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## Shortest Path Problems on a Polyhedral Surface

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### University of Texas at San Antonio. 2. Shortest Paths on a Polyhedral ... J. Canny and J.H. Reif. New Lower Bound Techniques for Robot Motion Planning Problems. ... – PowerPoint PPT presentation

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Title: Shortest Path Problems on a Polyhedral Surface

1
Shortest Path Problems on a Polyhedral Surface
Carola Wenk Atlas F. Cook IV University of Texas
at San Antonio
2
Shortest Paths on a Polyhedral Surface
• Why study shortest paths on a surface?
• Hike in the mountains
• Navigate the surface of Mars
• Convex polyhedral surfaces
• Arbitrary polyhedral surfaces

3
Our Results
• Convex polyhedron
• Give a linear-factor speedup for edge sequence
results of AARS97
• Compute a superset of all Q(M4) edge sequences
in O(M5) time
• Compute the set of all edge sequences in
O(M52a(M)log M) time
• Arbitrary polyhedral surface
• Maintain star unfolding over all edges in O(M4)
time
• Applications
• Fréchet distance (linear factor speedup for
convex polyh.)
• Diameter
• Shortest path maps

4
Edge Sequences
• An edge sequence is the
• sequence of edges crossed by a (shortest) path

End
E2
E1
Start
5
Edge Sequences Unfolding
• Sequence of edges crossed by a (shortest) path

E2
Unfold
E1
6
Edge Sequences Unfolding
• A shortest path on a convex 3D polyhedral surface
unfolds to a
• straight line segment in 2D.
• A shortest path on a non-convex 3D polyhedral
surface unfolds to a polygonal path in 2D.

7
Star Unfolding
s
s
v4
v3
v4
v4
v3
v3
v2
v1
v1
v1
v2
v2
Polyhedral Surface
(1) Compute shortest paths to all M vertices
(2) Unfold along shortest paths
8
Star Unfolding, Core, Anticore
• s maps to M points s1,,sM in the 2D unfolding

s1
s
v4
Star Unfolding
v3
s4
s2
v4
v3
v2
v1
core
v1
anticore
s3
v2
• One core region, and M anticore regions
• Compute star unfolding in O(M2) time

9
Star Unfolding
• t in anticore(si) d(s,t) si t
• t in core d(s,t) mini1,,M si t

s1
s
Star Unfolding
v4
v3
t
s4
s2
v4
v3
v2
v1
d(s,t)
v1
s3
v2
? Compute a Voronoi Diagram of all source images
to answer logarithmic shortest path queries
within the core
10
Star Unfolding
• If P is convex, then the Star Unfolding is a
simple polygon AR91
• If P is non-convex, the Star Unfolding can
self-overlap

11
Star Unfolding
• Lemma If P is non-convex, even the core of the
Star Unfolding can self-overlap

12
Star Unfolding
• O(M) polyhedron edges in the core
• O(M2) cut polyhedron edges in the anticore

13
Edgelets
• Subdivision of polygon edges to describe
combinatorially distinct star unfoldings with the
source on the edges
• Compute shortest paths between every two vertices
• ? O(M2) edgelets per edge, O(M3) edgelets total

14
Superset of Edge Sequences
• Theorem Construct a superset of all edge
sequences, on a convex polyhedron, in O(M5) time
• Proof
• O(M3) edgelets
• For each source image si
• construct edge sequences to all O(M) anticore
edges in anticore(si)
• construct edge sequences to all O(M) core edges

15
Edge Sequences Algorithm
• Algorithm
• Maintain the star unfolding as a source point s
varies continuously over all edges of P.
• Maintain the kinetic Voronoi diagram as a source
point s varies continuously over all edges of P.
• Extract shortest path edge sequences
• Continuous motion takes advantage of small

16
Maintaining the Star Unfolding
• Star Unfolding Event edgelet endpoint induces a
combinatorial change in the star unfolding
• Between events
• Source images move on line segments AARS97
• Other vertices are fixed

s2
s1
s2
s1
s
s3
s3
s5
s5
s4
s4
17
Maintaining the Star Unfolding
• Handling a star unfolding event
• One shortest path to a vertex changes
combinatorially
• Update two source images including their anticore
regions and all O(M) edges in these anticore
regions.
• Update O(m) edges in the core.

