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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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One-to-one Point Set Matchings for Grid Map Layout
- EUROCG
, 2012
"... We study several one-to-one point set matching prob-lems 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 one-to-one matching between A and B. We consider two optimisation criteria: minimising the sum of t ..."
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We study several one-to-one point set matching prob-lems 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 one-to-one matching between A and B. We consider two optimisation criteria: minimising the sum of the L1-distances between matched points, and maximis-ing the number of pairs of points in A for which the matching preserves the directional relation. We show how to minimise the total L1-distance under transla-tion or scaling in O(n6 log3 n) time, and under both translation and scaling in O(n10 log3 n) time. We further give a 4-approximation for preserving direc-tional relations by computing a minimum L1-distance 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. full-dimensional, shapes under translations and rigid motions. The shapes are subsets of Rd where d ≥ 2. The algorithms approximate with respect to an pre-specified 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. full-dimensional, shapes under translations and rigid motions. The shapes are subsets of Rd where d ≥ 2. The algorithms approximate with respect to an pre-specified 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
