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 article deals with a more complex model than
Unit Disk Graph.
First, I will explain Unit Disk Graph (UDG) and
gradually go to the more complicate model.
Once the new model is understood, we can define
some constructions on it, leading to efficient
algorithms and routing protocols.
From UDG to quasi-UDG
Abstraction of a quasi-UDG by a grid graph
An algorithm to find a small separator set in a
grid graph
Routing protocol with distance labelling on grid
graphs
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
A sensor network or a MANET network is
represented by a graph with vertices for systems
and edges if there is a communication.
It is quite easy to say that a vertex is a
system
but to express that two nodes are
communicating, we need a model.
This model must answers two fundamental
questions
Do all nodes have the same transmission range?
When can a node communicate with another one?
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
A sensor network or a MANET network is
represented by a graph with vertices for systems
and edges if there is a communication.
w
r
Yes, and we denote it by r.
uw gt r
hence no connection
This model is called the Unit Disk Graph (UDG).
A system u can communicate with a vertex v if
their distance is less than the communication
range.
Do all nodes have the same transmission range?
When can a node communicate with another one?
There is an edge (u, v) if uv ru for the
euclidian metric .
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Under the Unit Disk Model, the probability to
communicate with a vertex given the distance
looks as follows
guaranteed
maybe
too far
probability
distance
r
R
In the real world, the probability does not
have such a strong threshold.
When a node is a bit farther than r, the signal
does not suddenly disappear. It is just not
strong enough to guarantee the communication, but
maybe that we can still communicate.
In the quasi-UDG model, there is an edge for
uv r, not for uv gt R and maybe for r lt uv
R.
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
To model complex things, we need some
abstraction. The simpler the abstraction, the
easier we can develop results based on it.
Mr. UDG is happy because he has a simple model he
can deal with.
Mr. Quasi-UDG has a more accurate model, but no
so easy
The UDG model is quite a rough abstraction, but
the properties have been well understood.
The quasi-UDG model is more accurate, but there
has been so far quite a lack of understanding of
the properties.
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
One of the few properties available on the
quasi-UDG model is the link-crossing property
(Barrière, Fraigniaud and Narayanan 2001).
We know that if we are at distance r, we can
communicate.
r is the minimum transmission range.
We know that if we are at distance R, we
cannot communicate.
R is the maximum transmission range.
The authors have designed a protocol that
guarantees message delivery if R is at most 40
longer than r.
Message delivery is guaranteed for quasi-UDGs
where R v2 r.
In other words, we guarantee message delivery
if the ratio between R and r is at most v2.
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
This paper has two goals. As this presentation
is in two parts, this presentation will deal with
the first goal.
Remember what we said last time about
separators if a graph has a a small set of
separators then it is well separable and it
enables us to do some nice things (such as a
compact representation of the graph).
In particular, the local properties of
separable graphs can be used in many aspects of
networks routing, information retrieval,
monitoring
The authors will construct an abstraction of a
quasi-UDG and show that it is well separable,
with a separator size of O(vN).
As one of the applications, a compact routing
protocol will be described.
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
We impose a grid on the plane to build a grid
graph H.
A quasi-UDG can have a highly variable node and
edges densities, thus we cannot guarantee the
existence of small separators as such.
If there is at least one vertex in a cell, then
there is a vertex in the center of this cell for
H.
For two cells, if there is an edge connecting
vertices in those cells in G then the vertices in
the middle of the cells in H are connected.
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Theorem. The grid graph H constructed by the
algorithm GridGraph with a grid of cell size r/v2
r/v2 for any given quasi-UDG G is such that
inside any disk of radius y there are at most
O(y²/r²) vertices of H.
r/v2
By construction, the distance between any two
vertices is at least r/v2. On the scheme we
identify a vertex by the center of either the
blue or the red disk.
If we place a disk of radius r/(2v2) centered on
each vertex, no two disks will intersect. Blue
and red do not intersect.
r/v2
A disk D of size y can intersect with at most
O(y²/r²) disks.
The number of little disks that y is intersecting
with represents the number of vertices inside D.
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Theorem. The grid graph H constructed by the
algorithm GridGraph with a grid of cell size r/v2
r/v2 for any given quasi-UDG G is such that
the degree of each vertex of H is at most O(R²/r²)
We denote by v(U) the set of vertices of G inside
the cell U.
A vertex in v(U) can only be adjacent to a vertex
in v(V) is uv R r by construction.
Thus the number of vertices distant of R r from
U is at most O(R²/r²).
The number of vertices being the degree, we have
that the degree is upper bounded by O(R²/r²).
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Theorem. The grid graph H constructed by the
algorithm GridGraph with a grid of cell size r/v2
r/v2 for any given quasi-UDG G is such that
any given edge can be crossed by at most
O(R²/r4) edges.
Based on the fact that the number of vertices
within distance R r from any point on a line
(i.e. edge) is O(R²/r²) by the previous proof.
Our grid graph H is well defined with a set of
useful properties, but we need something more to
find a good separator
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
We impose a larger grid on the plane to create a
graph T.
Black vertex with a weight equal to the number of
vertices contains in its cell in H (i.e. order of
abstraction)
Red vertex, assigned to a weight of 0
The graph T is constructed from the grid graph H
with the same rules than H is made from G.
