Pattern Matching in Doubling Spaces
Corentin Allair, Antoine Vigneron

TL;DR
This paper studies the computational complexity of pattern matching in metric spaces with bounded doubling dimension, providing hardness results and near-linear time approximation algorithms for fixed pattern sizes.
Contribution
It establishes NP-hardness and W[1]-hardness for the problem, and introduces near-linear time approximation algorithms for fixed pattern sizes in doubling spaces.
Findings
NP-hardness and W[1]-hardness results for the problem
Near-linear time approximation algorithms for fixed pattern size
Extension of results to minimum distortion problem
Abstract
We consider the problem of matching a metric space of size with a subspace of a metric space of size , assuming that these two spaces have constant doubling dimension . More precisely, given an input parameter , the -distortion problem is to find a one-to-one mapping from to that distorts distances by a factor at most . We first show by a reduction from -clique that, in doubling dimension , this problem is NP-hard and W[1]-hard. Then we provide a near-linear time approximation algorithm for fixed : Given an approximation ratio , and a positive instance of the -distortion problem, our algorithm returns a solution to the -distortion problem in time . We also show how to extend these results to the minimum distortion…
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