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Title: University of Denver


1
  • University of Denver
  • Department of Mathematics
  • Department of Computer Science

2
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

3
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)
4
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
5
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?

6
  • 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?

7
1 . Minimum Disk Covering Problem (MDC)
Cover Blue points with unit disks centered at
Red points !! Use Minimum red disks!!
8
Other Variation
Cover all Blues with unit disks centered at blue
points !! Using Minimum Number of disks
9
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.

10
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

11
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

12
The method to cover triangle
13
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 ?
14
Covering a triangle cont
A
T1
T3
C
B
T2
15
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 ?

16
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

17
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
18
Result 108-approximation
  • Covered each triangle with approximation factor
    of 6
  • Optimal cover can intersect at most 18 triangles
  • Hence, 6 18 108 - approximation

19
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?

20
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

21
Split the tile into two regions
v0
n 2m 1 n 5 m 2
v1
v4
v3
v2
22
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

23
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

24
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?

25
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
26
Covering
  • Instead of tiling the plane, how about covering
    the plane?

27
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

28
  • 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?

29
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
30
Previous work (MFS)
  • Despite its simplicity, complexity is unknown
  • 3- and 6-approximation algorithms known
  • Algorithm is based on property P

31
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
32
Property P
  • Property P applies if the region is outside of
    disk radius

Unit disk
A
s
Q
33
Redundant points
  • Remove redundant points

Redundant point
x
y
s
34
Bell and Cover of node x
  • Remove points inside the Bell- Bell Elimination
    Algorithm (BEA)

35
Analysis
  • Assume points to be uniformly distributed
  • BEA eliminates all the points inside the disk of
    radius
  • Need about 75 points
  • Therefore exact solution

36
Empirical result
37
Distance of one-hop neighbors
  • Extra region

38
Approximation factor
39
  • 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?

40
Degree of at most 2
  • Two-hop to bipartite graph

2
1
3
1
2
3
4
4
41
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
42
  • 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?

43
4. Weighted region problem (WRP)
  • Objective - Find an optimal path from START to
    GOAL
  • Complexity of WRP is unknown

44
Planar Graphs
  • Planar sub-division considered as planar graph

45
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

46
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!!

47
Snells Law
  • Cost function

Optimal point of incidence
48
0/1/? Special case WRP
  • Construct a critical graph G
  • Run Dijkstra on G

Weight 0
49
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

50
Critical points
s
t
51
Snell points
t
s
52
Border points
t
s
53
  • 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?

54
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

55
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
56
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

57
Distortions - bends in raster paths
58
Approximate by a straight line
  • Reduce deviation errors

59
Compare vector vs. Raster
Raster 1178.68 50 secs
Straight 1130.56 65 secs
Vector(?.1) 1128.27 lt 1 sec
60
  • 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?

61
Conclusion
  • Improved approximation to MDC
  • Bell elimination algorithm
  • Two-hop realizability
  • Exact solutions to special cases of WRP
  • Straight optimal raster paths

62
  • 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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