The Complexity of Leader Election: A Chasm at Diameter Two.

Leader election is one of the fundamental problems in distributed computing. In its implicit version, only the leader must know who is the elected leader. This paper focuses on studying the message complexity of leader election in synchronous distributed networks, in particular, in networks of diameter two. Kutten et al. [JACM 2015] showed a fundamental lower bound of Omega (m) (m is the number of edges in the network) on the message complexity of (implicit) leader election that applied also to Monte Carlo randomized algorithms with constant success probability; this lower bound applies for graphs that have diameter at least three. On the other hand, for complete graphs (i.e., diameter 1), Kutten et al. [TCS 2015] established a tight bound of (Theta) over tilde(root n)(1) on the message complexity of randomized leader election (n is the number of nodes in the network). For graphs of diameter two, the complexity was not known. In this paper, we settle this complexity by showing a tight bound of (Theta) over tilde (n) on the message complexity of leader election in diameter two networks. We first give a simple randomized Monte-Carlo leader election algorithm that with high probability (i.e., probability at least 1 - n(-c), for some positive constant c) succeeds and uses O (n log(3) n) messages and runs in O(1) rounds; this algorithm works without knowledge of n (and hence needs no global knowledge). We then show that any algorithm (even Monte Carlo randomized algorithms with large enough constant success probability) needs Omega(n) messages (even when n is known), regardless of the number of rounds. We also present an O(n log n) messages deterministic algorithm that takes O(log n) rounds (but needs knowledge of n); we show that this message complexity is tight for deterministic algorithms. Our results show that leader election can be solved in diameter two graphs in (essentially) linear (in n) message complexity and thus the Omega(m) lower bound does not apply to diameter-two graphs. Together with the two previous results of Kutten et al., our results fully characterize the message complexity of leader election vis-a-vis the graph diameter.