GAMesh

Mesh Networks Goal Graph Drawing Simulations Conclusion
GAMesh: Automatic Placement of Wireless Mesh
Nodes via Genetic Algorithms
Giuseppe De Marco
Department of Information System Engineering,
Toyota Technological Institute, Nagoya, Japan
NbiS2008, Turin, Italy
September 1 - 5, 2008
GAMesh: Automatic Placement of Wireless Mesh Nodes via Genetic Algorithms Giuseppe De Marco

Outline
1 Mesh Networks
2 Main Objective
3 Graph Drawing
GAMESH: Coding
GAMesh: Fitness function
4 Simulation
5 Conclusions

Mesh Networks
• Proﬁtable Ad hoc networks
• Extend Internet connectivity via radio links (last mile problem)
• Rural villages, communities, nomadic users
• Very cost eﬀective: installation/area/bps
Antenna type Cost
Omnidirectional 10k ∼ 100k × km2
Directional 1k ∼ 10k × km2

Mesh Networks
Mesh Node (MN)
Mesh Portal
Internet connection
Test Point (TP) or Mesh Client
Usually divided in backhaul and access network:
• Backhaul has diﬀerent channels/ tx. power
• Bakchaul can have diﬀerent spatial reuse

Our goals
• How to displace the mesh nodes?
• Some Theoretical results
• Gupta & Kumar and al. → nodes density vs. transmission power
• Extended to directive antennae [H-N. Dai et al., INFOCOM2008]
• Which constraints do matter?

Our goals
• How to displace the mesh nodes?
• Some Theoretical results
• Gupta & Kumar and al. → nodes density vs. transmission power
• Extended to directive antennae [H-N. Dai et al., INFOCOM2008]
• Which constraints do matter?
• Connectivity of backhauls

Our goals
The displacement problem can be defined as follow.
Definition
In general, the problem is like a vertex-covering problem. Given a set Vacc
of m Test Points (TP), and a set V of m MNs to be deployed in A, find
the best arrangement of MNs s.t. the graph G(V, E) is connected and
every v ∈ Vacc has a link to an u ∈ V (covering)
TP
MN

Related work
• Choose a set of Candidate Site (CS) and ﬁnd the minimum subset
• Linear/Mixed Programming with some constraints1
• Cons: consider only a subset of available points. Measurement
campaign is needed
• Our Contribute: Consider all available CS and use a simpler
modeling
1
E. Amaldi, A. Capone, M. CESANA, I. Filippini, F. Malucelli, Optimization Models
and Methods for Planning Wireless Mesh Networks, Computer Networks, Elsevier, Vol.
52, Issue 11, August 2008, Page(s) 2159-2171

To our aim, we use GA. Why?
• Linear Programming unfeasible.
• Other techniques (force-directed drawing) hard to adapt to constraints
GAMesh synopsis:
• coding based on 2D-square of nodes
• cross-over based on squares-content swapping
• several mutation methods
• constraints via penalty functions

GAMesh: Coding
4
1
2
5
3
6
7
x
y
s = (A, x, y)
1 8
1
2 4 3 2 5 7 5
4 2 1 6 4 2 8
8A =













0 1 1 1 0 0 0 0
1 0 1 0 1 0 0 0
1 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 1 0 0 0 1 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0













Figure: Chromosome coding style.
A: adjacency matrix

GAMesh: Cross-over
1
1
8
8
2
5
3
6
7 4
1
6
5
3
4
6
5
5
3
6
3
1
1
4
4
7
2
2
7
1
7
2
rect1 rect1
rect2 rect2
parent 1 parent 2
child 1 child 2
Standard cross-over mechanisms (string based) not work (similar to TSP
problem).

GAMesh: Mutation
1 SingleMutate: picks up a single node and move it in a random
position in the plane.
2 RectMutate: picks up two squares in the plane and move the content
of one square to the other one.
3 SmallMutate: picks up a single node and move it by a ﬁxed step (=3).
4 SmallRectMutate: picks a square of nodes and move it by a ﬁxed step
(=3).

