2. Arindam K. Das CIA Lab University of Washington Seattle, WA MINIMUM POWER BROADCAST IN WIRELESS NETWORKS

3. with Robert J. Marks II & M.A. El-Sharkawi (UW CIA) Payman Arabshahi & Andrew Gray (JPL/NASA) MINIMUM POWER BROADCAST IN WIRELESS NETWORKS

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

5. Example : Minimum Power Broadcast A B C D E F Broadcast tree : A  B, C  D

6. 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.

7. Assumptions (2) The transmitter power is modeled as the ‘  ’ power of its distance from the receiver (2    4).

8. Proposed Approach We propose a GA based approach for solving the minimum power broadcast problem. Key question: Encoding of chromosomes

9. Some Definitions Power matrix, P: The ( i,j ) th element of the power matrix is defined as where r ij is the Euclidean distance between nodes i and j. P ij = r ij  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.

10. Examples P  P = [7 2 3 4 3 5 6]

11. 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.

12. Examples P  P = [7 2 3 4 3 5 6] t = [8 0 0 0 2 0 0]

13. 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.

14. Examples P  P = [7 2 3 4 3 5 6] t = [8 0 0 0 2 0 0]

15. 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

16. 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.

17. 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 m th element of the cut accordingly.

18. 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’.

19. 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) = H T   ( k )

20. Viability Lemma (3)  ( N -1) =  ( K ) = The iteration process terminates if Necessary and sufficient condition for a cut to be viable (assuming broadcast application)

21. 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.

22. 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

23. 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

24. 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

25. Generating the Initial Gene Pool Example Converting the transmission sequence to a cut vector,  P. 1  3 3  3 4  6 323656323656 123456123456

26. 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.

27. 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.