MINIMUM POWER
BROADCAST IN
WIRELESS NETWORKS
Arindam K. Das
CIA Lab
University of Washington
Seattle, WA
MINIMUM POWER
BROADCAST IN
WIRELESS NETWORKS
with
Robert J. Marks II & M.A. El-Sharkawi
(UW CIA)
Payman Arabshahi & Andrew Gray
(JPL/NASA)
Problem Statement
For a designated host and a
broadcast application, find the
connection tree which requires
minimum overall transmission
power.
Example : Minimum Power
Broadcast
Broadcast tree : A  B, C  D
F
C
E
A
D
B
Assumptions (1)
• We assume that there is a fixed source
node which wants to communicate with all
the other nodes in the wireless network
(broadcast).
• All nodes have omni-directional antennas.
• Power is expended for signal
transmission only. No power expenditure
for signal reception or processing.
Assumptions (2)
• The transmitter power is modeled as the
‘’ power of its distance from the receiver
(2    4).
PT  r
Proposed Approach
• We propose a GA based approach for
solving the minimum power broadcast
problem.
• Key question: Encoding of
chromosomes
Some Definitions
• Power matrix, P: The (i,j)th element of the
power matrix is defined as
Pij = rij
where rij is the Euclidean distance between
nodes i and j.
• Cut vector, P: The cut vector, referenced
to P, is an N-element integer vector. It
indicates the location of an element on each
row of the power matrix.
Examples
P
P = [7 2 3 4 3 5 6]
Some Definitions
• Threshold vector, t : An N-element vector
of the elements of P specified by the cut
vector. Represents power settings of the
individual nodes.
• Cost of a cut, c(P) : Sum of the elements
of the threshold vector.
Examples
P
P = [7 2 3 4 3 5 6]
t = [8 0 0 0 2 0 0]
Some Definitions
• Transfer matrix, H: The transfer matrix is
computed by thresholding the power matrix
as follows:
1, if Pij  ti
H ij  
0, otherwise
• Viability of a cut vector: A cut is viable if it
allows all destination nodes to be reached.
Otherwise, it is non-viable. A viable cut
vector has an associated connection tree.
Examples
P
P = [7 2 3 4 3 5 6]
t = [8 0 0 0 2 0 0]
Solution Approach
“As Implemented”
• GA based
• Chromosome encoding : cut vectors, P.
• Crossover : random 1-point crossover,
subject to a certain crossover probability.
• Parent selection : roulette wheel
• Fitness function : c(P)
• Mutation : none
• Elitism : yes
Viability of the Children
• Randomly generated cut vectors need not
be viable  the children created after
crossover and mutation need not correspond
to viable connection trees.
• Use the Viability Lemma to determine the
viability of a child.
- If viable, accept it.
- If not, reject it, or, apply a repair operator.
Viability of the Children
A Repair Strategy
• Suppose a node (say n) is not reached by
a cut.
• Identify the node closest to n (say m).
• Augment the power level of m so that node
n is reached and modify the mth element of
the cut accordingly.
Viability Lemma (1)
• Notation
k = iteration index
 = N-element binary node coverage vector
• Nodes which are reached are tagged by
a ‘1’ in the coverage vector. Nodes not
reached are tagged by a ‘0’.
Viability Lemma (2)
• Initialize (0) = [0 0 .. 1.. 0 0].
All elements, except that corresponding to
the source, are set to 0.
•   logical product of two matrices
(multiplications replaced by AND’s and
additions replaced by OR’s).
• Apply the iteration
(k+1) = HT  (k)
Viability Lemma (3)
• Necessary and sufficient condition for
a cut to be viable (assuming broadcast
application)

(N -1) = 1
• The iteration process terminates if
(K)

= 1, K  N  1
Generating the Initial Gene Pool
• The initial gene pool is generated using an
iterative, random node selection method
(the Stochastic Tree Generation
algorithm).
• Rules:
– First transmission must be from source.
– A node can transmit only once.
– A transmitting node, in general, can opt to be a
leaf, if choosing so does not render the tree
nonviable.
Generating the Initial Gene Pool
Example
Iteration 1
• Assume node 1 is the source.
1
Possible Transmitting
Nodes
2, 3, 4, 5, 6
Possible Destination
Nodes
• Transmitting node = 1
• Randomly chosen destination node = 3
Generating the Initial Gene Pool
Example
Iteration 2
• Assume 1  3 also reaches node 4.
3, 4
Possible Transmitting
Nodes
2, 3, 5, 6
Possible Destination
Nodes
• Randomly chosen transmitting node = 3
• Randomly chosen destination node = 3
Generating the Initial Gene Pool
Example
Iteration 3
• Assume 4  6 also reaches node 5.
4
Possible Transmitting
Nodes
[ …], 5, 6
Possible Destination
Nodes
• Randomly chosen transmitting node = 4
• Randomly chosen destination node = 6
Generating the Initial Gene Pool
Example
• Converting the transmission sequence to a
cut vector, P.
1
2
3
4
3
13
2
33
3
46
6
5
5
6
6
Simulation Results
• Simulations on 50 randomly generated 25-node
and 50-node networks show an improvement of
approximately 10% and 13% over the solutions
generated using the Broadcast Incremental
Power algorithm proposed by Wieselthier et al.
• Simulations were conducted using 100
chromosomes and 50 evolutions.
Summary
• Discussed a GA based search method for
solving the minimum power broadcast
problem in wireless networks.
• Discussed the Stochastic Tree Generation
algorithm for generating the initial population.
Solutions from other heuristics can be included
in the initial population.
• Discussed the computationally simple
Viability Lemma for determining the viability
of the children.