TY - GEN

T1 - Approximate nearest neighbor for curves - simple, efficient, and deterministic

AU - Filtser, Arnold

AU - Filtser, Omrit

AU - Katz, Matthew J.

N1 - Publisher Copyright:
© Arnold Filtser, Omrit Filtser, and Matthew J. Katz; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).

PY - 2020/6/1

Y1 - 2020/6/1

N2 - In the (1 + ε, r)-approximate near-neighbor 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 near-neighbor queries: Given a query curve Q, if there exists a curve C ∈ C such that δ(Q, C) ≤ r, then return a curve C0 ∈ C with δ(Q, C0) ≤ (1 + ε)r. There exists an efficient reduction from the (1 + ε)-approximate nearest-neighbor 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(1ε )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.

AB - In the (1 + ε, r)-approximate near-neighbor 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 near-neighbor queries: Given a query curve Q, if there exists a curve C ∈ C such that δ(Q, C) ≤ r, then return a curve C0 ∈ C with δ(Q, C0) ≤ (1 + ε)r. There exists an efficient reduction from the (1 + ε)-approximate nearest-neighbor 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(1ε )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.

KW - (asymmetric) approximate nearest neighbor

KW - Approximation algorithms

KW - Dynamic time warping

KW - Fréchet distance

KW - Polygonal curves

KW - Range counting

UR - http://www.scopus.com/inward/record.url?scp=85089352175&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ICALP.2020.48

DO - 10.4230/LIPIcs.ICALP.2020.48

M3 - Conference contribution

AN - SCOPUS:85089352175

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

A2 - Czumaj, Artur

A2 - Dawar, Anuj

A2 - Merelli, Emanuela

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

Y2 - 8 July 2020 through 11 July 2020

ER -