s2
s1
s
s3
s5
s4
18
Maintaining the Star Unfolding
• Theorem Maintain a star unfolding as a source
point s varies continuously over all edges of P
in O(M4) time.
• Proof
• O(M3) edgelets
• For each edgelet endpoint, update two source
images including their anticore regions, and all
O(M) edges in these anticore regions and in the
core.
• ? Linear factor faster than computing a new star
unfolding for each event.

19
Edge Sequences Algorithm
• Algorithm
• Maintain the star unfolding as a source point s
varies continuously over all edges of P.
• Maintain the kinetic Voronoi diagram as a source
point s varies continuously over all edges of P.
• Extract shortest path edge sequences
• Continuous motion takes advantage of small

20
Maintaining the Kinetic VD
• Maintain a kinetic Voronoi diagram as a source
point s varies continuously over all edges of P.
• Kinetic Voronoi diagram AMGR98
• Sites are allowed to move
• Sites source images s1,,sM of star unfolding
• Goal Maintain subdivision as sites move

21
Maintaining the Kinetic VD
• Between events
• Sites move on line segments
• Other vertices are fixed

s2
s1
s2
s1
s3
s
s3
s5
s5
s4
s4
22
Maintaining the Kinetic VD
• Voronoi Event
• A Voronoi edge appears or disappears.
• Each pair of moving sites defines O(M 2a(M))
Voronoi Events AMGR98.
• Thus, M moving sites define
• O(M2) pairs and
• O(M2 . M 2a(M)) Voronoi Events

s1
s2
s3
s4
23
Maintaining the Kinetic VD
• Star Unfolding Event
• Two source images are removed and inserted in a
different place (together with a new anticore
region)
• The new source images define new line segments,
which define O(M) new pairs of moving sites.
• ? Each Star Unfolding Event defines O(M . M
2a(M)) Voronoi Events.

24
Maintaining the Kinetic VD
Theorem A kinetic VD of the source images can be
maintained in O(M5 2a(M) log M) time as the
source point s varies continuously over all edges
of P.
• Proof
• Star Unfolding Events O(M3)
• Voronoi Events O(M3 . M2 2a(M))
• Time per Voronoi Event O(log M)
• Total Time O(M5 2a(M) log M)

25
Edge Sequences Algorithm
• Algorithm
• Maintain the star unfolding as a source point s
varies continuously over all edges of P.
• Maintain the kinetic Voronoi diagram as a source
point s varies continuously over all edges of P.
• Extract shortest path edge sequences
• Continuous motion takes advantage of small

26
Extract Shortest Path Edge Sequences
• Between events
• Each source image si defines a
• parameterized Voronoi cell.
• Let E be an edge of the star unfolding.
• The unique edge sequence from si to E corresponds
to a shortest path iff the parameterized Voronoi
cell for si ever intersects E.
• Upper envelope triangulation AARS97
• O(log M) time per parameterized Voronoi cell

E
si
27
Extract Shortest Path Edge Sequences
• Theorem Compute all ?(M4) shortest path edge
sequences on a convex polyhedral surface P with M
vertices in O(M5 2a(M) log M) time.
• Improves AARS97 by a linear factor by using
continuous motion to take advantage of small

28
Applications
29
Applications
• Diameter
• Compute the diameter of a convex polyhedron in
O(M7log M) time.
• Improves AARS97 by a linear factor
• Fréchet Distance
• Compute the Fréchet dist. between two polygonal
curves on a convex polyhedron in O(M6log2 M)
time.
• Improves MY05 by a linear factor
• Shortest Path Maps
• Support O(log n) queries from any point on a line
segment, with O(M42a(M) log M) preprocessing time.

