An Optimal-Time Algorithm for Shortest Paths on Realistic Polyhedra
We generalize our 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 to three realistic scenarios where P is a possibly nonconvex polyhedron. In the first scenario, ∂ P is a terrain whose maximum face...
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| Veröffentlicht in: | Discrete & computational geometry Jg. 43; H. 1; S. 21 - 53 |
|---|---|
| 1. Verfasser: | |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer-Verlag
01.01.2010
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0179-5376, 1432-0444 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | We generalize our 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
to three realistic scenarios where
P
is a possibly
nonconvex
polyhedron. In the first scenario,
∂
P
is a
terrain
whose maximum facet slope is bounded by any fixed constant. In the second scenario,
P
is an
uncrowded
polyhedron—each axis-parallel square
h
of side length
l
(
h
) whose smallest Euclidean distance to a vertex of
P
is at least
l
(
h
) is intersected by at most
O
(1) facets of
∂
P
—an input model which, as we show, is a generalization of the well-known
low-density
model. In the third scenario,
P
is
self-conforming
—here, for each edge
e
of
P
, there is a connected region
R
(
e
) of
O
(1) facets whose union contains
e
, so that the shortest path distance from
e
to any edge
e
′ of
∂
R
(
e
) is at least
c
⋅max {|
e
|,|
e
′|}, where
c
is some positive constant. In particular, it includes the case where each facet of
∂
P
is
fat
and each vertex is incident to at most
O
(1) facets of
∂
P
. In all the above cases the algorithm runs in
O
(
n
log
n
) time and space, where
n
is the number of edges of
P
, and produces an implicit representation of the shortest-path map, so that the shortest path from
s
to any query point
q
can be determined in
O
(log
n
) time. The constants of proportionality depend on the various parameters (maximum facet slope, crowdedness, etc.). We also note that the self-conforming model allows for a major simplification of the algorithm. |
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| Bibliographie: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 |
| ISSN: | 0179-5376 1432-0444 |
| DOI: | 10.1007/s00454-009-9136-8 |