### Abstract

We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M, induced by the L2 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 and g is defined over another such triangulation, the obvious naive algorithm requires Θ (n^{2}) arithmetic operations to compute this distance.We show that it is possible to compute it in O(nlog^{4} nlog log n) arithmetic operations by reducing the problem to multipoint evaluation of a certain type of polynomials. We also present several generalizations and an application to terrain matching.

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

Article number | 3 |

Journal | ACM Transactions on Algorithms |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2016 |

### Fingerprint

### Keywords

- Multipoint evaluation
- Piecewise-linear function
- Polyhedral terrain

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

*ACM Transactions on Algorithms*,

*12*(1), [3]. https://doi.org/10.1145/2847257

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

Research output: Contribution to journal › Article

*ACM Transactions on Algorithms*, vol. 12, no. 1, 3. https://doi.org/10.1145/2847257

}

TY - JOUR

T1 - Computing the distance between piecewise-linear bivariate functions

AU - Moroz, Guillaume

AU - Aronov, Boris

PY - 2016/2/1

Y1 - 2016/2/1

N2 - We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M, induced by the L2 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 and g is defined over another such triangulation, the obvious naive algorithm requires Θ (n2) arithmetic operations to compute this distance.We show that it is possible to compute it in O(nlog4 nlog log n) arithmetic operations by reducing the problem to multipoint evaluation of a certain type of polynomials. We also present several generalizations and 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, induced by the L2 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 and g is defined over another such triangulation, the obvious naive algorithm requires Θ (n2) arithmetic operations to compute this distance.We show that it is possible to compute it in O(nlog4 nlog log n) arithmetic operations by reducing the problem to multipoint evaluation of a certain type of polynomials. We also present several generalizations and an application to terrain matching.

KW - Multipoint evaluation

KW - Piecewise-linear function

KW - Polyhedral terrain

UR - http://www.scopus.com/inward/record.url?scp=84964424955&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84964424955&partnerID=8YFLogxK

U2 - 10.1145/2847257

DO - 10.1145/2847257

M3 - Article

VL - 12

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

SN - 1549-6325

IS - 1

M1 - 3

ER -