Pattern Matching In Doubling Spaces

We consider the problem of matching a metric space (X, d(X)) of size k with a subspace of a metric space (Y, d(Y)) of size n >= k, assuming that these two spaces have constant doubling dimension delta. More precisely, given an input parameter rho >= 1, the rho-distortion problem is to find a one-to-one mapping from X to Y that distorts distances by a factor at most rho. We first show by a reduction from k-clique that, in doubling dimension log(2) 3, 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 < epsilon <= 1, and a positive instance of the rho-distortion problem, our algorithm returns a solution to the (1 + epsilon)rho-distortion problem in time (rho/epsilon)(O(1))n log n. We also show how to extend these results to the minimum distortion problem, which is an optimization version of the rho-distortion problem where we allow scaling. For doubling spaces, we prove the same hardness results, and for fixed k, we give a (1+ epsilon)-approximation algorithm running in time (dist(X, Y)/epsilon)(O(1))n(2) log n, where dist(X, Y) denotes the minimum distortion between X and Y