# 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.

## References

- 1.Agarwal, P.K., Aronov, B., O’Rourke, J., Schevon, C.A.: Star unfolding of a polytope with applications. SIAM J. Comput.
**26**, 1679–1713 (1997)MathSciNetGoogle Scholar - 2.Alt, H., Buchin, M.: Semi-computability of the Fréchet distance between surfaces. In: Proceedings of the 21st European Workshop on Computational Geometry, pp. 45–48 (2005)Google Scholar
- 3.Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geom. Appl.
**5**, 75–91 (1995)zbMATHMathSciNetCrossRefGoogle Scholar - 4.Bennis, C., Vézien, J.-M., Iglésias, G., Gagalowicz, A.: Piecewise surface flattening for non-distorted texture mapping. In: Sederberg, T.W. (ed). Proceedings of SIGGRAPH ’91, vol. 25, pp. 237–246. ACM, New York (1991)Google Scholar
- 5.Brakatsoulas, S., Pfoser, D., Salas, R., Wenk, C.: On map-matching vehicle tracking data. In: Proceedings of 31st VLDB Conference, VLDB Endowment, pp. 853–864. Norwegian University of Science & Technology, Trondheim (2005)Google Scholar
- 6.Brightwell, G.A., Winkler, P.: Submodular percolation. SIAM J. Discrete Math.
**23**(3), 1149–1178 (2009)zbMATHMathSciNetCrossRefGoogle Scholar - 7.Buchin, K., Buchin, M., Gudmundsson, J.: Detecting single file movement. In: Proceedings of 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM GIS), pp. 288–297. ACM, New York (2008)Google Scholar
- 8.Buchin, K., Buchin, M., Gudmundsson, J., Maarten, L., Luo, J.: Detecting commuting patterns by clustering subtrajectories. In: Proceedings of the 19th International Symposium on Algorithms and Computation (ISAAC), pp. 644–655. Springer, Heidelberg (2008)Google Scholar
- 9.Buchin, M., Driemel, A., Speckmann, B.: Computing the Fréchet distance with shortcuts is NP-hard. In: Proceedings of the 30th Annual Symposium on Computational Geometry SOCG’14 (Kyoto, Japan), pp. 367–376. ACM, New York, NY (2014)Google Scholar
- 10.Chambers, E.W., Letscher, D.: On the height of a homotopy. In: Proceedings of the 21st Canadian Conference on Computational Geometry (CCCG), pp. 14. University of British Columbia, Vancouver (2009)Google Scholar
- 11.Chambers, E.W., Letscher, D.: Erratum for on the height of a homotopy. http://mathcs.slu.edu/~chambers/papers/hherratum.pdf (2010)
- 12.Chambers, G.R., Rotman, R.: Contracting loops on a Riemannian 2-surface (2013). arXiv:1311.2995
- 13.Chambers, E.W., Wang, Y.: Measuring similarity between curves on 2-manifolds via homotopy area. In: Proceedings of the 29th Annual Symposium on Computational Geometry (SoCG), pp. 425–434. ACM, New York (2013)Google Scholar
- 14.Chambers, E.W., Colin de Verdière, E., Erickson, J., Lazard, S., Lazarus, F., Thite, S.: Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time. Comput. Geom. Theory Appl.
**43**(3), 295–311 (2010)zbMATHCrossRefGoogle Scholar - 15.Chambers, E.W., Letscher, D., Ju, T., Liu, L.: Isotopic Fréchet distance. In: Proceedings of the 23rd Canadian Conference on Computational Geometry (CCCG), pp. 229–234. CCCG, Toronto (2011)Google Scholar
- 16.Cook, A.F., Wenk, C.: Geodesic Fréchet distance inside a simple polygon. ACM Trans. Algorithms
**7**, 9:1–9:19 (2010)MathSciNetGoogle Scholar - 17.Cook, A.F., Wenk, C.: Shortest path problems on a polyhedral surface. Algorithmica
**69**(1), 58–77 (2014)zbMATHMathSciNetCrossRefGoogle Scholar - 18.Cook, A.F., Driemel, A., Har-Peled, S., Sherette, J., Wenk, C.: Computing the Fréchet distance between folded polygons. In: Proceedings of the 12th Workshop Algorithms Data Structure (WADS), pp. 267–278. Springer, Berlin (2011)Google Scholar
- 19.Driemel, A., Har-Peled, S., Wenk, C.: Approximating the Fréchet distance for realistic curves in near linear time. Discrete Comput. Geom.
