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On the Auto-Correlation of Sets of Shapes (2008)
BibTeX
@MISC{Brise08onthe,
author = {Yves Brise and Bernd Gärtner and Trung Nguyen and Jean-daniel Boissonnat},
title = {On the Auto-Correlation of Sets of Shapes},
year = {2008}
}
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Abstract
Given a set of planar shapes (intervals, squares or disks) of the same size whose centers are fixed, we give the first polynomial-time algorithms to minimize (approximately) the common radius of the shapes under the requirement of large overlap with certain translated copies. The so-called auto-correlation function of a set of shapes at a vector x is the area of overlap between the set with its translated copy by vector x. The minimization is done with respect to an objective O and a threshold value µ, i.e. the auto-correlation function must not fall below µ within O. This type of problem is motivated by an application in optical interferometry, and the previous algorithms were of a heuristic nature, without any runtime or quality guarantees. Apart from fixing the centers, we had to restrict the original problem even further, in order to obtain the polynomial-time bounds. We consider the uniform case where all shapes are of the same size, and therefore our algorithms are not directly relevant for the motivating application. Rather, our research highlights the fact that even the simplified variants of the problem are difficult. The precise complexity status of these variants remains to be determined. Let n be the number of shapes involved. In particular, our results include an O(n2 logn) algorithm for minimizing the radius of uniform intervals in R2, an O(n4 logn) algorithm for axis-aligned uniform squares in R2, and a (10 9 + ǫ)-approximation algorithm for the case of uniform disk in R2 that runs in O(n6k2 lognk), where k is some sufficiently large integer depending on ǫ. The latter algorithm can be extended to a PTAS for uniform circles in R 2. Furthermore, in the appendix, we give some intuition why the problem of minimizing non-uniform sizes is difficult even for intervals in R1.
Keyphrases
optical interferometry simplified variant planar shape n2 logn previous algorithm n4 logn first polynomial-time algorithm certain translated copy quality guarantee non-uniform size motivating application axis-aligned uniform square polynomial-time bound auto-correlation function original problem large overlap uniform case large integer latter algorithm heuristic nature approximation algorithm n6k2 lognk uniform interval threshold value precise complexity status so-called auto-correlation function uniform circle uniform disk common radius
