Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs
We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in O ( n log 2 n ) time using linear space, where n is the number of the vertices of the grap...
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Published in | Discrete & computational geometry Vol. 64; no. 4; pp. 1141 - 1166 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
New York
Springer US
01.12.2020
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
ISSN | 0179-5376 1432-0444 |
DOI | 10.1007/s00454-020-00219-7 |
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Summary: | We revisit a classical graph-theoretic problem, the
single-source shortest-path
(SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in
O
(
n
log
2
n
)
time using linear space, where
n
is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejčič [CGTA’15] which uses
O
(
n
1
+
δ
)
time and
O
(
n
1
+
δ
)
space (for any constant
δ
>
0
) and the previous randomized algorithm by Kaplan et al. [SODA’17] which uses
O
(
n
log
12
+
o
(
1
)
n
)
expected time and
O
(
n
log
3
n
)
space. More specifically, we show that if the 2D offline insertion-only (additively) weighted nearest-neighbor problem with
k
operations (i.e., insertions and queries) can be solved in
f
(
k
) time, then the SSSP problem in weighted unit-disk graphs can be solved in
O
(
n
log
n
+
f
(
n
)
)
time. Using the same framework with some new ideas, we also obtain a
(
1
+
ε
)
-approximate algorithm for the problem, using
O
(
n
log
n
+
n
log
2
(
1
/
ε
)
)
time and linear space. This improves the previous
(
1
+
ε
)
-approximate algorithm by Chan and Skrepetos [SoCG’18] which uses
O
(
(
1
/
ε
)
2
n
log
n
)
time and
O
(
(
1
/
ε
)
2
n
)
space. More specifically, we show that if the 2D offline insertion-only weighted nearest-neighbor problem with
k
1
operations in which at most
k
2
operations are insertions can be solved in
f
(
k
1
,
k
2
)
time, then the
(
1
+
ε
)
-approximate SSSP problem in weighted unit-disk graphs can be solved in
O
(
n
log
n
+
f
(
n
,
O
(
ε
-
2
)
)
)
time. Because of the
Ω
(
n
log
n
)
-time lower bound of the problem (even when approximation is allowed), both of our algorithms are almost optimal. |
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Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
ISSN: | 0179-5376 1432-0444 |
DOI: | 10.1007/s00454-020-00219-7 |