Computing the Discrete Fréchet Distance in Subquadratic Time ∗ Pankaj

The Fréchet distance is a similarity measure between two curves A and B that takes into account the location and ordering of the points along the two curves: Informally, it is the minimum length of a leash required to connect a dog, walking along A, and its owner, walking along B, as they walk without backtracking along their respective curves from one endpoint to the other. The discrete Fréchet distance replaces the dog and its owner by a pair of frogs that can only reside on n and m specific stones on the curves A and B, respectively. These frogs hop from one stone to the next without backtracking, and the discrete Fréchet distance is the minimum length of a “leash” that connects the frogs and allows them to execute such a sequence of hops. It can be computed in quadratic time by a straightforward dynamic programming algorithm. We present the first subquadratic algorithm for computing the discrete Fréchet distance between two sequences of points in the plane. Assuming m ≤ n, the algorithm runs in O( log log n log n ) time, in the standard RAMmodel, using O(n) storage. Our approach uses the geometry of the problem in a subtle way to encode legal positions of the frogs as states of a finite automaton. ∗Work by Pankaj Agarwal is supported by NSF under grants CCF-09-40671, CCF-10-12254, and CCF-11-61359, by ARO grants W911NF-07-1-0376 and W911NF-08-1-0452, and by an ERDC contract W9132V-11-C-0003. Work by Haim Kaplan is supported by Grant 2006/204 from the U.S.–Israel Binational Science Foundation, and by Grant 822/10 from the Israel Science Fund. Work by Micha Sharir is supported by NSF Grant CCF-08-30272, by Grant 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work by Haim Kaplan and Micha Sharir is also supported by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Work by Rinat Ben Avraham is supported by the Israel Science Fund Grants 338/09 and 822/10. †Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129, USA; [email protected] ‡School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel; [email protected] §School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel; [email protected] ¶School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel; and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA; [email protected]