University of Denver Department of Mathematics Department of Computer Science
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University of Denver
Department of Mathematics
Department of Computer Science
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Geometric RoutingGeometric Routing
Applications Ad hoc Wireless networks Robot Route Planning in a terrain of varied types (ex:
grassland, brush land, forest, water etc.)
Geometric graphs Planar graph Unit disk graph
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General graphGeneral graph
A graph (network) consists of nodes and edges represented as G(V, E, W)
a b
c d
ee1(1)
e3(5)
e4(2)
e2(2)
e6(2)
e5(2)
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Planar Graphs Planar Graphs
A Planar graph is a graph that can be drawn in the plane such that edges do not intersect
a b
c d
e
Examples: Voronoi diagram and Delaunay triangulation
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AGENDAAGENDA
Topics:1. Minimum Disk Covering Problem (MDC)
2. Minimum Forwarding Set Problem (MFS)
3. Two-Hop Realizability (THP)
4. Exact Solution to Weighted Region Problem (WRP)
5. Raster and Vector based solutions to WRP Conclusion Questions?
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Topics:1. Minimum Disk covering Problem (MDC)
2. Minimum Forwarding Set Problem (MFS)
3. Two-Hop realizability (THP)
4. Exact solution to Weighted Region Problem (WRP)
5. Raster and vector based solutions to WRP Conclusion Questions?
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1 . Minimum Disk Covering Problem (MDC)
Cover Blue points with unit disks centered at Red points !! Use Minimum red disks!!
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Other VariationOther Variation
Cover all Blues with unit disks centered at blue points !! Using Minimum Number of disks
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ComplexityComplexity
MDC is known to be NP-complete Reference “Unit Disk Graphs”
Discrete Mathematics 86 (1990) 165–177, B.N. Clark, C.J. Colbourn and D.S. Johnson.
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Previous work (Cont…)Previous work (Cont…)
A 108-approximation factor algorithm for MDC is known
“Selecting Forwarding Neighbors in Wireless Ad-Hoc Networks”
Jrnl: Mobile Networks and Applications(2004)Gruia Calinescu ,Ion I. Mandoiu ,Peng-Jun Wan Alexander Z. Zelikovsky
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Previous methodPrevious method
Tile the plane with equilateral triangles of unit side
Cover Each triangle by solving a Linear program (LP)
Round the solution to LP to obtain a factor of 6 for each triangle
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The method to cover triangleThe method to cover triangle
1
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Covering a triangleCovering a triangle
IF No blue points in a triangle- NOTHING TO DO!!
IF ∆ contains RED + BLUE THEN Unit disk centered
at RED Covers the ∆
Assume BLUE + RED do not share a ∆
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Covering a triangle cont…Covering a triangle cont…
A
CB
T1
T2
T3
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Covering a triangle cont…Covering a triangle cont…
1. Using Skyline of disks
2. cover each of the 3 sides with 2-approximation
3. combine the result to get:
6-approximation for each ∆
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Desired PropertyDesired Property P
Skyline gives an approximation factor of 2
No two discs intersect more than once inside a triangle
No Two discs are tangent inside the triangle
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Unit disk intersects at most 18 trianglesUnit disk intersects at most 18 triangles
It can be easily verified that a Unit disk intersects at most 18 equilateral triangles in a tiling of a plane
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Result 108-approximationResult 108-approximation
Covered each triangle with approximation factor of 6
Optimal cover can intersect at most 18 triangles
Hence, 6 *18 = 108 - approximation
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ImprovementsImprovements
CAN WE use a larger tile? split the tile into two regions? get better than 6-approximation by different
tiling? cover the plane instead of tiling?
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Can we use a larger tile?Can we use a larger tile?
If tile is larger than a unit diameter !!
Unit disc inside Tile cannot cover the tile
Hence we cannot use previous method
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Split the tile into two regionsSplit the tile into two regions
v0
v1
v2
v3
v4
n = 2m +1
n = 5; m = 2
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Different shape Tile?Different shape Tile?
Each side with 2-approx. factor
Hence 8 for a square Unit disk can intersect
14 such squares 14 * 8 =112 No Gain by such
method
2
1
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Different shape Tile?Different shape Tile?
Each side with 2-approx. factor
Hence 12 for a hexagon Unit disk can intersect
12 such hexagons 12 * 12 =144 No Gain by such
method
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Our ApproachOur Approach
How about using a unit diameter hexagon as a tile
Split the tile into 3 regions around the hexagon
Does this give a better bound?
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Hexagon- split it into 3 regionsHexagon- split it into 3 regions
Partition Hexagon into 3 regions (Similar to triangle)
Obtain 2-approximation for each side 6-approximation for hexagon
Unit disk intersects 12 hexagons
Hence, 6 * 12 = 72-approximation
T1
T2
T3
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CoveringCovering
Instead of tiling the plane, how about covering the plane?
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Conclusion of MDCConclusion of MDC
Conjecture: A unit disk will intersect at least 12 tiles of any covering of R2 by unit diameter tiles
Each tile has an approximation of 6 by the known method
Cannot do better than 72 by the method used
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Topics:1. Minimum Disk covering Problem (MDC)
2. Minimum Forwarding Set Problem (MFS)
3. Two-Hop realizability (THP)
4. Exact solution to Weighted Region Problem (WRP)
5. Raster and vector based solutions to WRP Conclusion Questions?
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2. Minimum Forwarding Set Problem (MFS)2. Minimum Forwarding Set Problem (MFS)
Cover blue points with unit disks centered at red points, now all red points are inside a unit disk
s
A
ONE-HOP REGION
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Previous work (MFS)Previous work (MFS) Despite its simplicity, complexity is
unknown
3- and 6-approximation algorithms known Algorithm is based on property P
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Desired Property Desired Property P P AgainAgain
1. No two discs intersect more than once along their border inside a region Q
2. No Two discs are tangent inside a region Q
3. A disk intersect exactly twice along their border with Q
P1
P3
Q
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PropertyProperty P
Property P applies if the region is outside of disk radius 2
Unit disk 2
Q
s
A
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Redundant pointsRedundant points Remove redundant points
s
Redundant point
xy
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Bell and Cover of node Bell and Cover of node xx Remove points inside the Bell- Bell
Elimination Algorithm (BEA)
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AnalysisAnalysis
Assume points to be uniformly distributed BEA eliminates all the points inside the
disk of radius Need about 75 points Therefore exact solution
2
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80828486889092949698
100
50 60 75 85 97 140 200Number of Points
% success
Empirical result
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Distance of one-hop neighbors
Extra region
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Approximation factorApproximation factor
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Topics:1. Minimum Disk covering Problem (MDC)
2. Minimum Forwarding Set Problem (MFS)
3. Two-Hop realizability (THP)
4. Exact solution to Weighted Region Problem (WRP)
5. Raster and vector based solutions to WRP Conclusion Questions?
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Degree of at most 2Degree of at most 2
Two-hop to bipartite graph
s
ab
c
1 23
4
1
2
3
a
b
c
4
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3 . Two-hop realizability3 . Two-hop realizability
Result:A bipartite graph having a degree of at most 2 is two-hop realizable
1
2
3
a
b
c
4d
5one-hop neighbors two-hop neighbors
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Topics:1. Minimum Disk covering Problem (MDC)
2. Minimum Forwarding Set Problem (MFS)
3. Two-Hop realizability (THP)
4. Exact solution to Weighted Region Problem (WRP)
5. Raster and vector based solutions to WRP Conclusion Questions?
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4. Weighted region problem (WRP)4. Weighted region problem (WRP)
Objective - Find an optimal path from START to GOAL Complexity of WRP is unknown
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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Planar GraphsPlanar Graphs
Planar sub-division considered as planar graph
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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Shortest path G(V, E, W)
Dijkstra algorithm finds a shortest path from a source vertex to all other vertices
Running time O(|V| log |V| + |E|) Linear time for planar graphs
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WRP - General case WRP - General case
Notations
f = weight of face f
e = weight of edge e, where e = f f’ ≤ min {f, f’}
A weight of implies A path cannot cross that face or edge
Note that all optimal paths must be piecewise linear!!
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t ( x1,y1)
e
f (f)
f′ (f′)
s ( 0,-y0)
c (x, 0)
θ
θ′
f ≥ f′ > 0
Snell’s LawSnell’s Law
Cost function
f sinθ = α f ' sinθ '
Optimal point of incidence
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0/1/0/1/ Special case WRP Special case WRP
v wR
Construct a critical graph G Run Dijkstra on G
Weight 0
v w R
Weight
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Convex Polygon C
Exact path when s in C and t is arbitrary Construct “Exact Weighted” Graph Add edges that contribute to exact path Run Dijkstra shortest path Algorithm
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Critical points
θC
Critical edges
s
t
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Snell points
θ1
x
s
t
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Border points
t
s
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Topics:1. Minimum Disk covering Problem (MDC)
2. Minimum Forwarding Set Problem (MFS)
3. Two-Hop realizability (THP)
4. Exact solution to Weighted Region Problem (WRP)
5. Raster and vector based solutions to WRP Conclusion Questions?
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5. WRP - General case
-optimal path
• -optimal path from s to t is specified by users• path within a factor of (1+ ) from the optimal
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Transform weighted planar graph to uniform rectangular grid Make a graph with nodes and edges
- nodes : raster cells
- edges : the possible paths between the nodes Find the optimal path by running Dijkstra’s algorithm
Raster-based algorithms
8 connected 16 connected 32 connected
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Raster-based algorithms…
Advantages- Simple to
implement- Well suited for
grid input data- Easy to add other
cost criteria
Drawbacks- Errors in distance
estimate, since we measure grid distance instead of Euclidean distance
- Error factor :
4-connectivity:√2
8-connectivity:(√2+1)/5
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Distortions - bends in raster paths
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Approximate by a straight lineApproximate by a straight line
Reduce deviation errors
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Compare vector vs. Raster
Raster 1178.68 50 secs
Straight 1130.56 65 secs
Vector(=.1)
1128.27 < 1 sec
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Topics:1. Minimum Disk covering Problem (MDC)
2. Minimum Forwarding Set Problem (MFS)
3. Two-Hop realizability (THP)
4. Exact solution to Weighted Region Problem (WRP)
5. Raster and vector based solutions to WRP Conclusion Questions?
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ConclusionConclusion
Improved approximation to MDC Bell elimination algorithm Two-hop realizability Exact solutions to special cases of WRP Straight optimal raster paths
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Topics:1. Minimum Disk covering Problem (MDC)
2. Minimum Forwarding Set Problem (MFS)
3. Two-Hop realizability (THP)
4. Exact solution to Weighted Region Problem (WRP)
5. Raster and vector based solutions to WRP Conclusion Questions?