Reverse nearest neighbor queries in fixed dimension

Reverse nearest neighbor queries are defined as follows: Given an input point-set P, and a query point q, find all the points p in P whose nearest point in P U {q} \ {p} is q. We give a data structure to answer reverse nearest neighbor queries in fixed-dimensional Euclidean space. Our data structure uses O(n) space, its preprocessing time is O(n log n), and its query time is O(log n).Comment: 7 pages, 3 figures; typos corrected; more background material on compressed quadtree

Overlap of convex polytopes under rigid motion

We present an algorithm to compute a rigid motion that approximately maximizes the volume of the intersection of two convex polytopes P-1 and P-2 in R-3. For all epsilon is an element of (0, 1/2] and for all n >= 1/epsilon, our algorithm runs in O(epsilon(-3) n log(3.5) n) time with probability 1 - n(-O(1)). The volume of the intersection guaranteed by the output rigid motion is a (1 - epsilon)-approximation of the optimum, provided that the optimum is at least lambda . max{vertical bar P-1 vertical bar . vertical bar P-2 vertical bar} for some given constant lambda is an element of (0, 1]. (C) 2013 Elsevier B.V. All rights reserved.X1155Ysciescopu

Approximating Convex Shapes With Respect to Symmetric Difference Under Homotheties

The symmetric difference is a robust operator for measuring the error of approximating one shape by another. Given two convex shapes P and C, we study the problem of minimizing the volume of their symmetric difference under all possible scalings and translations of C. We prove that the problem can be solved by convex programming. We also present a combinatorial algorithm for convex polygons in the plane that runs in O((m+n) log^3(m+n)) expected time, where n and m denote the number of vertices of P and C, respectively

Finding largest common point sets

Let P and Q be two discrete point sets in ??>0d of sizes m and n, respectively, and let > 0 be a given input threshold. The largest common point set problem (LCP) seeks the largest subsets A ???P and B???Q such that |A| = |B| and there exists a transformation ??that makes the bottleneck distance between ??(A) and B at most??. We present two algorithms that solve a relaxed version of this problem under translations in Rd and under rigid motions in the plane, and that takes an additional input parameter??? > 0. Let ???be the largest subset size achievable for the given . Our algorithm finds subsets A ???P and B ??? Q of size |A| = |B|??? ???and a transformation ??such that the bottleneck distance between ?????(A) and B is at most (1 + n). For rigid motions in the plane, the running time is O(n2m2/2(n + m)log n). For translations inRd, the running time is O(nm\n(n + m)1.5log n), where ??= 1 for d = 2 and ??= 2d-1 for d ??? 3

Overlap of Convex Polytopes under Rigid Motion ∗

We present an algorithm to compute an approximate overlap of two convex polytopes P1 and P2 in R 3 under rigid motion. Given any ε ∈ (0, 1/2], our algorithm runs in O(ε −3 n log 3.5 n) time with probability 1 − n −O(1) and returns a (1 − ε)-approximate maximum overlap, provided that the maximum overlap is at least λ · max{|P1|, |P2|} for some given constant λ ∈ (0, 1]