Some algorithms have been devised for use in a method of constructing tree graphs that represent connections among the nodes of a wireless ommunication network. These algorithms provide for determining the viability of any given candidate connection tree and for generating an initial set of viable trees that can be used in any of a variety of search algorithms (e.g., a genetic algorithm) to find a tree that enables the network to broadcast from a source node to all other nodes while consuming the minimum amount of total power. The method yields solutions better than those of a prior algorithm known as the broadcast incremental power algorithm, albeit at a slightly greater computational cost.

It is not possible to give more than a highly abbreviated and oversimplified summary of the method within the space available for this article. However, to give meaning to even this brief summary, it is necessary to present some details of the underlying rules, simplifying assumptions, and mathematical constructs in the following two paragraphs.

Each node is equipped with an omnidirectional antenna. The minimum transmitter power that enables the jth node to send information to the ith node is proportional to r αij where rij is the distance from node j to node i and α is a channel-loss exponent that usually lies between 2 and 4, the exact value depending on the nature of the signal-propagation medium. Any node in a network can be used to relay a signal to any other node. For the purposes of this method, only the transmitter power levels are calculated: the receivers are assumed to draw negligible power.

For a network of N nodes, one constructs a power or cost matrix, which is an N-by-N matrix P, of which each element Pij is proportional to the minimum power needed for communication between nodes i and j as described above. To represent connections between nodes of the network, one uses an N-element vector, denoted a cut vector, each element of which represents the location of an element in a row of the power matrix. Associated with each cut vector is a threshold vector, the elements of which are the power-matrix elements picked out by the cut vector. The cost of a cut is defined as the sum of the elements of the threshold vector. A cut is regarded as viable if it enables a broadcast to reach all nodes; otherwise, it is regarded as not viable. Another means of characterizing the network is the transfer matrix, which is an N-by-N matrix that is a function of the threshold vector.

One element of the present method is a construct denoted the viability lemma, which provides a necessary and sufficient condition for a cut to be viable. The viability lemma is implemented by equations that require power operations on the transfer matrix. In the case of a large network, power operations on a matrix can be computationally expensive. An alternative algorithm for determining viability does not require matrix power operations; instead, it requires an iterative procedure that involves examination of the rows of the transfer matrix and that is guaranteed to reach an indication of either viability or non-viability in N – 1 or fewer iterations.

Another element of the present method is the connection matrix, which is a binary N-by-N matrix representation of a tree. The connection matrix is built iteratively from a cut that has been found to be viable by one of the means described above.

Yet another element of the present method is a stochastic tree-generation algorithm that can be used to generate a viable cut vector. This is an iterative algorithm that starts with a transmission from the source node to a randomly chosen destination node, followed by heuristic selection of the next destination node at each subsequent iteration, until all intended destination nodes are reached. This algorithm converges in N – 1 or fewer iterations.

This work was done by Payman Arabshahi, Andrew Gray, Arindam Das, Mohammed El- Sharkawi, and Robert Marks II of Caltech for NASA's Jet Propulsion Laboratory.

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Refer to NPO-30455, volume and number of this NASA Tech Briefs issue, and the page number.

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Finding Minimum-Power Broadcast Trees for Wireless Networks

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This article first appeared in the April, 2004 issue of NASA Tech Briefs Magazine.

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