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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| Published in: | Discrete & computational geometry Vol. 70; no. 2; pp. 426 - 454 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01.09.2023
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0179-5376, 1432-0444 |
| Online Access: | Get full text |
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| Summary: | 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. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0179-5376 1432-0444 |
| DOI: | 10.1007/s00454-022-00424-6 |