Balanced line separators of unit disk graphs

Paz Carmi; Man Kwun Chiu; Matthew J. Katz; Matias Korman; Yoshio Okamoto; André Van Renssen; Marcel Roeloffzen; Taichi Shiitada; Shakhar Smorodinsky

doi:https://doi.org/10.1007/978-3-319-62127-2_21

Balanced line separators of unit disk graphs

Paz Carmi, Man Kwun Chiu, Matthew J. Katz, Matias Korman, Yoshio Okamoto, André Van Renssen, Marcel Roeloffzen, Taichi Shiitada, Shakhar Smorodinsky

Ben-Gurion University of the Negev

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

Abstract

We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line ℓ such that ℓ intersects at most O(Formula presented) disks and each of the halfplanes determined by ℓ contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting O(Formula presented) disks exists, but each halfplane may contain up to 4n/5 disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m = 0. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(Formula presented m)).

Original language	American English
Title of host publication	Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings
Editors	Faith Ellen, Antonina Kolokolova, Jorg-Rudiger Sack
Publisher	Springer Verlag
Pages	241-252
Number of pages	12
ISBN (Print)	9783319621265
DOIs	https://doi.org/10.1007/978-3-319-62127-2_21
State	Published - 1 Jan 2017
Event	15th International Symposium on Algorithms and Data Structures, WADS 2017 - St. John’s, Canada Duration: 31 Jul 2017 → 2 Aug 2017

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	10389 LNCS

Conference

Conference	15th International Symposium on Algorithms and Data Structures, WADS 2017
Country/Territory	Canada
City	St. John’s
Period	31/07/17 → 2/08/17

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
General Computer Science

Access to Document

https://doi.org/10.1007/978-3-319-62127-2_21

Cite this

Carmi, P., Chiu, M. K., Katz, M. J., Korman, M., Okamoto, Y., Van Renssen, A., Roeloffzen, M., Shiitada, T., & Smorodinsky, S. (2017). Balanced line separators of unit disk graphs. In F. Ellen, A. Kolokolova, & J-R. Sack (Eds.), Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings (pp. 241-252). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10389 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-62127-2_21

Carmi, Paz ; Chiu, Man Kwun ; Katz, Matthew J. et al. / Balanced line separators of unit disk graphs. Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings. editor / Faith Ellen ; Antonina Kolokolova ; Jorg-Rudiger Sack. Springer Verlag, 2017. pp. 241-252 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{7eeb3ed01dfd4560a49c1ff2a56a1f14,

title = "Balanced line separators of unit disk graphs",

abstract = "We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line ℓ such that ℓ intersects at most O(Formula presented) disks and each of the halfplanes determined by ℓ contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting O(Formula presented) disks exists, but each halfplane may contain up to 4n/5 disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m = 0. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(Formula presented m)).",

author = "Paz Carmi and Chiu, {Man Kwun} and Katz, {Matthew J.} and Matias Korman and Yoshio Okamoto and {Van Renssen}, Andr{\'e} and Marcel Roeloffzen and Taichi Shiitada and Shakhar Smorodinsky",

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Carmi, P, Chiu, MK, Katz, MJ, Korman, M, Okamoto, Y, Van Renssen, A, Roeloffzen, M, Shiitada, T & Smorodinsky, S 2017, Balanced line separators of unit disk graphs. in F Ellen, A Kolokolova & J-R Sack (eds), Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10389 LNCS, Springer Verlag, pp. 241-252, 15th International Symposium on Algorithms and Data Structures, WADS 2017, St. John’s, Canada, 31/07/17. https://doi.org/10.1007/978-3-319-62127-2_21

Balanced line separators of unit disk graphs. / Carmi, Paz; Chiu, Man Kwun; Katz, Matthew J. et al.
Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings. ed. / Faith Ellen; Antonina Kolokolova; Jorg-Rudiger Sack. Springer Verlag, 2017. p. 241-252 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10389 LNCS).

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

TY - GEN

T1 - Balanced line separators of unit disk graphs

AU - Carmi, Paz

AU - Chiu, Man Kwun

AU - Katz, Matthew J.

AU - Korman, Matias

AU - Okamoto, Yoshio

AU - Van Renssen, André

AU - Roeloffzen, Marcel

AU - Shiitada, Taichi

AU - Smorodinsky, Shakhar

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line ℓ such that ℓ intersects at most O(Formula presented) disks and each of the halfplanes determined by ℓ contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting O(Formula presented) disks exists, but each halfplane may contain up to 4n/5 disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m = 0. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(Formula presented m)).

AB - We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line ℓ such that ℓ intersects at most O(Formula presented) disks and each of the halfplanes determined by ℓ contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting O(Formula presented) disks exists, but each halfplane may contain up to 4n/5 disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m = 0. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(Formula presented m)).

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DO - https://doi.org/10.1007/978-3-319-62127-2_21

M3 - Conference contribution

SN - 9783319621265

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 241

EP - 252

BT - Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings

A2 - Ellen, Faith

A2 - Kolokolova, Antonina

A2 - Sack, Jorg-Rudiger

PB - Springer Verlag

T2 - 15th International Symposium on Algorithms and Data Structures, WADS 2017

Y2 - 31 July 2017 through 2 August 2017

ER -

Carmi P, Chiu MK, Katz MJ, Korman M, Okamoto Y, Van Renssen A et al. Balanced line separators of unit disk graphs. In Ellen F, Kolokolova A, Sack J-R, editors, Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings. Springer Verlag. 2017. p. 241-252. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: https://doi.org/10.1007/978-3-319-62127-2_21

Balanced line separators of unit disk graphs

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