TY - GEN

T1 - Polychromatic coloring for half-planes

AU - Smorodinsky, Shakhar

AU - Yuditsky, Yelena

PY - 2010/7/21

Y1 - 2010/7/21

N2 - We prove that for every integer k, every finite set of points in the plane can be k-colored so that every half-plane that contains at least 2k-1 points, also contains at least one point from every color class. We also show that the bound 2k-1 is best possible. This improves the best previously known lower and upper bounds of 4/3k and 4k-1 respectively. As a corollary, we also show that every finite set of half-planes can be k colored so that if a point p belongs to a subset Hp of at least 4k-3 of the half-planes then Hp contains a half-plane from every color class. This improves the best previously known upper bound of 8k-3. Another corollary of our first result is a new proof of the existence of small size ε-nets for points in the plane with respect to half-planes.

AB - We prove that for every integer k, every finite set of points in the plane can be k-colored so that every half-plane that contains at least 2k-1 points, also contains at least one point from every color class. We also show that the bound 2k-1 is best possible. This improves the best previously known lower and upper bounds of 4/3k and 4k-1 respectively. As a corollary, we also show that every finite set of half-planes can be k colored so that if a point p belongs to a subset Hp of at least 4k-3 of the half-planes then Hp contains a half-plane from every color class. This improves the best previously known upper bound of 8k-3. Another corollary of our first result is a new proof of the existence of small size ε-nets for points in the plane with respect to half-planes.

KW - Discrete Geometry

KW - Geometric Hypergraphs

KW - Polychromatic Coloring

KW - ε-Nets

UR - http://www.scopus.com/inward/record.url?scp=77954629836&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-13731-0_12

DO - 10.1007/978-3-642-13731-0_12

M3 - Conference contribution

AN - SCOPUS:77954629836

SN - 364213730X

SN - 9783642137303

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 118

EP - 126

BT - Algorithm Theory - SWAT 2010 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, Proceedings

T2 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010

Y2 - 21 June 2010 through 23 June 2010

ER -