An optimal randomized ranking algorithm on the k-channel broadcast communication model

A broadcast communication model (BCM) is a distributed system with no central arbiter populated by n processing units referred to as stations. The stations can communicate by broadcasting/receiving data packets in one of k communication channels. We assume that the stations run on batteries and expands power while broadcasting/receiving a data packet. Thus, the most important measure to evaluate algorithms on the BCM is the number of awake time slots, in which a station is broadcasting/receiving a data packet. We also assume that the stations are identical and have no unique ID number, and no station knows the number n of the stations. For given n keys one for each station, the ranking problem asks each station to determine the number of keys in the BCM smaller than its own key. The main contribution of the paper is to present an optimal randomized ranking algorithm on the k-channel BCM. Our algorithm solves the ranking problem, with high probability, in O(n/k+log n) time slots with no station being awake for more than O(log n) time slots. We also prove that any randomized ranking algorithm is required to run in expected Ω(n/k+log n) time slots with at least one station being awake for expected Ω(log n) time slots. Therefore, our ranking algorithm is optimal.