Incremental algorithms for finding the convex hulls of circles and the lower envelopes of parabolas

Olivier Devillers 1 Mordecai Golin 1
1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : The existing $O(n \log n)$ algorithms for finding the convex hulls of circles and the lower envelope of parabolas follow the divide-and-conquer paradigm. The difficulty with developing incremental algorithms for these problems is that the introduction of a new circle or parabola can cause $\Theta(n)$ structural changes, leading to $\Theta(n^2)$ total structural changes during the running of the algorithm. In this note we examine the geometry of these problems and show that, if the circles or parabolas are first sorted by appropriate parameters before constructing the convex hull or lower envelope incrementally, then each new addition may cause at most 3 changes in an amortized sense. These observations are then used to develop $O(n \log n)$ incremental algorithms for these problems.
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Submitted on : Wednesday, May 24, 2006 - 3:12:34 PM
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Olivier Devillers, Mordecai Golin. Incremental algorithms for finding the convex hulls of circles and the lower envelopes of parabolas. [Research Report] RR-2280, INRIA. 1994. ⟨inria-00074391⟩

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2006