Partial Construction of an Arrangement of Lines and Its Application to Optimal Partitioning of Bichromatic Point Set

Tetsuo ASANO  Takeshi TOKUYAMA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A   No.4   pp.595-600
Publication Date: 1994/04/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
arrangement of lines,  clustering,  computational geometry,  duality transform,  topological walk,  

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This paper presents an efficient algorithm for constructing at-most-k levels of an arrangement of n lines in the plane in time O(nk+n log n), which is optimal since Ω(nk) line segments are included there. The algorithm can sweep the at-most-k levels of the arrangement using O(n) space. Although Everett et al. recently gave an algorithm for constructing the at-most-k levels with the same time complexity independently, our algorithm is superior with respect to the space complexity as a sweep algorithm. Then, we apply the algorithm to a bipartitioning problem of a bichromatic point set: For r red points and b blue points in the plane and a directed line L, the figure of demerit fd(L) associated with L is defined to be the sum of the number of blue points below L and that of red ones above L. The problem we are going to consider is to find an optimal partitioning line to minimize the figure of demerit. Given a number k, our algorithm first determines whether there is a line whose figure of demerit is at most k, and further finds an optimal bipartitioning line if there is one. It runs in O(kn+n log n) time (n=r+b), which is subquadratic if k is sublinear.