Abstract
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 $ℝ^d$ 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.
Highlights
We design and analyze simple probabilistic algorithms for matching solid, i.e. full-dimensional, shapes in d-dimensional Euclidean space (d ≥ 2), under translations and rigid motions
We show that the algorithms approximate the maximal volume of overlap under translations in the following sense
Cheong et al [4] introduce a general probabilistic framework, which they use for approximating the maximal area of overlap of two unions of n and m triangles in the plane, with pre-specified absolute error ε, in time O(m + (n2/ε4)(log n)2) for translations and in time O(m + (n3/ε8)(log n)5) for rigid motions
Summary
We design and analyze simple probabilistic algorithms for matching solid, i.e. full-dimensional, shapes in d-dimensional Euclidean space (d ≥ 2), under translations and rigid motions. Cheong et al [4] introduce a general probabilistic framework, which they use for approximating the maximal area of overlap of two unions of n and m triangles in the plane, with pre-specified absolute error ε, in time O(m + (n2/ε4)(log n)2) for translations and in time O(m + (n3/ε8)(log n)5) for rigid motions. The latter time bound is smaller in their paper, due to a calculation error in the final derivation of the time bound, as was noted in [17].
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