An optimal algorithm for intersecting three-dimensional convex polyhedra
SIAM Journal on Computing
Cutting hyperplanes for divide-and-conquer
Discrete & Computational Geometry
On the area of overlap of translated polygons
Computer Vision and Image Understanding
On linear-time deterministic algorithms for optimization problems in fixed dimension
Journal of Algorithms
Similarity and Symmetry Measures for Convex Shapes Using Minkowski Addition
IEEE Transactions on Pattern Analysis and Machine Intelligence
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
State of the art in shape matching
Principles of visual information retrieval
Maximizing the overlap of two planar convex sets under rigid motions
Computational Geometry: Theory and Applications
Maximum overlap and minimum convex hull of two convex polyhedra under translations
Computational Geometry: Theory and Applications
Geometric optimization and sums of algebraic functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Overlap of convex polytopes under rigid motion
Computational Geometry: Theory and Applications
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We study the problem of maximizing the overlap of two convex polytopes under translation in R^d for some constant d=3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any @e0, finds an overlap at least the optimum minus @e and reports the translation realizing it. The running time is O(n^@?^d^/^2^@?^+^1log^dn) with probability at least 1-n^-^O^(^1^), which can be improved to O(nlog^3^.^5n) in R^3. The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. The perturbation causes an additive error @e, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. The running time bounds, the probability bound, and the big-O constants in these bounds are independent of @e.