Separability and Topology Control of Quasi Unit Disk Graphs

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Separability and Topology Control of Quasi Unit
Disk Graphs
Philippe Giabbanelli CMPT 880 Spring 2008
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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
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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.
guaranteed
maybe
too far
probability
distance
r
R
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
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What is a k-spanner?
All pairs are connected by a path of length 1.
Now all pairs are connected by paths of length 2
21, hence it is a 2-spanner.
dG(u,v)
dT(u,v)
Lets consider a graph G.
T is a k-spanner of G if dG(u, v) lt k.dT(u, v)
and T is a tree.
A k-spanner is a tree such that the distance
between any two vertices is at most k times their
distance in G.
k tells you how much longer are the shortest
paths in the spanner.
k is called the stretch factor.
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Spanners Backbone Algorithm
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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 youd 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
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Construction presented in this paper
Done!
?We want the stretching factor to be constant.
Bounded by 3 e where e 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
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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
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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
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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 point in k cones.
The number of short edges is reduced to k 5
where k is the number of cones.
For each region, select the shortest edge.
As short edges are being deleted, the minimum
communication costs increases by at most 1 (2
sin(p/k))ß.
For every maximal sequence of l empty regions,
select the first l/2 unselected clockwise and the
first l/2 unselected anti-clockwise.
4/2 2
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Spanners Backbone Algorithm
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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
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Algorithm to build the backbone
Graph G under the Quasi-UDG model
Make it planar azraz
Modified Yao Step
Gabriel Subgraph of G
Reduce the number of short edges
Grid
Backbone of small constant stretching factor
Reduce the number of long edges
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Spanners Backbone Algorithm
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Networks topologies used in the experiments
Lets 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
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Networks topologies used in the experiments
Lets have a plane of size 1500 x 1500. We
randomly set N vertices.
By varying N from 1000 to 2000, we measure the
impact of the density.
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.
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
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Properties of the Backbone
1000 nodes
Max. degree in G
Max. degree
Average degree in G
Average degree
1500 nodes
Avg. crossings in G
Average crossings
Overall, a good reduction of the degree and
number of crossings.
2000 nodes
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Spanners Backbone Algorithm
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Properties of the Backbone
1000 nodes
Very small stretch factor (i.e. not a big
increase of the length of the shortest paths).
1500 nodes
Overall, a good reduction of the degree and
number of crossings.
2000 nodes
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Spanners Backbone Algorithm
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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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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
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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)
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Spanners Backbone Algorithm
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Can we do a complementary analysis?
Between r and R we have a  maybe  for the
connection
guaranteed
maybe
too far
probability
distance
r
R
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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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
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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 wants to design a wonderful adaptive
routing algorithm taking advantages of properties
in some part of the network?
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
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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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