Arindam K. Das

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MINIMUM POWER BROADCAST IN WIRELESS NETWORKS

• Arindam K. Das
• CIA Lab
• University of Washington
• Seattle, WA

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MINIMUM POWER BROADCAST IN WIRELESS NETWORKS

• with
• Robert J. Marks II M.A. El-Sharkawi (UW CIA)
• Payman Arabshahi Andrew Gray (JPL/NASA)

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Problem Statement
For a designated host and a broadcast
application, find the connection tree which
requires minimum overall transmission power.
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Example Minimum Power Broadcast

Broadcast tree A ? B, C ? D
F
C
E
A
D
B
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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.

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Assumptions (2)

• The transmitter power is modeled as the ? power
  of its distance from the receiver (2 ? ? ? 4).

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Proposed Approach

• We propose a GA based approach for solving the
  minimum power broadcast problem.
• Key question Encoding of chromosomes

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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.

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Examples

P
?P 7 2 3 4 3 5 6
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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.

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Examples

t 8 0 0 0 2 0 0
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Some Definitions

• Transfer matrix, H The transfer matrix is
  computed by thresholding the power matrix as
  follows

• 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.

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Examples

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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

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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.

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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.

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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.

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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 ANDs and additions
  replaced by ORs).
• Apply the iteration

?(k1) HT ? ?(k)
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Viability Lemma (3)

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

• The iteration process terminates if

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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.

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Generating the Initial Gene PoolExample

• Iteration 1
• Assume node 1 is the source.

• Transmitting node 1

• Randomly chosen destination node 3

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Generating the Initial Gene PoolExample

• Iteration 2
• Assume 1 ? 3 also reaches node 4.

• Randomly chosen transmitting node 3

• Randomly chosen destination node 3

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Generating the Initial Gene PoolExample

• Iteration 3
• Assume 4 ? 6 also reaches node 5.

• Randomly chosen transmitting node 4

• Randomly chosen destination node 6

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Generating the Initial Gene PoolExample

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

1 2 3 4 5 6
3 2 3 6 5 6
1 ? 3 3 ? 3 4 ? 6
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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.

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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.