On computing the centroid of the vertices of an arrangement and related problems

Authors:
Deepak Ajwani;Saurabh Ray;Raimund Seidel;Hans Raj Tiwary
Affiliations:
Max-Planck-Institut für Informatik, Saarbrücken, Germany and Universität des Saarlandes, Saarbrücken, Germany;Max-Planck-Institut für Informatik, Saarbrücken, Germany and Universität des Saarlandes, Saarbrücken, Germany;Max-Planck-Institut für Informatik, Saarbrücken, Germany and Universität des Saarlandes, Saarbrücken, Germany;Max-Planck-Institut für Informatik, Saarbrücken, Germany and Universität des Saarlandes, Saarbrücken, Germany
Venue:
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Year:
2007

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Abstract

We consider the problem of computing the centroid of all the vertices in a non-degenerate arrangement of n lines. The trivial approach requires the enumeration of all (n 2) vertices. We present an O(n log2 n) algorithm for computing this centroid. For arrangements of n segments we give an O(n4/3+ε) algorithm for computing the centroid of its vertices. For the special case that all the segments of the arrangement are chords of a simply connected planar region we achieve an O(n log5 n) time bound. Our bounds also generalize to certain natural weighted versions of those problems.