On the Energy Consumption of Fast Convergecast in Wireless Networks

Kesselman and Kowalski (JPDC ´06) have proposed a simple distributed randomized algorithm for fast convergecast in wireless networks. The Kesselman-Kowalski algorithm works in a synchronized mode under the assumptions that all wireless nodes have variable transmission range, each can learn the distance to the closest active neighbor at any time, and all have a special collision detection (CD) capability so that a transmitting node can detect a collision within its transmission range.They prove that their algorithm finishes in O(log n) expected timeslots and that, assuming the nodes are located in a Euclidean plane, the total energy used is O(n log n) times the minimum energy of a (not necessarily fast) convergecast. They also provide instances where an Omega(n/log n) gap exists between the energy of any fast (that is, with O(log n) timeslots) convergecast, and the minimum energy convergecast. We prove that, when compared to the most energy efficient fast convergecast, the Kesselman-Kowalski algorithm uses O(log² n) times the optimum energy. With the additional assumption that n, or a good estimate of it, is known, we modify the Kesselman-Kowalski algorithm such that it also finishes in O(log n) expected timeslots, and uses O(n) times the energy of a minimum energy convergecast. We also provide a centralized algorithm which produces a fast convergecast using n/log n times the minimum energy convergecast. All the above bounds (including those from Kesselman-Kowalski) assume that the energy spent for sending a packet from a node to another is proportional to |u, u²| (the square of the Euclidean distance).