**48**, 94–127 (2012)zbMATHMathSciNetCrossRefGoogle Scholar - 20.Efrat, A., Guibas, L.J., Har-Peled, S., Mitchell, J.S.B., Murali, T.M.: New similarity measures between polylines with applications to morphing and polygon sweeping. Discrete Comput. Geom.
**28**, 535–569 (2002)zbMATHMathSciNetCrossRefGoogle Scholar - 21.Eiter, T., Mannila, H.: Computing discrete Fréchet distance. Technical Report CD-TR 94/64, Christian Doppler Laboratory for Expert Systems, TU, Vienna (1994)Google Scholar
- 22.Floater, M.S.: Parameterization and smooth approximation of surface triangulations. Comput. Aided Geom. Des.
**14**(4), 231–250 (1997)zbMATHCrossRefGoogle Scholar - 23.Frechét, M.: Sur la distance de deux surfaces. Ann. Soc. Polonaise Math.
**3**, 4–19 (1924)Google Scholar - 24.Godau, M.: On the complexity of measuring the similarity between geometric objects in higher dimensions. PhD Thesis, Free University of Berlin (1999)Google Scholar
- 25.Har-Peled, S., Raichel, B.: The Fréchet distance revisited and extended. In: Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG), pp. 448–457. ACM Press, New York (2011)Google Scholar
- 26.Har-Peled, S., Nayyeri, A., Salavatipour, M., Sidiropoulos, A.: How to walk your dog in the mountains with no magic leash. In: Proceedings of the 28th Annual ACM Symposium on Computational Geometry (SoCG), pp. 121–130. ACM, New York (2012)Google Scholar
- 27.Henzinger, M.R., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. J. Comput. Syst. Sci.
**55**, 3–23 (1997)zbMATHMathSciNetCrossRefGoogle Scholar - 28.Keogh, E.J., Pazzani, M.J.: Scaling up dynamic time warping to massive dataset. In: Proceedings of the Third European Conference on Principles of Data Mining and Knowledge Discovery, pp. 1–11. Springer, Prague (1999)Google Scholar
- 29.Kim, M.S., Kim, S.W., Shin, M.: Optimization of subsequence matching under time warping in time-series databases. In: Proceedings of the ACM Symposium on Applied Computing, pp. 581–586. ACM, New York (2005)Google Scholar
- 30.Lewis, P.M., II, Stearns, R.E., Hartmanis, J.: Memory bounds for recognition of context-free and context-sensitive languages. In: Proceedings of the 6th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 191–202. ACM, New York (1965)Google Scholar
- 31.Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math.
**36**, 177–189 (1979)zbMATHMathSciNetCrossRefGoogle Scholar - 32.Mascret, A., Devogele, T., Le Berre, I., Hénaff, A.: Coastline matching process based on the discrete Fréchet distance. In: Proceedings of the 12th International Symposium on Spatial Data Handling, pp. 383–400. Springer, Berlin (2006)Google Scholar
- 33.Mitchell, J.S.B., Mount, D.M., Papadimitriou, C.H.: The discrete geodesic problem. SIAM J. Comput.
**16**, 647–668 (1987)zbMATHMathSciNetCrossRefGoogle Scholar - 34.Papasoglu, P.: Contracting thin disks (2013). arXiv:1309.2967
- 35.Piponi, D., Borshukov, G.: Seamless texture mapping of subdivision surfaces by model pelting and texture blending. In: Proceedings of the SIGGRAPH 2000, pp. 471–478. ACM, New York (2000)Google Scholar
- 36.Serrà, J., Gómez, E., Herrera, P., Serra, X.: Chroma binary similarity and local alignment applied to cover song identification. IEEE Trans. Audio Speech Lang. Process.
**16**(6), 1138–1151 (2008)CrossRefGoogle Scholar - 37.Sheffer, A., de Sturler, E.: Surface parameterization for meshing by triangulation flattening. In: Proceedings of the 9th International Meshing Roundtable, pp. 161–172. Springer, Berlin (2000)Google Scholar
- 38.Sherette, J., Wenk, C.: Simple curve embedding. CoRR (2013). arXiv:1303.0821
- 39.Wenk, C., Salas, R., Pfoser, D.: Addressing the need for map-matching speed: Localizing global curve-matching algorithms. In: Proceedings of the 18th Conference on Scientific and Statistical Database Management, pp. 879–888. Springer, Berlin (2006)Google Scholar