Translating a Convex Polygon to Contain a Maximum Number of Points

Abstract Given a set S of n points in the plane and a convex polygon P with m vertices, we consider the problem of finding a translation of P that contains the maximum number of points in S . We present two different solutions. Our first algorithm uses standard line-sweep techniques and requires O( nk log( nm ) + m ) time where k is the maximum number of points contained. Our second algorithm requires O( nk log( mk ) + m ) time, which is the asymptotically fastest known algorithm for this problem. Both algorithms require optimal O( m + n ) space. The algorithms also solve in the same running time the bichromatic variant of the problem, where we are given two point sets A and B and the goal is to maximize the number of points covered from A while minimizing the number of points covered from B .

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