30
Diameter
Theorem Compute the diameter of a convex
polyhedral surface in O(M7log M) time.
• Proof
• O(M4) ridge free regions Induced by all shortest
paths between pairs of vertices same star
unfolding
• Compute kinetic VD for first region in O(M4) time
• Process adjacent ridge free regions
• Update two sites ? O(M) pairs of sites in kinetic
VD, each defining O(M2) new Voronoi vertices
AARS97
• O(M3) new Voronoi vertices
• Find max of all O(M7) Voronoi vertex distance
functions in O(log n) time AARS97

31
Fréchet Distance
Theorem Compute the Fréchet dist. between two
poly-gonal curves on a convex polyhedron in
O(M6log2 M) time.
• Proof sketch
• Partition one polygonal curve into O(M3)
edgelets.
• Maintain star unfolding over all edgelets in
O(M4) time
• Free space cell for an anticore edge and an
edgelet
• Constant complexity, despite parameterization of
anticore edge. O(M5) total complexity.
• Free space cell for a core edge and an edgelet
• Union of M ellipses ? O(M2) complexity
• O(M3) complexity over all O(M) core edges, and
O(M6) total

32
Shortest Path Map
Theorem A shortest path map from a line segment
on a convex polyhedron can be built in O(M42a(M)
log M) time and supports queries in O(log2 n)
time.
• Proof
• Partition the line segment into O(M2) edgelets
• Maintain kinetic VD in O(M4 2a(M) log M) time.
• Queries take O(log2 n) time by DGKS96.

33
Conclusions
• Convex polyhedron
• Compute a superset of all Q(M4) edge sequences
in O(M5) time
• Compute the set of all edge sequences in
O(M52a(M)log M) time
• Arbitrary polyhedral surface
• Maintain star unfolding over all edges in O(M4)
time
• Applications
• Fréchet distance (linear factor speedup for
convex polyh.)
• Diameter, Shortest path maps
• Open question
• Can one compute all Q(M4) edge sequences in o(M5)
time?

34
Conclusion
• Our Main Result
• The ?(M4) shortest path edge sequences on a
convex polyhedral surface with M vertices can be
computed a linear factor faster than Agarwal97.
• Applications
• Diameter, Fréchet Distance, Shortest Path Maps
• Future Work
• Can the ?(M4) shortest path edge sequences be
computed in o(M5) time?

35
References
• AARS97
• P. K. Agarwal, B. Aronov, J. ORourke, and C.
Schevon. Star Unfolding of a Polytope with
Applications. SIAM Journal on Computing, Society
for Industrial and Applied Mathematics, 1997, 26,
1689-1713
• AMGR98
• G. Albers, J. S. B. Mitchell, L. J. Guibas, and
T. Roos. Voronoi diagrams of moving points.
International Journal of Computational Geometry
and Applications, 1998, 8365380.
• AR91
• B. Aronov, J. ORourke. Nonoverlap of the Star
Unfolding. SoCG, 1991, 105-114.
• CR87
• J. Canny and J.H. Reif. New Lower Bound
Techniques for Robot Motion Planning Problems.
Proc. 28th IEEE Annual Symp. Foundations of
Computer Science, 1987, 49-60.
• CHK04
• V. Chandru, R. Hariharan, and N. M. Krishnakumar.
Short-cuts on star, source and planar unfoldings.
Foundations of Software Technology and
Theoretical Computer Science (FSTTCS), 2004,
174185.

36
References
• CH96
• J. Chen and Y. Han. Shortest paths on a
polyhedron. International Journal of
Computational Geometry and Applications, 1996,
6127-144.
• DGKS96
• O. Devillers, M. Golin, K. Kedem, S. Schirra.
Queries on Voronoi diagrams of moving points.
CGTA 6(5) 315-327, 1996.
• MY05
• A. Maheshwari and J. Yi. On computing Fréchet
distance of two paths on a convex polyhedron.
21st European Workshop on Computational Geometry
(EuroCG), 2005.
• M90
• D. M. Mount. The number of shortest paths on the
surface of a polyhedron. SIAM Journal on
Computing, 1990, 19(4)593611.

37
Results
38
Maintaining the Star Unfolding
• Handling a star unfolding event
• One shortest path to a vertex changes
combinatorially