To make T planar, a virtual vertex is added in
the middle of each diagonal edge (it eliminates
all crossings).
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Algorithm to separate a grid graph H
Let T be the auxiliary graph of H and T a copy
of T.
Step 1 Create a tree
Select a vertex from the outer face as root to
launch the BFS process.
The undiscovered neighbours are visited in
clockwise order.
1
When a new vertex u is discovered
It is added to the BFS
For every face F containing u
2
Add edges to all other vertices of F as long as
T remains planar
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Algorithm to separate a grid graph H
Let T be the auxiliary graph of H and T a copy
of T.
Step 2 Split the tree
With the BFS process, we have a tree.
A cycle formed by one edge that is not in the
tree and some edges of the tree is called a
fundamental cycle.
We look for the fundamental cycle that splits
T in the most balanced way.
In other words, we are facing an optimization
problem of finding ST such that it splits T in
T1 and T2 and minimizes w(T1) w(T2).
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Algorithm to separate a grid graph H
Let T be the auxiliary graph of H and T a copy
of T.
Step 3 Virtual vertices in the split
For each virtual vertex u (i.e. red) in the
separator set ST
If all the neighbours of u outside of ST are in
T1, then we move u to T1
If all the neighbours of u outside of ST are in
T2, then we move u to T2
T1
T2
If the neighbours of u outside of ST are both
in T1 and T2, then
ST
move u to T1
move all the neighbours in ST
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Algorithm to separate a grid graph H
Let T be the auxiliary graph of H and T a copy
of T.
Step 4 Results in the grid graph H
Let SH be the set of vertices in the cells
given by the real (i.e. black) vertices in the
separator set ST.
Similarly, let H1 and H2 be the set of vertices
in the cells given by the real (i.e. black)
vertices in T1 and T2.
T1
ST
T2
H1
SH
H2
Clearly, SH separates H into H1 and H2.
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Algorithm to separate a grid graph H
Graph G under the Quasi-UDG model
Graph H with bounded degree
Planar Graph T aztzet
Abstraction by a grid and virtual vertices
Abstraction by a grid
Build a BFS tree and change T
For a a minimum hop distance h(u, v) between u
and v, we have a routing algorithm in 2h(u, v)
1 hops and a routing table of size O(vN.log N) at
each node.
Find a fundamental cycle to split well T
Take virtual vertices out of the split
Separator of size O(vN) for H
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Principles of Distance labeling in Graphs
A representation of a network is often global
If we want to know something such as the distance
between two vertices, we need an access to the
entire data structure.
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110
5
111
How far is 1 from 5?
How far is 000 from 111?
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100
4
101
000 and 111 differ on 3 positions. Distance 3.
8
011
010
7
1
3
000
001
If we label the nodes in some way, we can guess
things such as the adjacency of two nodes
directly by looking at the label. There is no
need to access the whole data structure.
We can use label of any arbitrary large size to
encode an information
but we want efficiency short label and fast
information deduction!
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Principles of Distance labeling in Graphs
Distance labeling is useful in communication
networks, especialy for memory-free routing
schemes routing schemes with very little data to
be stored locally.
It needs a function L to label the nodes
(node-labeling), and f to compute the distance
between two labels (distance-labeling).
f does not need to access the whole graph, as
we pointed out. It only needs to know the family
of the graph.
For general graphs, the encoding of the
labelling is of size ?(n) and the distance can be
computed in O(log log n).
For graphs with a separator of size k, the
encoding of the labelling is of size O(k log n
log²n) and the distance can be computed in O(log
n).
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
A small separator set can be found in the grid
graph H.
The routing algorithm will send messages
between cells of H, using the shortest path.
We know that the real nodes represented by a
cell are fully connected, hence going from one
cell to an adjacent one takes at most 2 hops in G.
How to find the shortest path without a big
overhead on the routing table?
Each cell remembers the distance to the
vertices in the separator set. They work like a
gateway.
Two vertices in different halves have to go
communicate through a gateway thus they can know
their shortest path distance.
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UDG and quasi-UDGs Grid-graph
Splitting Algorithm Routing
Now that you have the idea on the overall
graph, just think that we apply the same one
recursively to cut the graph into O(log n) levels.
Let denote by basic block the lowest level.
Each node needs to store
minimum distance to the gateways on the
boundaries of all the partitions the node is in
O(vN.log n)
constant
the neighboring vertices through which to get
to other cells
a shortest path routing for the basic blocks
where it resides
constant
Let d(p) be the number of hops of a path and
c(p) the number of times it changes cells.
Let p be the optimal path and p the one by our
algorithm.
c(p) d(p) and c(p) c(p)
p travels from one cell to another in at most 2
hops hence
d(p) 2c(p) 1
d(p) 2d(p) 1
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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
Robust position-based routing in wireless ad hoc
networks with irregular transmission ranges (Lali
Barrière, Pierre Fraigniaud, Lata Narayanan,
Jaroslav Opatrny, Wireless Com. and Mobile
Computing 2003)
Distance labeling in graphs (Cyril Gavoille,
David Peleg, Stéphane Pérennes, Ran Raz, Elsevier
2004)
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