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An upper bound on the volume of the symmetric difference of a body and a congruent copy
, 2010
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Onetoone Point Set Matchings for Grid Map Layout
 EUROCG
, 2012
"... We study several onetoone point set matching problems which are motivated by layout problems for grid maps. We are given two sets A and B of n points in the plane, and we wish to compute an optimal onetoone matching between A and B. We consider two optimisation criteria: minimising the sum of t ..."
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We study several onetoone point set matching problems which are motivated by layout problems for grid maps. We are given two sets A and B of n points in the plane, and we wish to compute an optimal onetoone matching between A and B. We consider two optimisation criteria: minimising the sum of the L1distances between matched points, and maximising the number of pairs of points in A for which the matching preserves the directional relation. We show how to minimise the total L1distance under translation or scaling in O(n6 log3 n) time, and under both translation and scaling in O(n10 log3 n) time. We further give a 4approximation for preserving directional relations by computing a minimum L1distance matching in O(n2 log3 n) time.
Matching solid shapes in arbitrary dimension via random sampling ∗
, 2012
"... We give simple probabilistic algorithms that approximately maximize the volume of overlap of two solid, i.e. fulldimensional, shapes under translations and rigid motions. The shapes are subsets of Rd where d ≥ 2. The algorithms approximate with respect to an prespecified additive error and succeed ..."
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We give simple probabilistic algorithms that approximately maximize the volume of overlap of two solid, i.e. fulldimensional, shapes under translations and rigid motions. The shapes are subsets of Rd where d ≥ 2. The algorithms approximate with respect to an prespecified additive error and succeed with high probability. Apart from measurability assumptions, we only require that points from the shapes can be generated uniformly at random. An important example are shapes given as finite unions of simplices that have pairwise disjoint interiors. 1