GAMesh uses combinatorial optimization, i.e. the optimization is tied to
the combination of nodes. What function to optimize?
Hints (constraints)
1 The network must be connected
2 The network must cover all TPs
3 The network should satisfy other constraints
• Minimum interference
• Minimum installation cost, K

Some notes
• Giant component C0 of G(V, E) → the largest connected component
• Degree Di = j aij
Figure: |C0| = 5
8
i

It’s very simple to recognize that the key parameter is maxi(D) and C0
• ↓ maxi(D) ⇒↓ K
• For omnidirectional antenna
• ↓ maxi(D) ⇒↓ Di ⇒↓ Interference
• For directive antenna we suppose that interference is neglectable
We can set also contraints on D to make redundant the network (not a
tree)

Fitness function
We use a threshold Th on C0: bad individuals have C0 < 1
We can derive the ﬁtness function by imposing that
−
max(D)
n − 1
+ w(p0) > 1 (1)
∂w
∂p0
< 0 (2)
f(s) =
|C0|
n , if |C0| < Th
−max(D)
n−1 + 2(2 − p0)2, if |C0| ≥ Th
(3)
n number of MNs
p0 = #isolated
m
Th threshold

Constraints
Use of a penalty function
f(s) = −
max(D)
n − 1
+ 2(2 − p0)2
+
O
i=1
ǫiΦi , |C0| ≥ Th (4)
In this work:
• Constraint on the access network capacity, θi
• Constraint on the backhaul network capacity, ρij

By assuming perfect scheduling and ρij = ρ, θi = θ:
Φ1 =

1 −
1
ρnD (i,j)∈E
(fi + fj)


Φ2 = 1 −
λ0d
θ
where fi is the traﬃc demand of access net.
Di=2
id =5

Simulation
Table: GAMesh parameters
Initial pop. Mutation rate Crossover square(m) Grid step(m)
50 0.02 8 20
Table: Simulation parameters. Bit rates are in Mbps.
Service Area(m2) Position of TPs n MPs λ0 θ ρ
L2 = 8002 l1 = L
2 , l2 = L 15 1 0.5 11 54
User traﬃc demand λ0, access traﬃcf = λ0θd. GAMesh uses a
termination condition based on relative improvements.

Simulations: Other parameters
Radio parameters
L ≤ LTh Pt − PRX ,
where PRX is the sensitivity of the receiver end.
We set LTh = 55dB and LTh = 48dB, for the BN and the AN, respectively.

Simulations: Sample topologies
0 100 200 300 400 500 600 700 800
0
100
200
300
400
500
600
700
800
1
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3 4
5
6 7
8
9
10 11
12 13
14 15
1617
(a) 2nd
0 100 200 300 400 500 600 700 800
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(b) 12th
0 100 200 300 400 500 600 700 800
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(c) 20th
0 100 200 300 400 500 600 700 800
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1617
(d) 30th (last)GAMesh: Automatic Placement of Wireless Mesh Nodes via Genetic Algorithms Giuseppe De Marco

Other metrics
0 50 100 150 200
5
6
7
8
9
10
max(f(s))
iteration
0 50 100 150 200
0
0.2
0.4
0.6
0.8
1
%ofconverged
max(f(s))
(a)
0 50 100 150 200
0
5
10
mean(d)
iteration
0 50 100 150 200
2
4
6
max(D)
mean(d)
(b)
Figure: Metrics.

0 20 40 60 80 100 120 140 160 180 200
0
10
20
30
40
50
60
70
80
90
100
%
iteration
min(n
0
)
Median n
0
Median(1−Tav
)
Median(1−T
v
)
Table: GAMesh iterations
(ite) for diﬀerent values of
the grid size.
grid size ite Var(ite)
50 158.60 42.3089
80 180 80.1
100 179 60.0462

Conclusions
In this work, we presented GAMesh, a framework for wireless mesh
network design
• Algorithm features
• Graph combination through single objective function
• GAMesh explores all points in the plane
• The running time depends on n and pop. size
• It does not depend on the grid size!
• Multiple constraints embedded in f(s)
• In regard to wireless mesh net
• Few parameters, D and p0, can describe the problem (interference,
cost, traﬃc...)
• No need to make measurements (in theory)

Conclusions
Further investigations
• Consider the Digital Elevation Map of the area...
• ...consider also the height of the MN
• Abstract model of radio propagation
• Flow graph (for testing routing feasibility)
• Multiobjective, vertex-covering + GAMesh, i.e. automatic setting of
the minimum of n

GAMesh

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