Distributed Algorithms for Constructing Approximate Minimum Spanning Trees in Wireless Networks

While there are distributed algorithms for the Minimum Spanning Tree (MST) problem, these algorithms require relatively large number of messages and time, and are fairly involved, making them impractical for resource-constrained networks such as wireless sensor networks. In such networks, a sensor has very limited power, and any algorithm needs to be simple, local, and energy efficient. Motivated by these considerations, we design and analyze a class of simple and local distributed algorithms called Nearest Neighbor Tree (NNT) algorithms for energy-efficient construction of an approximate MST in wireless networks. Assuming that the nodes are uniformly distributed, we show provable bounds on both the quality of the spanning tree produced and the energy needed to construct them. We show that while NNT produces a close approximation to the MST, it consumes asymptotically less energy than the classical message-optimal distributed MST algorithm due to Gallager, Humblet, and Spira. Further, the NNTs can be maintained dynamically with polylogarithmic rearrangements under node insertions/deletions. We also perform extensive simulations, which show that the bounds are much better in practice. Our results, to the best of our knowledge, demonstrate the first tradeoff between the quality of approximation and the energy required for building spanning trees on wireless networks, and motivate similar considerations for other important problems.