TY - JOUR

T1 - Approximate Nearest Neighbor for Curves

T2 - Simple, Efficient, and Deterministic

AU - Filtser, Arnold

AU - Filtser, Omrit

AU - Katz, Matthew J.

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2023

Y1 - 2023

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 C′∈ C with δ(Q, C′) ≤ (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 C′∈ C with δ(Q, C′) ≤ (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 - Dynamic time warping

KW - Fréchet distance

KW - Nearest neighbor search

KW - Polygonal curves

UR - http://www.scopus.com/inward/record.url?scp=85143771132&partnerID=8YFLogxK

U2 - 10.1007/s00453-022-01080-1

DO - 10.1007/s00453-022-01080-1

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AN - SCOPUS:85143771132

SN - 0178-4617

VL - 85

SP - 1490

EP - 1519

JO - Algorithmica

JF - Algorithmica

IS - 5

ER -