Efficient nearest-neighbor query and clustering of planar curves

Boris Aronov; Omrit Filtser; Michael Horton; Matthew J. Katz; Khadijeh Sheikhan

doi:10.1007/978-3-030-24766-9_3

Efficient nearest-neighbor query and clustering of planar curves

Boris Aronov, Omrit Filtser, Michael Horton, Matthew J. Katz, Khadijeh Sheikhan

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We study two fundamental problems dealing with curves in the plane, namely, the nearest-neighbor problem and the center problem. Let C be a set of n polygonal curves, each of size m. In the nearest-neighbor problem, the goal is to construct a compact data structure over C, such that, given a query curve Q, one can efficiently find the curve in C closest to Q. In the center problem, the goal is to find a curve Q, such that the maximum distance between Q and the curves in C is minimized. We use the well-known discrete Fréchet distance function, both under L_∞ and under L₂, to measure the distance between two curves. For the nearest-neighbor problem, despite discouraging previous results, we identify two important cases for which it is possible to obtain practical bounds, even when m and n are large. In these cases, either Q is a line segment or C consists of line segments, and the bounds on the size of the data structure and query time are nearly linear in the size of the input and query curve, respectively. The returned answer is either exact under L_∞, or approximated to within a factor of 1 + ε under L₂. We also consider the variants in which the location of the input curves is only fixed up to translation, and obtain similar bounds, under L_∞. As for the center problem, we study the case where the center is a line segment, i.e., we seek the line segment that represents the given set as well as possible. We present near-linear time exact algorithms under L_∞, even when the location of the input curves is only fixed up to translation. Under L₂, we present a roughly O(n²m³) -time exact algorithm.

Original language	English
Title of host publication	Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings
Editors	Zachary Friggstad, Mohammad R. Salavatipour, Jörg-Rüdiger Sack
Publisher	Springer Verlag
Pages	28-42
Number of pages	15
ISBN (Print)	9783030247652
DOIs	https://doi.org/10.1007/978-3-030-24766-9_3
State	Published - 2019
Externally published	Yes
Event	16th International Symposium on Algorithms and Data Structures, WADS 2019 - Edmonton, Canada Duration: 5 Aug 2019 → 7 Aug 2019

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	11646 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	16th International Symposium on Algorithms and Data Structures, WADS 2019
Country/Territory	Canada
City	Edmonton
Period	5/08/19 → 7/08/19

Bibliographical note

Funding Information:
B. Aronov was supported by NSF grants CCF-12-18791 and CCF-15-40656, and by grant 2014/170 from the US-Israel Binational Science Foundation. O. Filtser was supported by the Israeli Ministry of Science, Technology & Space, and by grant 2014/170 from the US-Israel Binational Science Foundation. Most of the work on this project by M. Horton was performed while visiting the Department of Computer Science and Engineering at the Tandon School of Engineering, New York University in the spring/summer of 2018, partially supported by NSF grant CCF-12-18791. M. Katz was supported by grant 1884/16 from the Israel Science Foundation and by grant 2014/170 from the US-Israel Binational Science Foundation. Part of the work on this project by M. Katz was performed while visiting the Department of Computer Science and Engineering at the Tandon School of Engineering, New York University in the spring of 2018, partially supported by NSF grants CCF-12-18791 and CCF-15-40656. Work of K. Sheikhan on this paper was performed while at the Tandon School of Engineering, New York University, supported by NSF grant CCF-12-18791.

Funding Information:
Acknowledgements. B. Aronov was supported by NSF grants CCF-12-18791 and CCF-15-40656, and by grant 2014/170 from the US-Israel Binational Science Foundation. O. Filtser was supported by the Israeli Ministry of Science, Technology & Space, and by grant 2014/170 from the US-Israel Binational Science Foundation. Most of the work on this project by M. Horton was performed while visiting the Department of Computer Science and Engineering at the Tandon School of Engineering, New York University in the spring/summer of 2018, partially supported by NSF grant CCF-12-18791. M. Katz was supported by grant 1884/16 from the Israel Science Foundation and by grant 2014/170 from the US-Israel Binational Science Foundation. Part of the work on this project by M. Katz was performed while visiting the Department of Computer Science and Engineering at the Tandon School of Engineering, New York University in the spring of 2018, partially supported by NSF grants CCF-12-18791 and CCF-15-40656. Work of K. Sheikhan on this paper was performed while at the Tandon School of Engineering, New York University, supported by NSF grant CCF-12-18791.

