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

Sariel Har-Peled

TL;DR

The paper generalizes the Combination Lemma to overlays of t arc arrangements, bounding the complexity of faces in the overlay by linking it to the total boundary complexity of the input arrangements and the number of arrangements. It develops a single-face bound using lower envelopes and Davenport–Schinzel sequences, establishing a tight bound and deriving a key corollary for polygonal arrangements: the maximum complexity of a single face is Θ(n α(k)). The authors then extend these ideas to multiple marked faces, obtaining near-tight bounds for the total complexity of k faces in the overlay. Finally, they derive improved bounds for sparse arrangements via chromatic-number arguments and discuss several applications and variants relevant to computational geometry and motion planning.

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 $O \left( \sqrt{m} λ_{s+2}( n ) \right)$ 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 $O \left( (n + \sqrt{m}\sqrt{w}) \frac{λ_{s+2}(n)}{n} \right)$, where $w$ is the total complexity of the arrangement. Several other applications and variants of the Combination Lemma are also presented.

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

TL;DR

The paper generalizes the Combination Lemma to overlays of t arc arrangements, bounding the complexity of faces in the overlay by linking it to the total boundary complexity of the input arrangements and the number of arrangements. It develops a single-face bound using lower envelopes and Davenport–Schinzel sequences, establishing a tight bound and deriving a key corollary for polygonal arrangements: the maximum complexity of a single face is Θ(n α(k)). The authors then extend these ideas to multiple marked faces, obtaining near-tight bounds for the total complexity of k faces in the overlay. Finally, they derive improved bounds for sparse arrangements via chromatic-number arguments and discuss several applications and variants relevant to computational geometry and motion planning.

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 simple polygons with a total of sides is , where is the inverse of Ackermann's function. We also give a new and simpler proof of the bound on the total number of edges of faces in an arrangement of Jordan arcs, each pair of which intersect in at most points, where is the maximum length of a Davenport-Schinzel sequence of order with symbols. We extend this result, showing that the total number of edges of faces in a sparse arrangement of Jordan arcs is , where is the total complexity of the arrangement. Several other applications and variants of the Combination Lemma are also presented.

Paper Structure

This paper contains 10 sections, 29 theorems, 36 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Complexity of a Single Face in an Overlay of Arrangements
  3. Introduction
  4. The Complexity of the Overlay of LowerEnvelopes
  5. Restricted Davenport-Schinzel Sequences
  6. The Complexity of a Single Face in an Overlay of Arrangements
  7. The Complexity of a Face in Certain Arrangements of Line Segments
  8. Complexity of Many Faces in Arrangements
  9. Many Faces in the Overlay of Many Arrangements
  10. The Complexity of Many Faces in Sparse Arrangements

Key Result

Lemma 1.1

Given two sets $\Gamma_{1}, \Gamma_{2}$ of Jordan arcs as above, and a set $P$ of $k$ marking points, the total complexity of all the marked faces in $A(\Gamma_{1} \cup \Gamma_{2})$ is $O( r + b + k)$, where $r, b$ are the total complexities of all marked faces in $A(\Gamma_{1}), A(\Gamma_{2})$, res

Figures (5)

  • Figure 1.1: The arrangement of $\left\{ { \gamma_1, \gamma_2, \gamma_3, \gamma_4 } \right\}$.
  • Figure 1.2: Overlay of two arrangements with two marked faces
  • Figure 2.1: Overlay of the lower envelopes of the red and blue functions
  • Figure 2.2: Traversing a face boundary and the resulting sequence $S = \left\{ \gamma_{1}^{+} \right.$$\gamma_{2}^{-}$$\gamma_{2}^{+}$$\gamma_{1}^{+}$$\gamma_{7}^{-} \gamma_{3}^{-} \gamma_{6}^{+} \gamma_{6}^{-} \gamma_{3}^{-} \gamma_{5}^{+} \gamma_{5}^{-} \gamma_{3}^{-} \gamma_{4}^{+} \gamma_{2}^{-} \left. \gamma_{1}^{-} \right\}$.
  • Figure 2.3: The subsequence $\cdots p \cdots q \cdots p \cdots q \cdots$ is impossible

Theorems & Definitions (38)

  • Lemma 1.1
  • Lemma 1.2: GSS89
  • Theorem 2.1: Single Face Combination Theorem
  • Lemma 2.2
  • Remark 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6: The Consistency Lemma GSS89
  • Theorem 2.7: The Complexity of a Single Face
  • Lemma 2.8
  • ...and 28 more