University of Denver

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• University of Denver
• Department of Mathematics
• Department of Computer Science

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Geometric 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 graph

• A graph (network) consists of nodes and edges
  represented as G(V, E, W)

e3(5)
b
a
e4(2)
e6(2)
e1(1)
e
e5(2)
d
c
e2(2)
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Planar Graphs

• A Planar graph is a graph that can be drawn in
  the plane such that edges do not intersect

Examples Voronoi diagram and Delaunay
triangulation
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AGENDA

• Topics
• Minimum Disk Covering Problem (MDC)
• Minimum Forwarding Set Problem (MFS)
• Two-Hop Realizability (THP)
• Exact Solution to Weighted Region Problem (WRP)
• Raster and Vector based solutions to WRP
• Conclusion
• Questions?

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• Topics
• Minimum Disk covering Problem (MDC)
• Minimum Forwarding Set Problem (MFS)
• Two-Hop realizability (THP)
• Exact solution to Weighted Region Problem (WRP)
• 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 Variation
Cover all Blues with unit disks centered at blue
points !! Using Minimum Number of disks
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Complexity

• MDC is known to be NP-complete
• Reference Unit Disk GraphsDiscrete
  Mathematics 86 (1990) 165177, B.N. Clark, C.J.
  Colbourn and D.S. Johnson.

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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 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 triangle
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Covering 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
A
T1
T3
C
B
T2
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Covering a triangle cont

• Using Skyline of disks
• cover each of the 3 sides with 2-approximation
• combine the result to get
• 6-approximation for each ?

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Desired 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 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-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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Improvements

• 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?

• 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 regions
v0
n 2m 1 n 5 m 2
v1
v4
v3
v2
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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

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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 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 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
T3
T2
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Covering

• Instead of tiling the plane, how about covering
  the plane?

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Conclusion 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
• Minimum Disk covering Problem (MDC)
• Minimum Forwarding Set Problem (MFS)
• Two-Hop realizability (THP)
• Exact solution to Weighted Region Problem (WRP)
• Raster and vector based solutions to WRP
• Conclusion
• Questions?

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2. Minimum Forwarding Set Problem (MFS)
s
ONE-HOP REGION
A
Cover blue points with unit disks centered at
red points, now all red points are inside a unit
disk
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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 P Again

• No two discs intersect more than once along their
  border inside a region Q
• No Two discs are tangent inside a region Q
• A disk intersect exactly twice along their border
  with Q

P1
Q
P3
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Property P

• Property P applies if the region is outside of
  disk radius

Unit disk
A
s
Q
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Redundant points

• Remove redundant points

Redundant point
x
y
s
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Bell and Cover of node x

• Remove points inside the Bell- Bell Elimination
  Algorithm (BEA)

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Analysis

• Assume points to be uniformly distributed
• BEA eliminates all the points inside the disk of
  radius
• Need about 75 points
• Therefore exact solution

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Empirical result
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Distance of one-hop neighbors

• Extra region

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Approximation factor
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• Topics
• Minimum Disk covering Problem (MDC)
• Minimum Forwarding Set Problem (MFS)
• Two-Hop realizability (THP)
• Exact solution to Weighted Region Problem (WRP)
• Raster and vector based solutions to WRP
• Conclusion
• Questions?

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Degree of at most 2

• Two-hop to bipartite graph

2
1
3
1
2
3
4
4
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3 . Two-hop realizability

• ResultA bipartite graph having a degree of at
  most 2 is two-hop realizable

two-hop neighbors
one-hop neighbors
1
3
4
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• Topics
• Minimum Disk covering Problem (MDC)
• Minimum Forwarding Set Problem (MFS)
• Two-Hop realizability (THP)
• Exact solution to Weighted Region Problem (WRP)
• Raster and vector based solutions to WRP
• Conclusion
• Questions?

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4. Weighted region problem (WRP)

• Objective - Find an optimal path from START to
  GOAL
• Complexity of WRP is unknown

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Planar Graphs

• Planar sub-division considered as planar graph

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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

• 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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Snells Law

• Cost function

Optimal point of incidence
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0/1/? Special case WRP

• Construct a critical graph G
• Run Dijkstra on G

Weight 0
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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
s
t
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Snell points
t
s
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Border points
t
s
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• Topics
• Minimum Disk covering Problem (MDC)
• Minimum Forwarding Set Problem (MFS)
• Two-Hop realizability (THP)
• Exact solution to Weighted Region Problem (WRP)
• 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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Raster-based algorithms

• 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 Dijkstras
  algorithm

32 connected
8 connected
16 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-connectivityv2
• 8-connectivity(v21)/5

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Distortions - bends in raster paths
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Approximate 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 lt 1 sec
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• Topics
• Minimum Disk covering Problem (MDC)
• Minimum Forwarding Set Problem (MFS)
• Two-Hop realizability (THP)
• Exact solution to Weighted Region Problem (WRP)
• Raster and vector based solutions to WRP
• Conclusion
• Questions?

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Conclusion

• 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
• Minimum Disk covering Problem (MDC)
• Minimum Forwarding Set Problem (MFS)
• Two-Hop realizability (THP)
• Exact solution to Weighted Region Problem (WRP)
• Raster and vector based solutions to WRP
• Conclusion
• Questions?