Bisections of Two Sets of Points in the Plane Lattice
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.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E92-A No.2 pp.502-507
- Publication Date
- 2009/02/01
- Publicized
- Online ISSN
- 1745-1337
- DOI
- 10.1587/transfun.E92.A.502
- Type of Manuscript
- PAPER
- Category
- Algorithms and Data Structures
Authors
Keyword
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Miyuki UNO, Tomoharu KAWANO, Mikio KANO, "Bisections of Two Sets of Points in the Plane Lattice" in IEICE TRANSACTIONS on Fundamentals,
vol. E92-A, no. 2, pp. 502-507, February 2009, doi: 10.1587/transfun.E92.A.502.
Abstract: 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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E92.A.502/_p
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@ARTICLE{e92-a_2_502,
author={Miyuki UNO, Tomoharu KAWANO, Mikio KANO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Bisections of Two Sets of Points in the Plane Lattice},
year={2009},
volume={E92-A},
number={2},
pages={502-507},
abstract={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.},
keywords={},
doi={10.1587/transfun.E92.A.502},
ISSN={1745-1337},
month={February},}
Copy
TY - JOUR
TI - Bisections of Two Sets of Points in the Plane Lattice
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 502
EP - 507
AU - Miyuki UNO
AU - Tomoharu KAWANO
AU - Mikio KANO
PY - 2009
DO - 10.1587/transfun.E92.A.502
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E92-A
IS - 2
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - February 2009
AB - 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.
ER -
English
