Bisections of Two Sets of Points in the Plane Lattice

Miyuki UNO  Tomoharu KAWANO  Mikio KANO 

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences  Vol.E92-A  No.2  pp.502-507
Publication Date: 2009/02/01
Online ISSN: 1745-1337
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Algorithms and Data Structures
Keyword: 
red point,  blue point,  lattice,  bisector,  rectangular cut,  semi-rectangular cut,  two sets of points, 

Full Text: PDF(312.3KB)

Summary: 
Assume that 2m red points and 2n blue points are given on the lattice Z2 in the plane R2. We show that if they are in general position, that is, if at most one point lies on each vertical line and horizontal line, then there exists a rectangular cut that bisects both red points and blue points. Moreover, if they are not in general position, namely if some vertical and horizontal lines may contain more than one point, then there exists a semi-rectangular cut that bisects both red points and blue points. We also show that these results are best possible in some sense. Moreover, our proof gives O(N log N), N=2m+2n, time algorithm for finding the desired cut.