Computing similarity between piecewise-linear functions

Authors:
Pankaj K. Agarwal;Boris Aronov;Marc van Kreveld;Maarten Löffler;Rodrigo I. Silveira
Affiliations:
Duke University, Durham, NC, USA;Polytechnic Institute of NYU, New York, NY, USA;Utrecht University, Utrecht, Netherlands;University of California, Irvine, Irvine, CA, USA;Universitat Politècnica de Catalunya, Barcelona, Spain
Venue:
Proceedings of the twenty-sixth annual symposium on Computational geometry
Year:
2010

Citing 6
Cited 1

Implicitly representing arrangements of lines or segments

Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Cutting hyperplanes for divide-and-conquer

Discrete & Computational Geometry
Las Vegas algorithms for linear and integer programming when the dimension is small

Journal of the ACM (JACM)
Overlaying simply connected planar subdivisions in linear time

Proceedings of the eleventh annual symposium on Computational geometry
Local and Global Comparison of Continuous Functions

VIS '04 Proceedings of the conference on Visualization '04
I/O-efficient map overlay and point location in low-density subdivisions

ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation

Computing the distance between piecewise-linear bivariate functions

Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms

Quantified Score

Hi-index	0.00

Visualization

Abstract

We study the problem of computing the similarity between two piecewise-linear bivariate functions defined over a common domain, where the surfaces they define in 3D - polyhedral terrains - can be transformed vertically by a linear transformation of the third coordinate (scaling and translation). We present a randomized algorithm that minimizes the maximum vertical distance between the graphs of the two functions, over all linear transformations of one of the terrains, in O(n4/3 polylog n) expected time, where n is the total number of vertices in the graphs of the two functions. We also study the computation of similarity between two univariate or bivariate functions by minimizing the area or volume between their graphs. For univariate functions we give a (1+ε)-approximation algorithm for minimizing the area that runs in O(n/√ε) time, for any fixed ε 0. The (1 + ε)- approximation algorithm for the bivariate version, where volume is minimized, runs in O(n/ε2) time, for any fixed ε 0, provided the two functions are defined over the same triangulation of their domain.