Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points
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.
Computational geometry
geometric optimization
rectangles
perimeter
area
52:1-52:13
Regular Paper
Haim
Kaplan
Haim Kaplan
Sasanka
Roy
Sasanka Roy
Micha
Sharir
Micha Sharir
10.4230/LIPIcs.ESA.2017.52
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