Arindam K. DasCIA 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)

MINIMUM POWER BROADCAST IN WIRELESS NETWORKS

Problem Statement

For a designated host and a broadcast application, find the connection tree which requires minimum overall transmission power.

Example : Minimum Power Broadcast

A

B

C

D

E

F

Broadcast tree : A B, C D

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).

rPT

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

where rij is the Euclidean distance between nodes i and j.

Pij = rij

• 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:

otherwise

PifH

,0

,1 iijij

t

• 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 ChildrenA 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)

(N -1) = 1

(K) = 1,1 NK

• The iteration process terminates if

• Necessary and sufficient condition for a cut to be viable (assuming broadcast application)

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 PoolExample

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 PoolExample

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 PoolExample

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 PoolExample

• Converting the transmission sequence to a cut vector, P.

1 3

3 3

4 6

3

2

3

6

5

6

1

2

3

4

5

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.