Placing Text Boxes on Graphs

A Fast Approximation Algorithm for Maximizing Overlap of a Square and a Simple Polygon
  • Sjoerd van Hagen
  • Marc van Kreveld
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)

Abstract

In this paper we consider the problem of placing a unit square on a face of a drawn graph bounded by n vertices such that the area of overlap is maximized. Exact algorithms are known that solve this problem in O(n2) time. We present an approximation algorithm that—for any given ε> 0—places a (1 + ε)-square on the face such that the area of overlap is at least the area of overlap of a unit square in an optimal placement. The algorithm runs in \(O(\frac{1}{\epsilon}\, n\log^2 n)\) time. Extensions of the algorithm solve the problem for unit discs, using \(O(\frac{\log (1/\epsilon)}{\epsilon\sqrt{\epsilon}}\, n\log ^2n)\) time, and for bounded aspect ratio rectangles of unit area, using \(O(\frac{1}{\epsilon^2}\, n\log^2 n)\) time.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Sjoerd van Hagen
    • 1
  • Marc van Kreveld
    • 1
  1. 1.Department of Information and Computing SciencesUtrecht UniversityThe Netherlands

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