Separability and Topology Control of Quasi Unit Disk Graphs
Philippe Giabbanelli
CMPT 880 – Spring 2008
This presentation deals with the half of the material that was left over.
We will present the theoretical concepts that we need, the algorithm and
the overall simulation results.
Some hints will be explained for possible follow-up works.
Spanners: why, what?
How to build a good backbone
Simulations results
Potential research problems
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Spanners
Backbone Algorithm
Simulations
Research problems
What is a k-spanner?
• Last time we saw that quasi-UDGs are a generalization of the UDG
model, but properties are not as well understood yet.
probability
guaranteed
maybe
r
too far
R
distance
• We showed the existence of small separators and its application to
routing as a first property.
• The second property concerns spanners.
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Spanners
Backbone Algorithm
Simulations
Research problems
What is a k-spanner?
All pairs are connected
by a path of length 1.
dG(u,v)
dT(u,v)
Now all pairs are
connected by paths of
length 2 = 2*1, hence
it is a 2-spanner.
Let’s
G.v) and T is a tree.
T is a k-spanner of
G ifconsider
dG(u, v)a<graph
k.dT(u,
k-spanner
a treelonger
such that
anyspanner.
two
kA
tells
you howismuch
are the
the distance
shortest between
paths in the
vertices is at most k times their distance in G.
k is called the stretch factor.
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Spanners
Backbone Algorithm
Simulations
Research problems
Desirable properties of spanners
• You do not want the length of the longest path to increase by a factor of
10 for some paths and by 2 for some others… you’d like it to be constant
►We want the stretching factor to be constant.
• A spanner is used as backbone: a sub-network for communications.
By reducing the number of edges we make the routing tables smaller
and have less interferences.
As the paths get longer, it costs more energy to do the transmission.
►We want the stretching factor to be small.
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Spanners
Backbone Algorithm
Simulations
Research problems
Construction presented in this paper
Done!
►We want the stretching factor to be constant.
Bounded by 3 + ε where ε can be made arbitrarily small
For suitable routing operations, we also make it nearly planar.
►We want the stretching factor to be small.
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Spanners
Backbone Algorithm
Simulations
Research problems
Algorithm to build the backbone
Let G be the graph with parameters R and r.
Step 1 – Planarize
• For every edge (uv) between u and v
∙ If there is no common neighbour in
the disk of diameter (uv), we take it.
∙ Otherwise, there is a neighbour w.
∙ Drop (uv).
∙ Repeat the process with (uw).
This construction is called Gabriel Graph.
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Spanners
Backbone Algorithm
Simulations
Research problems
Algorithm to build the backbone
Let G be the graph with parameters R and r.
Step 1 – Planarize
• For every edge (uv) between u and v
∙ If there is no common neighbour in
the disk of diameter (uv), we take it.
∙ Otherwise, there is a neighbour w.
∙ Drop (uv).
∙ Repeat the process with (uw).
This construction is called Gabriel Graph.
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Spanners
Backbone Algorithm
Simulations
Research problems
Algorithm to build the backbone
Let G be the graph with parameters R and r.
Step 2 – Reduce the number of short edges
4/2 = 2
• Direct the edges so that the graph is
acyclic and of maximum in-degree 5.
• Perform a modified Yao Step.
∙ Divide the region around each
• The
of short edges is reduced to
pointnumber
in k cones.
k + 5 where k is the number of cones.
∙ For each region, select the shortest edge.
• As
short
edges
are being
deleted,
∙ For
every
maximal
sequence
of the
l empty regions,
minimum
costsclockwise
increases and
by the
select thecommunication
first l/2 unselected
at first
mostl/2
1 +unselected
(2 sin(π/k))^β.
anti-clockwise.
4/2 = 2
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Spanners
Backbone Algorithm
Simulations
Research problems
Algorithm to build the backbone
Let G be the graph with parameters R and r.
Step 3 – Reduce the number of long edges
• Put a grid on the plane.
• An edge is considered long if it
connects vertices from different cells.
• For each pair of cells
∙ Keep the smallest edge between them.
(i.e. smallest of the longest)
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Spanners
Backbone Algorithm
Simulations
Research problems
Algorithm to build the backbone
Graph G under the
Quasi-UDG model
Gabriel Subgraph of G
Make it planar
azraz
Modified Yao Step
Reduce the number
of short edges
Backbone of small
constant stretching
factor
Grid
Reduce the number
of long edges
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Spanners
Backbone Algorithm
Simulations
Research problems
Networks topologies used in the experiments
• Let’s have a plane of size 1500 x 1500. We randomly set N vertices.
• Position randomly a big hole of radius chosen in [R, 2R] and five
small holes of radius chosen in [0, R].
• If the distance between two vertices is in [r, R] we set a link randomly.
