# 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 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 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<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)n log n. We also show how to extend these results to the minimum distortion problem in 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)n^2log n, where dist(X,Y) denotes the minimum distortion between X and Y.

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