Pattern Matching in Doubling Spaces

Abstract

We consider the problem of matching a metric space (X,dX) of size k with a subspace of a metric space (Y,dY) of size n⩾k , assuming that these two spaces have constant doubling dimension δ . More precisely, given an input parameter ρ⩾1 , the ρ -distortion problem is to find a one-to-one mapping from X to Y that distorts distances by a factor at most ρ . We first show by a reduction from k-clique that, in doubling dimension log23 , this problem is NP-hard and W[1]-hard. Then we provide a near-linear time approximation algorithm for fixed k: Given an approximation ratio 0<ε⩽1 , and a positive instance of the ρ -distortion problem, our algorithm returns a solution to the (1+ε)ρ -distortion problem in time (ρ/ε)O(1)nlogn . We also show how to extend these results to the minimum distortion problem, which is an optimization version of the ρ -distortion problem where we allow scaling. For doubling spaces, we prove the same hardness results, and for fixed k, we give a (1+ε) -approximation algorithm running in time (dist(X,Y)/ε)O(1)n2logn , where dist(X,Y) denotes the minimum distortion between X and Y.