Random Tree Optimization for Energy-Efficient Broadcast in All-Wireless Networks
We study optimization methods for source-initiated energy-efficient broadcast in all-wireless networks. Past studies on energy-efficient broadcast [1,3,5] focus on deterministic optimization to construct an energy-efficient broadcast tree. We present a Random Tree Optimization (RTO) approach that transforms the deterministic optimization problem into a related stochastic one. We apply the cross-entropy (CE) method [2,4] to this problem. Preliminary results show that it achieves considerable power savings compared with state-of-theart approaches. I. The Problem Formulation Given a source node and a group of intended destination nodes, say N destination nodes, in a wireless ad hoc network, the problem can be stated as how to construct a broadcast tree such that the total required energy is minimal. This problem has been proved to be NPcomplete [1]. We assume that the power level of a transmission can be chosen within a given range of values and the use of omni-directional antennas at each node. Thus, all nodes within communication range of a transmitting node can receive its transmission, which is also known as wireless broadcast advantage (WBA) [5]. II. The RTO Process We call the presented approach Random Tree Optimization (RTO) algorithm. As we will see that the algorithm operates iteratively by randomly generating improved sample trees till the stochastic process converges based on our predefined termination criteria and the performance function. The basic idea is to translate the deterministic optimization problem into a related stochastic optimization one and then use Rare Event Simulation (RES) techniques to find the solution. First, we define the performance function ) (tree F as the total required power of a tree. There are two key components in RTO algorithm: (1) the random generation of the sample trees; (2) the update of the transition probability matrix at each iteration based on the performance in the previous round. The basic idea is that if it performs well for a given transmission in the previous round, it will have higher probability to transmit for the next round. We use a Markov chain to construct a sample tree. We define )) 1 ( ) 1 (( , ) ( + × + = N N j i q Q as the one-step transition matrix, where N is the number of destination nodes and j i q , denotes the probability that there is a transmission from node i to node j . 2.1. Initialization of Transition Probability Matrix The initial matrix 0 Q can be set as follows: (a) the column corresponding to transmissions to the root node and the diagonal elements are zeros as no node transmits to itself and no node transmits to the root node;(b) for other elements j i q , , we associate it with the reciprocal of the required power for this transmission and normalize it for each column. 2.2. Random Tree Generation The random tree generation algorithm proceeds by randomly choosing a parent node based on the transition probability matrix for a given non-parented node (except the root node) among its non-descendent nodes till each destination node has a parent node. 2.3. Update of Transition Probability Matrix At each iteration of the RTO algorithm based on the CE method, we need to calculate the benchmark value of t γ as follows: } ) ) ( ( : min{ 1 ρ γ ≥ ≤ = − f T F Q f t t , (1) where ρ normally takes a value of 0.01 so that the event of obtaining high performance is not too rare. There are several choices to set the termination conditions. Normally, If for some , l t ≥ say 5 = l , l t t t − − = = = γ γ γ ... 1 , (2) then stop the optimization process. The updated value of j i q , can be estimated as: