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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| Vydané v: | Discrete & computational geometry Ročník 43; číslo 1; s. 21 - 53 |
|---|---|
| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
New York
Springer-Verlag
01.01.2010
Springer Nature B.V |
| Predmet: | |
| ISSN: | 0179-5376, 1432-0444 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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, [partial differential] 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 [partial differential] 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 [partial differential] R(e) is at least cmax{|e|,|e'|}, where c is some positive constant. In particular, it includes the case where each facet of [partial differential] P is fat and each vertex is incident to at most O(1) facets of [partial differential] P. In all the above cases the algorithm runs in O(nlogn) 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(logn) 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. [PUBLICATION ABSTRACT] |
| Author | Schreiber, Yevgeny |
| Author_xml | – sequence: 1 givenname: Yevgeny surname: Schreiber fullname: Schreiber, Yevgeny email: yevgeny.schreiber@cs.tau.ac.il organization: School of Computer Science, Tel Aviv University |
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| Cites_doi | 10.1016/S0020-0190(96)00154-8 10.1142/S0218195996000095 10.1137/S0097539797325223 10.1007/s00453-001-0027-5 10.1137/0216038 10.1007/s00454-007-9031-0 10.1137/S0097539795289604 10.1137/S0097539799352759 10.1137/0215014 10.1137/0216045 10.1007/s004530010047 10.1007/PL00009402 10.1007/s00453-002-0961-x 10.1007/BF02187877 10.1007/BF01758853 10.1145/263867.263869 10.1007/978-3-540-45077-1_23 10.1145/262839.262983 10.1145/301250.301449 10.1007/BFb0024013 10.1145/177424.177453 10.1145/1137856.1137885 10.1016/0925-7721(95)00005-8 10.21236/ADA166246 10.1016/S0925-7721(96)00016-8 |
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| Keywords | Shortest path map Continuous Dijkstra Terrain Conforming subdivision Realistic polyhedral surface Wavefront |
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| Snippet | 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... 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... |
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| SubjectTerms | Algorithms Combinatorics Computational Mathematics and Numerical Analysis Geometry Mathematics Mathematics and Statistics Polyhedra |
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| Title | An Optimal-Time Algorithm for Shortest Paths on Realistic Polyhedra |
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