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Dynamic maintenance of 2-d convex hulls and order decomposable problems

MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44728

Kapoor,  Sanjiv
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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MPI-I-95-1-015.pdf
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Kapoor, S.(1995). Dynamic maintenance of 2-d convex hulls and order decomposable problems (MPI-I-1995-1-015). Saarbrücken: Max-Planck-Institut für Informatik.

In this paper, we consider dynamic data structures for order decomposable problems. This class of problems include the convex hull problem, the Voronoi diagram problem, the maxima problem, and intersection of halfspaces. This paper first describes a scheme for maintaining convex hulls in the plane dynamically in $O(\log n)$ amortized time for insertions and $O(\log^2 n)$ time for deletions. $O(n)$ space is used. The scheme improves on the time complexity of the general scheme by Overmars and Van Leeuwen. We then consider the general class of Order Decomposable Problems. We show improved behavior for insertions in the presence of deletions, under some assumptions. The main assumption we make is that the problems are required to be {\em change sensitive}, i.e., updates to the solution of the problem at an insertion can be obtained in time proportional to the changes.