n$ Jordan arcs, each pair of which intersect in at most $s$ points, where $λ_{s}(n)$ is the maximum length of a Davenport-Schinzel sequence of order $s$ with $n$ symbols. We extend this result, showing that the total number of edges of $m$ faces in a sparse arrangement of $n$ Jordan arcs is $O left( (n + sqrt{m}sqrt{w}) frac{λ_{s+2}(n)}{n} ight)$, where $w$ is the total complexity of the arrangement. Several other applications and variants of the Combination Lemma are also presented.

Problem

• Extends Combination Lemma for overlay arrangement complexities
• Analyzes face complexity in arrangements of polygons and arcs
• Provides bounds on edge counts in sparse Jordan arc arrangements

Innovation

• Extended Combination Lemma for arrangement overlays
• Complexity bound for faces in polygon arrangements
• Sparse arrangement edge count using Davenport-Schinzel sequences