15 research outputs found
Computing the Maximum Overlap of Two Convex Polygons Under Translations.
International audienceLet P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices. We prove that the maximum number of combinatorially distinct placements of Q with respect to P under translations is O(n^2+m^2+min(nm^2+n^2m)), and we give an example showing that this bound is tight in the worst case. Second, we present an O((n+m)log(n+m)) algorithm for determining a translation of Q that maximizes the area of overlap of P and Q. We also prove that the position which translates the centroid of Q on the centroid of P always realizes an overlap of 9/25 of the maximum overlap and that this overlap may be as small as 4/9 of the maximum
Cache-Oblivious Selection in Sorted X+Y Matrices
Let X[0..n-1] and Y[0..m-1] be two sorted arrays, and define the mxn matrix A
by A[j][i]=X[i]+Y[j]. Frederickson and Johnson gave an efficient algorithm for
selecting the k-th smallest element from A. We show how to make this algorithm
IO-efficient. Our cache-oblivious algorithm performs O((m+n)/B) IOs, where B is
the block size of memory transfers
Approximating the Maximum Overlap of Polygons under Translation
Let and be two simple polygons in the plane of total complexity ,
each of which can be decomposed into at most convex parts. We present an
-approximation algorithm, for finding the translation of ,
which maximizes its area of overlap with . Our algorithm runs in
time, where is a constant that depends only on and .
This suggest that for polygons that are "close" to being convex, the problem
can be solved (approximately), in near linear time
Probabilistic Matching of Planar Regions
We analyze a probabilistic algorithm for matching shapes modeled by planar
regions under translations and rigid motions (rotation and translation). Given
shapes and , the algorithm computes a transformation such that with
high probability the area of overlap of and is close to maximal. In
the case of polygons, we give a time bound that does not depend significantly
on the number of vertices
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
Computing the maximum overlap of two convex polygons under translations
Theme 2 - Genie logiciel et calcul symbolique. Projet PrismeSIGLEAvailable from INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1996 n.2832 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc
Computing the Maximum Overlap of Two Convex Polygons Under Translations
Let P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices. We prove that the maximum number of combinatorially distinct place- ments of Q with respect to P under translations is O(n 2 + m + rain(rim + nm)), and we give an example showing that this bound is tight in the worst case. Second, we present an O((n + m) log(n + m)) algorithm for determining a translation of Q that maximizes the area of overlap of P and Q
Computing the maximum overlap of two convex polygons under translations
Let P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices, We prove that the maximum number of combinatorially distinct placements of Q with respect to P under translations is O (n(2) + m(2) + min(nm(2) + n(2)m)), and we give an example showing that this bound is tight in the worst case. Second, we present an O((n + m) log(n + m)) algorithm for determining a translation of Q that maximizes the area of overlap of P and Q, We also prove that the placement of Q that makes the centroids of Q and P coincide realizes an overlap of at least 9/25 of the maximum possible overlap. Pls an upper bound, we show an example where the overlap in this placement is 4/9 of the maximum possible overlap,=