# How to Walk Your Dog in the Mountains with No Magic Leash

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## Abstract

We describe a \(O(\log n )\)-approximation algorithm for computing the homotopic Fréchet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms were known only for curves on the Euclidean plane with polygonal obstacles. A key technical ingredient in our analysis is a \(O(\log n)\)-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic Fréchet distance, or the minimum height of a homotopy, is in NP.

## Keywords

Fréchet distance Approximation algorithms Homotopy height## Mathematics Subject Classification

68W25## Notes

### Acknowledgments

The authors thank Jeff Erickson and Gary Miller for their comments and suggestions. The authors also thank the anonymous referees for their detailed and insightful reviews. S. Har-Peled was partially supported by NSF AF awards CCF-0915984, CCF-1421231, and CCF-1217462. M. Salavatipour was supported by NSERC and Alberta Innovates and also the part of this work was done while visiting Toyota Technological Institute at Chicago. A. Sidiropoulos was supported in part by the NSF grants CCF 1423230 and CAREER 1453472.

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