Energy-Optimal Distributed Algorithms for Minimum Spanning Trees

Abstract

Traditionally, the performance of distributed algorithms has been measured in terms of time and message complexity.Message complexity concerns the number of messages transmitted over all the edges during the course of the algorithm. However, in energy-constrained ad hoc wireless networks (e.g., sensor networks), energy is a critical factor in measuring the efficiency of a distributed algorithm. Transmitting a message between two nodes has an associated cost (energy) and moreover this cost can depend on the two nodes (e.g., the distance between them among other things). Thus in addition to the time and message complexity, it is important to consider energy complexity that accounts for the total energy associated with the messages exchanged among the nodes in a distributed algorithm. This paper addresses the minimum spanning tree (MST) problem, a fundamental problem in distributed computing and communication networks. We study energy-efficient distributed algorithms for the Euclidean MST problem assuming random distribution of nodes. We show a non-trivial lower bound of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ω(log n)</i> on the energy complexity of any distributed MST algorithm. We then give an energy-optimal distributed algorithm that constructs an optimal MST with energy complexity <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log n)</i> on average and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log n log log n)</i> with high probability. This is an improvement over the previous best known bound on the average energy complexity of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ω(log²). Our energy-optimal algorithm exploits a novel property of the giant component of sparse random geometric graphs. All of the above results assume that nodes do not know their geometric coordinates. If the nodes know their own coordinates, then we give an algorithm with <i>O(1)</i> energy complexity (which is the best possible) that gives an <i>O(1)</i> approximation to the MST.</i>