A Population Protocol for Exact Majority with O(log5/3 n) Stabilization Time and Theta(log n) States

Reaching Agreement in Competitive Microbial Systems

“…Alistarh et al [2] gave the first protocols with expected converge time of the order polylog n using polylog n states. This result has been gradually improved [3,11,12], and recently, Doty et al [17] gave protocols that solve majority in O(log n) time using O(log n) states. In the spatial setting, Alistarh et al [4] showed how to solve exact majority in polylog n parallel time using polylog n states per node in regular graphs with constant conductance.…”

“…Time complexity in this model is studied under a stochastic scheduler that picks two individuals to interact uniformly at random. The (parallel) time complexity is given by the expected number of steps to reach a stable configuration divided by the total population size n. A long series of papers have investigated how fast majority can be computed by protocols that use a given number of states [11,12,17,18]. Many of the recent results and techniques in this area are surveyed by Elsässer and Radzik [19] and Alistarh and Gelashvili [1].…”

A survey of size counting in population protocols

“…Stabilization can take Θ(n) time in the worst case, but it takes only O(log n) time for all agents to hold three consecutive values (two of which are ⌊m/n⌋ or ⌈m/n⌉) [25,29]. The discrete averaging technique has been crucial in a number of polylogarithmic-time protocols for problems such as population size counting [22,23] and majority related problems [13,17,28,29] and its time complexity has been tightly analyzed [17,25,26,30].…”

“…In the original model of population protocols [4], the states and transitions are constant with respect to the population size n. However, recent studies use a variant of the model allowing the number of states and transitions to grow with n. One motivation to study population protocols with ω(1) states is the existence of impossibility results showing that no constant-state protocol can stabilize in sublinear time with probability 1 for problems such as leader election [5], majority [6], or computation of more general predicates and integer-valued functions [7]. 3 There have been recent algorithmic advances using non-constant states [6,[8][9][10][11][12][13][14][15][16][17][18][19][20], that lead to time-and space-optimal solutions for the problems of leader election [19] and majority [18]. However, most of these solutions [6,8,9,[11][12][13][14][15][16][17][18][19][20] propose a nonuniform protocol.…”

“…3 There have been recent algorithmic advances using non-constant states [6,[8][9][10][11][12][13][14][15][16][17][18][19][20], that lead to time-and space-optimal solutions for the problems of leader election [19] and majority [18]. However, most of these solutions [6,8,9,[11][12][13][14][15][16][17][18][19][20] propose a nonuniform protocol. Rather than being a single set of transitions, a nonuniform protocol is actually a family of protocols, where the set of transition rules 2 For the problems of leader election and population size computation, all agents start in the same state in a valid initial configuration.…”