SWAT 2012: Algorithm Theory – SWAT 2012 pp 13-23

# Partial Matching between Surfaces Using Fréchet Distance

• Jessica Sherette
• Carola Wenk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)

## Abstract

Computing the Fréchet distance for surfaces is a surprisingly hard problem. We introduce a partial variant of the Fréchet distance problem, which for given surfaces P and Q asks to compute a surface R ⊆ Q with minimum Fréchet distance to P. Like the Fréchet distance, the partial Fréchet distance is NP-hard to compute between terrains and also between polygons with holes. We restrict P, Q, and R to be coplanar simple polygons. For this restricted class of surfaces, we develop a polynomial time algorithm to compute the partial Fréchet distance and show that such an R ⊆ Q can be computed in polynomial time as well. This is the first algorithm to address a partial Fréchet distance problem for surfaces and extends Buchin et al.’s algorithm for computing the Fréchet distance between simple polygons.

## Keywords

Computational Geometry Shape Matching Fréchet Distance

## References

1. 1.
Alt, H., Buchin, M.: Can we compute the similarity between surfaces? Discrete and Computational Geometry 43, 78–99 (2010)
2. 2.
Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry and Applications 5, 75–91 (1995)
3. 3.
Buchin, K., Buchin, M., Schulz, A.: Fréchet Distance of Surfaces: Some Simple Hard Cases. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part II. LNCS, vol. 6347, pp. 63–74. Springer, Heidelberg (2010)
4. 4.
Buchin, K., Buchin, M., Wang, Y.: Exact algorithm for partial curve matching via the Fréchet distance. In: Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pp. 645–654 (2009)Google Scholar
5. 5.
Buchin, K., Buchin, M., Wenk, C.: Computing the Fréchet distance between simple polygons in polynomial time. In: 22nd Symposium on Computational Geometry (SoCG), pp. 80–87 (2006)Google Scholar
6. 6.
Cook IV, A.F., Driemel, A., Har-Peled, S., Sherette, J., Wenk, C.: Computing the Fréchet Distance between Folded Polygons. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 267–278. Springer, Heidelberg (2011)
7. 7.
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 and Computational Geometry 28(4), 535–569 (2002)
8. 8.
Godau, M.: On the Difficulty of Embedding Planar Graphs with Inaccuracies. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 254–261. Springer, Heidelberg (1995)
9. 9.
Godau, M.: On the complexity of measuring the similarity between geometric objects in higher dimensions. PhD thesis, Freie Universität Berlin, Germany (1998)Google Scholar
10. 10.
Guibas, L.J., Hershberger, J.: Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences 39(2), 126–152 (1989)
11. 11.
Hershberger, J.: A new data structure for shortest path queries in a simple polygon. Inf. Process. Lett. 38(5), 231–235 (1991)