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Spanners
Backbone Algorithm
Simulations
Research problems
Networks topologies used in the experiments
• Let’s have a plane of size 1500 x 1500. We randomly set N vertices.
varyingrandomly
N from 1000
2000,
we measure
thefive
density.
•By
Position
a bigtohole
of radius
chosenthe
in impact
[R, 2R]ofand
small holes of radius chosen in [0, R].
• If the distance between two vertices is in [r, R] we set a link randomly.
A random network is too uniform, thus holes are a simple attempt to
simulate non-trivial topologies.
The values for the ratio R/r go from 1 to 10 to measure the impacts
of different connectivity models.
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Spanners
Backbone Algorithm
Simulations
Research problems
Properties of the Backbone
1000
nodes
1500
nodes
2000
nodes
Max. degree in G
Max. degree
Average degree in G
Average degree
Avg. #crossings in G
Average #crossings
Overall, a good
reduction of the
degree and number of
crossings.
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Spanners
Backbone Algorithm
Simulations
Research problems
Properties of the Backbone
1000
nodes
1500
nodes
2000
nodes
Very small stretch factor
(i.e. not a big increase of
the length of the shortest
paths).
Overall, a good
reduction of the
degree and number of
crossings.
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Spanners
Backbone Algorithm
Simulations
Research problems
Labels and stretch factor
The backbones are
sparser than the original
graph thus we have
smaller separators.
Thanks to the smallest
separators, the distance
labelling is better on a
backbone.
However, the hop
stretch factor is larger
on the backbone.
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Spanners
Backbone Algorithm
Simulations
Research problems
Summary
• Compared to a classic greedy-forwarding plus local flooding algorithm,
our protocol performs much better.
• We have small routing tables, reduced interferences and still an
efficient local routing.
• In this serie of presentations, we have learnt new tools and concepts:
∙ Separators (with applications to compact structures) and spanners
∙ Quasi-UDG model (and its link-crossing properties)
∙ Use of grid graphs and virtual vertices to get simpler structures
∙ Distance labelling, gabriel graphs and yao step.
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Spanners
Backbone Algorithm
Simulations
Research problems
Main article used in this presentation
Separability and Topology Control of Quasi Unit Disk Graphs (Chen,
Jiang, Kanj, Xia and Zhang, IEEE 2007)
Other articles used to provide a better understanding
Improved Stretch Factor for Bounded-Degree Planar Power Spanners of
Wireless Ad-Hoc Networks (Iyad Kanj & Ljubomir Perkovic,
ALGOSENSORS 2006)
On geometric spanners of euclidian graphs and their applications in
wireless networks (Iyad Kanj & Ljubomir Perkovic, Technical Report
DePaul University 2007)
T HAN K
YOU
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Spanners
Backbone Algorithm
Simulations
Research problems
Can we do a complementary analysis?
• Between r and R we have a « maybe » for the connection…
probability
guaranteed
maybe
r
too far
R
distance
• The analysis presented in this paper are only the worst case analysis.
• A straightforward complement would be to do an average case analysis.
• Several possible distribution of probabilities could be use to represent
different situations.
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Spanners
Backbone Algorithm
Simulations
Research problems
Can we have more specific situations?
• The network topology used in this experiment is a random network.
• That might be good enough for sensor networks deployed uniformly on
a battlefield or in a forest.
• However, for MANET such as laptop devices, the network is probably
not that random.
∙ There are places with more users than others (cafe…) which
create clusters
∙ Some systems have more bandwidth or power than others thus
there should be an incentive to use them more often.
• Models for non-random real-world networks have been developped
during the last 8 years (scale-free, small-world, hierarchical, …).
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Spanners
Backbone Algorithm
Simulations
Research problems
Can we have more specific situations?
• What is the influence of the properties of the topology on the results?
• Can we use the properties from the topology to create better algorithms?
∙ Routing algorithms under scale-free and small-world topologies
have been published 5 years ago.
∙ They were still pretty basic without the use of labels or explicit
separators.
• A network is not of one kind only, thus trade-off should be developped.
∙ Who
to design
a wonderful
adaptive
algorithm
• Models
forwants
non-random
real-world
networks
haverouting
been developped
advantages
of properties
in some part
of the network?
duringtaking
the last
8 years (scale-free,
small-world,
hierarchical,
…).
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Spanners
Backbone Algorithm
Simulations
Research problems
Can we have more specific situations?
• For an excellent review of properties of real-world networks:
Structure and function of complex networks, M. E. J. Newman, 2003
• Greedy routing with tree-decomposition for small-world graphs:
A new perspective on the Small-World Phenomenon: Greedy Routing
in Tree-Decomposed Graphs, Pierre Fraigniaud, Report 2005
Distributed routing in small-world networks, Oskar Sandberg 2005
• Properties of transport in scale-free graphs:
Anomalous Transport in Scale-Free Networks, Eugene Stanley,
Physical Review Letters 2005
Search in power-law networks, Adamic & Lukose, Phys. Rev. E 2001
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