An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions
We present an optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source s on the surface of a convex polytope P in three dimensions. Our algorithm runs in O ( n log n ) time and requires O ( n log n ) space, where n is the number of edges of P ....
Saved in:
| Published in: | Discrete & computational geometry Vol. 39; no. 1-3; pp. 500 - 579 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer-Verlag
01.03.2008
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0179-5376, 1432-0444 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | We present an optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source
s
on the surface of a convex polytope
P
in three dimensions. Our algorithm runs in
O
(
n
log
n
) time and requires
O
(
n
log
n
) space, where
n
is the number of edges of
P
. The algorithm is based on the
O
(
n
log
n
) algorithm of Hershberger and Suri for shortest paths in the plane (Hershberger, J., Suri, S. in SIAM J. Comput. 28(6):2215–2256,
1999
), and similarly follows the continuous Dijkstra paradigm, which propagates a “wavefront” from
s
along
∂
P
. This is effected by generalizing the concept of conforming subdivision of the free space introduced by Hershberger and Suri and by adapting it for the case of a convex polytope in ℝ
3
, allowing the algorithm to accomplish the propagation in discrete steps, between the “transparent” edges of the subdivision. The algorithm constructs a dynamic version of Mount’s data structure (Mount, D.M. in Discrete Comput. Geom. 2:153–174,
1987
) that implicitly encodes the shortest paths from
s
to all other points of the surface. This structure allows us to answer single-source shortest-path queries, where the length of the path, as well as its combinatorial type, can be reported in
O
(log
n
) time; the actual path can be reported in additional
O
(
k
) time, where
k
is the number of polytope edges crossed by the path.
The algorithm generalizes to the case of
m
source points to yield an implicit representation of the geodesic Voronoi diagram of
m
sites on the surface of
P
, in time
O
((
n
+
m
)log (
n
+
m
)), so that the site closest to a query point can be reported in time
O
(log (
n
+
m
)). |
|---|---|
| Bibliography: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23 |
| ISSN: | 0179-5376 1432-0444 |
| DOI: | 10.1007/s00454-007-9031-0 |