Energy-Efficient Distributed Constructions of Minimum Spanning Tree for Wireless Ad-hoc Networks

The Minimum Spanning Tree (MST) problem is one of the most important and commonly occurring primitive in the design and operation of data and communication networks. While there are distributed algorithms for the MST problem these require relatively large number of messages and time, and are fairly involved, require synchronization and a lot of book keeping; this makes these algorithms impractical for emerging technologies such as ad hoc and sensor networks. In such networks, a sensor has very limited power, and any algorithm needs to be simple, local and energy efficient for being practical. Motivated by these considerations, we study the performance of a class of simple and local algorithms called Nearest Neighbor Tree (NNT) algorithms for energy-efficient construction of MSTs in a wireless ad hoc setting. These employ a very simple idea to eliminate the work involved in cycle detection in other MST algorithms: each node chooses a distinct rank, and connects to the closest node of higher rank. We consider two variants of the NNT algorithms, obtained by two ways of choosing the ranks: (i) Random NNT, in which each node chooses a rank randomly, and (ii) Directional NNT, in which each node uses directional information for choosing the rank. We show provable bounds on the performance of these algorithms in instances obtained by uniformly distributed points in the unit square. Finally, we perform extensive simulations of our algorithms. We tested our algorithms on both uniformly random distributions of points, and on realistic distributions of points in an urban setting. The cost of the tree found by the NNT algorithms is within a factor of 2 of the MST, but there is more than a ten-fold saving on the energy and about a five fold saving on the number of messages sent. Also, our algorithms are significantly simpler to implement compared to, for instance, the GHS algorithm, which is essentially optimal with regards to the message complexity. Thus, our results demonstrate the first such tradeoff between the quality of approximation and the energy cost for spanning trees on ad hoc networks, and motivates similar considerations for M. Khan and G. Pandurangan are with the Department of Computer Sciences, Purdue University, West Lafayette, IN 47907, USA. E-mail: {mmkhan, gopal}@cs.purdue.edu. V.S. Anil Kumar is with the Basic and Applied Simulation Science (CCS-5) and National Infrastructure and Analysis Center (NISAC), Los Alamos National Laboratory, MS M997, P.O. Box 1663, Los Alamos, NM 87545. E-mail: [email protected] other important problems.