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

Reading time: 2 minute
...

📝 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.

📄 Full Content

...(본문 내용이 길어 생략되었습니다. 사이트에서 전문을 확인해 주세요.)

Start searching

Enter keywords to search articles

↑↓
ESC
⌘K Shortcut