### Abstract

We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L
_{2}-norm, that is ||f - g||
_{2} = √∫∫
_{M}(f - g)
^{2}. If f is defined by linear interpolation over a triangulation of M with n triangles, while g is defined over another such triangulation, the obvious naïve algorithm requires Θ(n
^{2}) arithmetic operations to compute this distance. We show that it is possible to compute it in O(n log
^{4} n) arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.

Original language | English (US) |
---|---|

Title of host publication | Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 |

Pages | 288-293 |

Number of pages | 6 |

State | Published - 2012 |

Event | 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan Duration: Jan 17 2012 → Jan 19 2012 |

### Other

Other | 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 |
---|---|

Country | Japan |

City | Kyoto |

Period | 1/17/12 → 1/19/12 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012*(pp. 288-293)

**Computing the distance between piecewise-linear bivariate functions.** / Moroz, Guillaume; Aronov, Boris.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012.*pp. 288-293, 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, 1/17/12.

}

TY - GEN

T1 - Computing the distance between piecewise-linear bivariate functions

AU - Moroz, Guillaume

AU - Aronov, Boris

PY - 2012

Y1 - 2012

N2 - We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L 2-norm, that is ||f - g|| 2 = √∫∫ M(f - g) 2. If f is defined by linear interpolation over a triangulation of M with n triangles, while g is defined over another such triangulation, the obvious naïve algorithm requires Θ(n 2) arithmetic operations to compute this distance. We show that it is possible to compute it in O(n log 4 n) arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.

AB - We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L 2-norm, that is ||f - g|| 2 = √∫∫ M(f - g) 2. If f is defined by linear interpolation over a triangulation of M with n triangles, while g is defined over another such triangulation, the obvious naïve algorithm requires Θ(n 2) arithmetic operations to compute this distance. We show that it is possible to compute it in O(n log 4 n) arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.

UR - http://www.scopus.com/inward/record.url?scp=84860166197&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860166197&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84860166197

SN - 9781611972108

SP - 288

EP - 293

BT - Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012

ER -