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

📝 Original Info

• Title: The Complexity of One or Many Faces in the Overlay of Many Arrangements
• ArXiv ID: 2512.11445
• Date: 2025-12-12
• Authors: Sariel Har-Peled

📝 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(nα(k)), where α(•) is the inverse of Ackermann's function. We also give a new and simpler proof of the bound O ( √ mλ s+2 (n)) 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 , where w is the total complexity of the arrangement. Several other applications and variants of the Combination Lemma are also presented.

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