An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons
Given in the plane a set S of m point sites in a simple polygon P of n vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for S in P . It is known that the problem has an Ω ( n + m log m ) time lower bound. Previously, a randomized algorithm was proposed [Barb...
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| Veröffentlicht in: | Discrete & computational geometry Jg. 70; H. 2; S. 426 - 454 |
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
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Springer US
01.09.2023
Springer Nature B.V |
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| ISSN: | 0179-5376, 1432-0444 |
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| Abstract | Given in the plane a set
S
of
m
point sites in a simple polygon
P
of
n
vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for
S
in
P
. It is known that the problem has an
Ω
(
n
+
m
log
m
)
time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that solves the problem in
O
(
n
+
m
log
m
)
expected time. The previous best deterministic algorithms solve the problem in
O
(
n
log
log
n
+
m
log
m
)
time [Oh, Barba, and Ahn, SoCG 2016] or in
O
(
n
+
m
log
m
+
m
log
2
n
)
time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm that takes
O
(
n
+
m
log
m
)
time, which is optimal. This answers affirmatively an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago. |
|---|---|
| AbstractList | Given in the plane a set
S
of
m
point sites in a simple polygon
P
of
n
vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for
S
in
P
. It is known that the problem has an
Ω
(
n
+
m
log
m
)
time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that solves the problem in
O
(
n
+
m
log
m
)
expected time. The previous best deterministic algorithms solve the problem in
O
(
n
log
log
n
+
m
log
m
)
time [Oh, Barba, and Ahn, SoCG 2016] or in
O
(
n
+
m
log
m
+
m
log
2
n
)
time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm that takes
O
(
n
+
m
log
m
)
time, which is optimal. This answers affirmatively an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago. Given in the plane a set S of m point sites in a simple polygon P of n vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for S in P. It is known that the problem has an Ω(n+mlogm) time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that solves the problem in O(n+mlogm) expected time. The previous best deterministic algorithms solve the problem in O(nloglogn+mlogm) time [Oh, Barba, and Ahn, SoCG 2016] or in O(n+mlogm+mlog2n) time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm that takes O(n+mlogm) time, which is optimal. This answers affirmatively an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago. |
| Author | Wang, Haitao |
| Author_xml | – sequence: 1 givenname: Haitao orcidid: 0000-0001-8134-7409 surname: Wang fullname: Wang, Haitao email: haitao.wang@usu.edu organization: Department of Computer Science, Utah State University |
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| Cites_doi | 10.1007/BF02189321 10.1137/S0097539793253577 10.1016/0022-0000(89)90045-7 10.1007/BF02187751 10.1137/S0097539795289604 10.1007/BF01553882 10.1016/0020-0190(91)90064-O 10.1016/j.comgeo.2015.10.009 10.1007/s00453-019-00651-z 10.1007/s00454-019-00063-4 10.1007/s00454-016-9796-0 10.1007/s00454-013-9527-8 10.1016/0022-0000(89)90041-X 10.1016/0166-218X(82)90065-8 10.1007/s00453-019-00624-2 10.1016/0020-0190(80)90064-2 10.1007/BF01377183 10.1145/322276.322289 10.1007/PL00009199 10.1109/SFCS.1982.58 10.1016/B978-044482537-7/50016-4 10.1137/1.9781611975482.25 10.1145/1542362.1542402 10.1016/B978-0-12-386870-1.50010-1 |
| ContentType | Journal Article |
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| Keywords | 68Q25 68W40 Voronoi diagrams Geodesic distance Simple polygons 68U05 Shortest paths Farthest sites |
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| Snippet | Given in the plane a set
S
of
m
point sites in a simple polygon
P
of
n
vertices, we consider the problem of computing the geodesic farthest-point Voronoi... Given in the plane a set S of m point sites in a simple polygon P of n vertices, we consider the problem of computing the geodesic farthest-point Voronoi... |
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| SubjectTerms | Algorithms Apexes Combinatorics Computational geometry Computational Mathematics and Numerical Analysis Geometry Handbooks Lower bounds Mathematics Mathematics and Statistics Polygons Voronoi graphs |
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| Title | An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons |
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