Fast algorithms for approximate Fréchet matching queries in geometric trees

Let T be a tree in Rd and let ∆ > 0 be a real number. The aim is to preprocess T into a data structure, such that for any polygonal query path Q, we can decide if T contains a path P whose Fréchet distance δF (Q,P ) to Q is at most ∆. For any real number ε > 0, we present an efficient data structure that solves an approximate version of this problem for the case when T is c-packed and each of the edges of T and Q has length Ω(∆): If the query algorithm returns NO, then there is no such path P . If the query algorithm returns YES, then T contains a path P for which δF (Q,P ) ≤ (1+ε)∆ if Q is a line segment, and δF (Q,P ) ≤ 3(1 + ε)∆ otherwise.