### Abstract

All known algorithms for the Fr\'echet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; then they use this oracle to find the optimum among a finite set of critical values. We present a novel approach that avoids the detour through the decision version. We demonstrate its strength by presenting a quadratic time algorithm for the Fr\'echet distance between polygonal curves in R^d under polyhedral distance functions, including L_1 and L_infty. We also get a (1+epsilon)-approximation of the Fr\'echet distance under the Euclidean metric. For the exact Euclidean case, our framework currently gives an algorithm with running time O(n^2 log^2 n). However, we conjecture that it may eventually lead to a faster exact algorithm.

Original language | English |
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Number of pages | 13 |

Publication status | Published - 2013 |

### Publication series

Name | arXiv.org |
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Volume | 1306.5527 [cs.CG] |

## Cite this

Buchin, K., Buchin, M., Leusden, van, R., Meulemans, W., & Mulzer, W. (2013).

*Computing the Fréchet distance with a retractable leash*. (arXiv.org; Vol. 1306.5527 [cs.CG]).