A Parallel Algorithm for Constructing Strongly Convex Superhulls of Points

Carla Denise CASTANHO  Wei CHEN  Koichi WADA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E83-A   No.4   pp.722-732
Publication Date: 2000/04/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
computational geometry,  convexity,  strongly convex superhull,  parallel algorithm,  divide-and-conquer,  

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Let S be a set of n points in the plane and CH(S) be the convex hull of S. We consider the problem of constructing an approximate convex hull which contains CH(S) with strong convexity. An ε-convex δ-superhull of S is a convex polygon P satisfying the following conditions: (1) P has at most O(n) vertices, (2) P contains S, (3) no vertex of P lies farther than δ outside CH(S), and (4) P remains convex even if its vertices are perturbed by as much as ε. The parameters ε and δ represent the strength of convexity of P and the degree of approximation of P to CH(S), respectively. This paper presents the first parallel method for the problem. We show that an ε-convex (8+42)ε-superhull of S can be constructed in O(log n) time using O(n) processors, or in O(log n) time using O(n/log n) processors if S is sorted, in the EREW-PRAM model. We implement the algorithm and find that the average performance is even much better: the results are more strongly convex and much more approximate to CH(S) than the theoretical analysis shows.