Largest Empty Square Queries in Rectilinear Polygons

  • Michael GesterEmail author
  • Nicolai Hähnle
  • Jan Schneider
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9155)


Given a rectilinear polygon P and a point \(p \in P\), what is a largest axis-parallel square in P that contains p? This question arises in VLSI design from physical limitations of manufacturing processes. Related problems with disks instead of squares and point sets instead of polygons have been studied previously.

We present an efficient algorithm to preprocess P in time O(n) for simple polygons or \(O(n \log n)\) if holes are allowed. The resulting data structure of size O(n) can be used to answer largest square queries for any point in P in time \(O(\log n)\). Given a set of points Q instead of a rectilinear polygon, the same algorithm can be used to find a largest square containing a given query point but not containing any point in Q in its interior.


Voronoi diagram Computational geometry Rectilinear polygon VLSI design 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Michael Gester
    • 1
    Email author
  • Nicolai Hähnle
    • 1
  • Jan Schneider
    • 1
  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

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