Computing the discrete Fréchet distance in subquadratic time

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(mn log log n/log n) time, in the standard RAM model, 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.