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Partial Construction of an Arrangement of Lines and Its Application to Optimal Partitioning of Bichromatic Point Set
Tetsuo ASANO Takeshi TOKUYAMA
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E77A
No.4
pp.595600 Publication Date: 1994/04/25 Online ISSN:
DOI: Print ISSN: 09168508 Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications) Category: Keyword: arrangement of lines, clustering, computational geometry, duality transform, topological walk,
Full Text: PDF(538.1KB)>>
Summary:
This paper presents an efficient algorithm for constructing atmostk 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 atmostk levels of the arrangement using O(n) space. Although Everett et al. recently gave an algorithm for constructing the atmostk 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 f_{d}(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.

