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Shortest Path Problems on a Polyhedral Surface

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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
    changes between adjacent events.

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
    changes between adjacent events.

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
    changes between adjacent events.

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
    changes between adjacent events.

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
  • This causes two source images to jump to new
    positions and start moving on new line segments.

s
39
Maintaining the Kinetic VD
  • Star Unfolding Event
  • Two sites start moving on new line segments
  • Both of the new line segments define M-1 new
    pairs of moving sites.
  • ? Each Star Unfolding Event defines O(M . M
    2a(M)) Voronoi Events.
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