Filtser, Arnold ;
Filtser, Omrit ;
Katz, Matthew J.
Approximate Nearest Neighbor for Curves  Simple, Efficient, and Deterministic
Abstract
In the (1+ε,r)approximate nearneighbor problem for curves (ANNC) under some similarity measure δ, the goal is to construct a data structure for a given set 𝒞 of curves that supports approximate nearneighbor queries: Given a query curve Q, if there exists a curve C ∈ 𝒞 such that δ(Q,C)≤ r, then return a curve C' ∈ 𝒞 with δ(Q,C') ≤ (1+ε)r. There exists an efficient reduction from the (1+ε)approximate nearestneighbor problem to ANNC, where in the former problem the answer to a query is a curve C ∈ 𝒞 with δ(Q,C) ≤ (1+ε)⋅δ(Q,C^*), where C^* is the curve of 𝒞 most similar to Q.
Given a set 𝒞 of n curves, each consisting of m points in d dimensions, we construct a data structure for ANNC that uses n⋅ O(1/ε)^{md} storage space and has O(md) query time (for a query curve of length m), where the similarity measure between two curves is their discrete Fréchet or dynamic time warping distance. Our method is simple to implement, deterministic, and results in an exponential improvement in both query time and storage space compared to all previous bounds.
Further, we also consider the asymmetric version of ANNC, where the length of the query curves is k ≪ m, and obtain essentially the same storage and query bounds as above, except that m is replaced by k. Finally, we apply our method to a version of approximate range counting for curves and achieve similar bounds.
BibTeX  Entry
@InProceedings{filtser_et_al:LIPIcs:2020:12455,
author = {Arnold Filtser and Omrit Filtser and Matthew J. Katz},
title = {{Approximate Nearest Neighbor for Curves  Simple, Efficient, and Deterministic}},
booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
pages = {48:148:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771382},
ISSN = {18688969},
year = {2020},
volume = {168},
editor = {Artur Czumaj and Anuj Dawar and Emanuela Merelli},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12455},
URN = {urn:nbn:de:0030drops124555},
doi = {10.4230/LIPIcs.ICALP.2020.48},
annote = {Keywords: polygonal curves, Fr{\'e}chet distance, dynamic time warping, approximation algorithms, (asymmetric) approximate nearest neighbor, range counting}
}
29.06.2020
Keywords: 

polygonal curves, Fréchet distance, dynamic time warping, approximation algorithms, (asymmetric) approximate nearest neighbor, range counting 
Seminar: 

47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Issue date: 

2020 
Date of publication: 

29.06.2020 