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A fast planar partition algorithm, II

Published:03 January 1991Publication History
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Abstract

Randomized, optimal algorithms to find a partition of the plane induced by a set of algebraic segments of a bounded degree, and a set of linear chains of a bounded degree, are given. This paper also provides a new technique for clipping, called virtual clipping, whose overhead per window W depends logarithmically on the number if intersections between the borders of W and the input segments. In contrast, the overhead of the conventional clipping technique depends linearly on this number of intersections. As an application of virtual clipping, a new simple and efficient algorithm for plannar point location is given.

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  1. A fast planar partition algorithm, II

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            Ralph Walter Wilkerson

            In these two papers, Mulmuley discusses in considerable detail algorithms for finding the partition induced by a set of linear segments in the plane and selected applications. Two of the algorithms presented for solving the planar partition problem are randomized, optimal algorithms, and the third is based on the classical divide and conquer strategy. All three algorithms run in time O ( m + n log n ), where m is the number of points of intersection and n is the number of input segments. Detailed analyses of all algorithms are included. The main analytical tool is the probabilistic analysis of a certain game. The second paper continues this study by examining several applications of the above algorithms. In particular, this paper develops a new clipping algorithm, called virtual clipping, where cost per window W depends logarithmically on the number of intersections between the borders of W and the input segments. As an application of virtual clipping, the author gives a simple and efficient algorithm for planar point location. The technique is extended to obtain a hidden surface removal algorithm.

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