Publisher Copyright:
© Springer Nature Switzerland AG 2019.

Keywords

(Approximation) algorithms
Clustering
Data structures
Fréchet distance
Nearest-neighbor queries
Polygonal curves

Access to Document

10.1007/978-3-030-24766-9_3

Cite this

Aronov, B., Filtser, O., Horton, M., Katz, M. J., & Sheikhan, K. (2019). Efficient nearest-neighbor query and clustering of planar curves. In Z. Friggstad, M. R. Salavatipour, & J-R. Sack (Eds.), Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings (pp. 28-42). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11646 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-24766-9_3

Aronov, Boris ; Filtser, Omrit ; Horton, Michael et al. / Efficient nearest-neighbor query and clustering of planar curves. Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings. editor / Zachary Friggstad ; Mohammad R. Salavatipour ; Jörg-Rüdiger Sack. Springer Verlag, 2019. pp. 28-42 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "We study two fundamental problems dealing with curves in the plane, namely, the nearest-neighbor problem and the center problem. Let C be a set of n polygonal curves, each of size m. In the nearest-neighbor problem, the goal is to construct a compact data structure over C, such that, given a query curve Q, one can efficiently find the curve in C closest to Q. In the center problem, the goal is to find a curve Q, such that the maximum distance between Q and the curves in C is minimized. We use the well-known discrete Fr{\'e}chet distance function, both under L∞ and under L2, to measure the distance between two curves. For the nearest-neighbor problem, despite discouraging previous results, we identify two important cases for which it is possible to obtain practical bounds, even when m and n are large. In these cases, either Q is a line segment or C consists of line segments, and the bounds on the size of the data structure and query time are nearly linear in the size of the input and query curve, respectively. The returned answer is either exact under L∞, or approximated to within a factor of 1 + ε under L2. We also consider the variants in which the location of the input curves is only fixed up to translation, and obtain similar bounds, under L∞. As for the center problem, we study the case where the center is a line segment, i.e., we seek the line segment that represents the given set as well as possible. We present near-linear time exact algorithms under L∞, even when the location of the input curves is only fixed up to translation. Under L2, we present a roughly O(n2m3) -time exact algorithm.",

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author = "Boris Aronov and Omrit Filtser and Michael Horton and Katz, {Matthew J.} and Khadijeh Sheikhan",

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Aronov, B, Filtser, O, Horton, M, Katz, MJ & Sheikhan, K 2019, Efficient nearest-neighbor query and clustering of planar curves. in Z Friggstad, MR Salavatipour & J-R Sack (eds), Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11646 LNCS, Springer Verlag, pp. 28-42, 16th International Symposium on Algorithms and Data Structures, WADS 2019, Edmonton, Canada, 5/08/19. https://doi.org/10.1007/978-3-030-24766-9_3

Efficient nearest-neighbor query and clustering of planar curves. / Aronov, Boris; Filtser, Omrit; Horton, Michael et al.
Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings. ed. / Zachary Friggstad; Mohammad R. Salavatipour; Jörg-Rüdiger Sack. Springer Verlag, 2019. p. 28-42 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11646 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Aronov B, Filtser O, Horton M, Katz MJ, Sheikhan K. Efficient nearest-neighbor query and clustering of planar curves. In Friggstad Z, Salavatipour MR, Sack J-R, editors, Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings. Springer Verlag. 2019. p. 28-42. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-030-24766-9_3

Efficient nearest-neighbor query and clustering of planar curves

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