TY - GEN

T1 - Finding axis-parallel rectangles of fixed perimeter or area containing the largest number of points

AU - Kaplan, Haim

AU - Roy, Sasanka

AU - Sharir, Micha

N1 - Funding Information:
∗ Work by Haim Kaplan has been supported by Grant 1161/2011 from the German-Israeli Science Foundation, by Grant 1841-14 from the Israel Science Foundation, and by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11). Work by Micha Sharir has been supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11), by the Blavatnik Research Fund in Computer Science at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - Let P be a set of n points in the plane in general position, and consider the problem of finding an axis-parallel rectangle with a given perimeter, or area, or diagonal, that encloses the maximum number of points of P. We present an exact algorithm that finds such a rectangle in O(n5/2 log n) time, and, for the case of a fixed perimeter or diagonal, we also obtain (i) an improved exact algorithm that runs in O(nk3/2 log k) time, and (ii) an approximation algorithm that finds, in O ( n + n/kϵ5 log5/2 n/k log(1/ϵ log n/ k )) time, a rectangle of the given perimeter or diagonal that contains at least (1 - ϵ)k points of P, where k is the optimum value. We then show how to turn this algorithm into one that finds, for a given k, an axis-parallel rectangle of smallest perimeter (or area, or diagonal) that contains k points of P. We obtain the first subcubic algorithms for these problems, significantly improving the current state of the art.

AB - Let P be a set of n points in the plane in general position, and consider the problem of finding an axis-parallel rectangle with a given perimeter, or area, or diagonal, that encloses the maximum number of points of P. We present an exact algorithm that finds such a rectangle in O(n5/2 log n) time, and, for the case of a fixed perimeter or diagonal, we also obtain (i) an improved exact algorithm that runs in O(nk3/2 log k) time, and (ii) an approximation algorithm that finds, in O ( n + n/kϵ5 log5/2 n/k log(1/ϵ log n/ k )) time, a rectangle of the given perimeter or diagonal that contains at least (1 - ϵ)k points of P, where k is the optimum value. We then show how to turn this algorithm into one that finds, for a given k, an axis-parallel rectangle of smallest perimeter (or area, or diagonal) that contains k points of P. We obtain the first subcubic algorithms for these problems, significantly improving the current state of the art.

KW - Area

KW - Computational geometry

KW - Geometric optimization

KW - Perimeter

KW - Rectangles

UR - http://www.scopus.com/inward/record.url?scp=85030527532&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2017.52

DO - 10.4230/LIPIcs.ESA.2017.52

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AN - SCOPUS:85030527532

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 25th European Symposium on Algorithms, ESA 2017

A2 - Sohler, Christian

A2 - Sohler, Christian

A2 - Pruhs, Kirk

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 25th European Symposium on Algorithms, ESA 2017

Y2 - 4 September 2017 through 6 September 2017

ER -