eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-09-01
52:1
52:13
10.4230/LIPIcs.ESA.2017.52
article
Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points
Kaplan, Haim
Roy, Sasanka
Sharir, Micha
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(n^{5/2} 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(nk^{3/2} log k) time, and (ii) an approximation algorithm that finds, in O(n+(n/(k epsilon^5))*log^{5/2}(n/k)log((1/epsilon) log(n/k))) time, a rectangle of the given perimeter or diagonal that contains at least (1-epsilon)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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol087-esa2017/LIPIcs.ESA.2017.52/LIPIcs.ESA.2017.52.pdf
Computational geometry
geometric optimization
rectangles
perimeter
area