# Pattern Matching in Doubling Spaces

@inproceedings{Allair2020PatternMI, title={Pattern Matching in Doubling Spaces}, author={Corentin Allair and Antoine Vigneron}, booktitle={Workshop on Algorithms and Data Structures}, year={2020} }

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 log2 3, this problem is NP-hard and W[1]hard. Then we provide a near…

## References

SHOWING 1-10 OF 27 REFERENCES

### Bypassing the embedding: algorithms for low dimensional metrics

- Computer Science, MathematicsSTOC '04
- 2004

This paper explores the option of bypassing the embedding of metrics with low doubling dimension and shows the following for low dimensional metrics: Quasi-polynomial time (1+ε)-approximation algorithm for various optimization problems such as TSP, k-median and facility location.

### Space-Time Tradeoffs for Proximity Searching in Doubling Spaces

- Computer ScienceESA
- 2008

The objective is to build a data structure so that given any query point q in the space, it is possible to efficiently determine a point of S whose distance from q is within a factor of (1 + i¾?) of the distance between q and its nearest neighbor in S.

### Reality Distortion: Exact and Approximate Algorithms for Embedding into the Line

- Computer Science2015 IEEE 56th Annual Symposium on Foundations of Computer Science
- 2015

Algorithms for the problem of minimum distortion embeddings of finite metric spaces into the real line (or a finite subset of the line) are described, which yields a quasipolynomial running time for constant δ, and polynomial D.

### Bounded geometries, fractals, and low-distortion embeddings

- Mathematics, Computer Science44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
- 2003

This work considers both general doubling metrics, as well as more restricted families such as those arising from trees, from graphs excluding a fixed minor, and from snowflaked metrics, which contains many families of metrics that occur in applied settings.

### Metric Spaces with Expensive Distances

- Computer Science, MathematicsInt. J. Comput. Geom. Appl.
- 2020

This model is motivated by metric spaces that appear in the context of topological data analysis, and it is proved that this strategy returns an approximate nearest neighbor after a logarithmic number of distance queries.

### Searching dynamic point sets in spaces with bounded doubling dimension

- Computer ScienceSTOC '06
- 2006

A new data structure is presented that facilitates approximate nearest neighbor searches on a dynamic set of points in a metric space that has a bounded doubling dimension and finds a (1+ε)-approximate nearest neighbor in time O(log n) + (1/ε)O(1).

### Low distortion maps between point sets

- Mathematics, Computer ScienceSTOC '04
- 2004

This work presents a polynomial time algorithm that finds an optimal bijection between two line metrics, provided the distortion is less than 3+2√2.

### Fast construction of nets in low dimensional metrics, and their applications

- Computer Science, MathematicsSCG
- 2005

We present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension. This data-structure is then applied to obtain improved algorithms…

### New Doubling Spanners: Better and Simpler

- Computer Science, MathematicsSIAM J. Comput.
- 2015

This work presents a simpler construction of spanners for doubling metrics with the above guarantees, and extends in a simple and natural way to provide k-fault tolerant spanners with maximum degree O(k2), hop-diameter O(logn) and lightness O( k2 logn).

### On Geometric Alignment in Low Doubling Dimension

- Computer ScienceAAAI
- 2019

This paper proposes an effective framework to compress the high dimensional geometric patterns and approximately preserve the alignment quality and adopts the widely used notion "doubling dimension" to measure the extents of compression and the resulting approximation.