Fast Fréchet Queries

Abstract

Inspired by video analysis of team sports, we study the following problem. Let P be a polygonal path in the plane with n vertices. We want to preprocess P into a data structure that can quickly count the number of inclusion-minimal subpaths of P whose Fréchet Distance to a given query segment Q is at most some threshold value ε. We present a data structure that solves an approximate version of this problem: it counts all subpaths whose Fréchet Distance is at most ε, but this count may also include subpaths whose Fréchet Distance is up to $(2+3\sqrt{2})\varepsilon$. For any parameter n ≤ s ≤ n ², our data structure can be tuned such that it uses O(s polylog n) storage and has $O((n/\sqrt{s}){\rm polylog} n)$ query time. For the special case where we wish to count all subpaths whose Fréchet Distance to Q is at most ε·length(Q), we present a structure with O(n polylog n) storage and O(polylog n) query time.