Finding the largest empty disk containing a query point

Haim Kaplan; Micha Sharir

doi:https://doi.org/10.1142/S021819591360008X

Finding the largest empty disk containing a query point

Haim Kaplan, Micha Sharir

School of Computer Science

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

Let P be a set of n points in the plane. We present an efficient algorithm for preprocessing P, so that, for a given query point q, we can quickly report the largest disk that contains q but its interior is disjoint from P. The storage required by the data structure is O(n log n), the preprocessing cost is O(n log² n), and a query takes O(log² n) time. We also present an alternative solution with an improved query cost and with slightly worse storage and preprocessing requirements.

Original language	English
Pages (from-to)	335-355
Number of pages	21
Journal	International Journal of Computational Geometry and Applications
Volume	23
Issue number	4-5
DOIs	https://doi.org/10.1142/S021819591360008X
State	Published - 2013

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Geometry and Topology
Computational Theory and Mathematics
Computational Mathematics
Applied Mathematics

Access to Document

https://doi.org/10.1142/S021819591360008X

Cite this

@article{369658104da542bb8078c0b1761deedf,

title = "Finding the largest empty disk containing a query point",

abstract = "Let P be a set of n points in the plane. We present an efficient algorithm for preprocessing P, so that, for a given query point q, we can quickly report the largest disk that contains q but its interior is disjoint from P. The storage required by the data structure is O(n log n), the preprocessing cost is O(n log2 n), and a query takes O(log2 n) time. We also present an alternative solution with an improved query cost and with slightly worse storage and preprocessing requirements.",

author = "Haim Kaplan and Micha Sharir",

note = "Funding Information: ∗Work on this paper was partially supported by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Work by Haim Kaplan was also supported by Grant 2006/204 from the U.S.–Israel Binational Science Foundation, and by Grant 822/10 from the Israel Science Fund. Work by Micha Sharir was also supported by NSF Grant CCR-08-30272, by Grant 2006/194 from the U.S.–Israel Binational Science Foundation, by Grant 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper appeared in Ref. 10.",

year = "2013",

doi = "https://doi.org/10.1142/S021819591360008X",

language = "الإنجليزيّة",

volume = "23",

pages = "335--355",

journal = "International Journal of Computational Geometry and Applications",

issn = "0218-1959",

publisher = "World Scientific",

number = "4-5",

}

TY - JOUR

T1 - Finding the largest empty disk containing a query point

AU - Kaplan, Haim

AU - Sharir, Micha

N1 - Funding Information: ∗Work on this paper was partially supported by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Work by Haim Kaplan was also supported by Grant 2006/204 from the U.S.–Israel Binational Science Foundation, and by Grant 822/10 from the Israel Science Fund. Work by Micha Sharir was also supported by NSF Grant CCR-08-30272, by Grant 2006/194 from the U.S.–Israel Binational Science Foundation, by Grant 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper appeared in Ref. 10.

PY - 2013

Y1 - 2013

N2 - Let P be a set of n points in the plane. We present an efficient algorithm for preprocessing P, so that, for a given query point q, we can quickly report the largest disk that contains q but its interior is disjoint from P. The storage required by the data structure is O(n log n), the preprocessing cost is O(n log2 n), and a query takes O(log2 n) time. We also present an alternative solution with an improved query cost and with slightly worse storage and preprocessing requirements.

AB - Let P be a set of n points in the plane. We present an efficient algorithm for preprocessing P, so that, for a given query point q, we can quickly report the largest disk that contains q but its interior is disjoint from P. The storage required by the data structure is O(n log n), the preprocessing cost is O(n log2 n), and a query takes O(log2 n) time. We also present an alternative solution with an improved query cost and with slightly worse storage and preprocessing requirements.

UR - http://www.scopus.com/inward/record.url?scp=84904346242&partnerID=8YFLogxK

U2 - https://doi.org/10.1142/S021819591360008X

DO - https://doi.org/10.1142/S021819591360008X

M3 - مقالة

SN - 0218-1959

VL - 23

SP - 335

EP - 355

JO - International Journal of Computational Geometry and Applications

JF - International Journal of Computational Geometry and Applications

IS - 4-5

ER -

Finding the largest empty disk containing a query point

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this