Leader Election Requires Logarithmic Time in Population Protocols

Abstract

In this paper, it is shown that any leader election problem requires logarithmic stabilization time in the population protocol model. This lower bound holds even if each agent has knowledge of the exact size of a population and we can use arbitrarily large number of agent states. This lower bound concludes that the protocol given in [Sudo et al., PODC 2019] is optimal in terms of stabilization time.

Logarithmic Expected-Time Leader Election in Population Protocol Model

“…Stabilization Time [Ang+06] O(1) O(n) [AG15] O(log 3 n) O(log 3 n) [Ali+17] O(log 2 n) O(log 5.3 n · log log n) [AAG18] O(log n) O(log 2 n) [GS18] O(log log n) O(log 2 n) [GSU18] O(log log n) O(log n · log log n) [MST18] O(n) O(log n) This work O(log n) O(log n) O(1) Ω(n) [Ali+17] < 1/2 log log n Ω(n/(polylog n)) [SM19] any large Ω(log n)…”

“…The stabilization time of [MST18] is optimal; any leader election algorithm requires Ω(log n) parallel time if it uses any large number of states and assumes the exact knowledge of population size n [SM19]. At the beginning of an execution, all the agents are in the same initial state specified by a protocol.…”

“…Therefore, simple analysis on Coupon Collector's problem shows that we cannot achieve o(log n) parallel stabilization time if an agent in the initial state is a leader. The lower bound of [SM19] shows that we cannot achieve o(log n) parallel time even if we define the initial state such that all the agents are non-leaders initially.…”

Time-Optimal Leader Election in Population Protocols

“…Agents are strongly anonymous: they do not have identifiers and they cannot distinguish their neighbors with the same states. As with the majority of studies on population protocols [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], we assume that the network of agents is a complete graph and that the scheduler selects an interacting pair of agents at each step uniformly at random.…”

“…The stabilization time of [9] is optimal; any leader election protocol requires Vðlog nÞ parallel time even if it uses any large number of states and assumes the exact knowledge of population size n [11]. At the beginning of an execution, all the agents are in the same initial state specified by a protocol.…”