TY - GEN

T1 - Computing similarity between piecewise-linear functions

AU - Agarwal, Pankaj K.

AU - Aronov, Boris

AU - Van Kreveld, Marc

AU - Löffler, Maarten

AU - Silveira, Rodrigo I.

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Approximation algorithm

KW - Piecewise-linear function

KW - Polyhedral terrain

KW - Randomized algorithm

KW - Similarity

UR - http://www.scopus.com/inward/record.url?scp=77954897132&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954897132&partnerID=8YFLogxK

U2 - 10.1145/1810959.1811020

DO - 10.1145/1810959.1811020

M3 - Conference contribution

AN - SCOPUS:77954897132

SN - 9781450300162

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 375

EP - 383

BT - Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10

T2 - 26th Annual Symposium on Computational Geometry, SoCG 2010

Y2 - 13 June 2010 through 16 June 2010

ER -