Fast Graphical Population Protocols

Abstract

Let G be a graph on n nodes. In the stochastic population protocol model, a collection of n indistinguishable, resource-limited nodes collectively solve tasks via pairwise interactions. In each interaction, two randomly chosen neighbors first read each other's states, and then update their local states. A rich line of research has established tight upper and lower bounds on the complexity of fundamental tasks, such as majority and leader election, in this model, when G is a clique. Specifically, in the clique, these tasks can be solved fast, i.e., in n polylog n pairwise interactions, with high probability, using at most polylog n states per node.In this work, we consider the more general setting where G is an arbitrary graph, and present a technique for simulating protocols designed for fully-connected networks in any connected regular graph. Our main result is a simulation that is efficient on many interesting graph families: roughly, the simulation overhead is polylogarithmic in the number of nodes, and quadratic in the conductance of the graph. As a sample application, we show that, in any regular graph with conductance φ, both leader election and exact majority can be solved in φ −2 • n polylog n pairwise interactions, with high probability, using at most φ −2 • polylog n states per node. This shows that there are fast and space-efficient population protocols for leader election and exact majority on graphs with good expansion properties. We believe our results will prove generally useful, as they allow efficient technology transfer between the well-mixed (clique) case, and the under-explored spatial setting.

Population Protocols for Graph Class Identification Problems

“…In most distributed systems, it is essential to understand properties of the communication graph in order to design efficient algorithms. Actually, in the population protocol model, efficient protocols are proposed with limited communication graphs (e.g., ring graphs and regular graphs) [2,6,13,14]. In the population protocol model, the computability of the graph property was first considered in [3].…”

“…In the population protocol model, researchers studied various fundamental problems such as leader election problems [1,11,15,18], counting problems [7,8,9], majority problems [5,10,17], etc. In [2,6,13,14], researchers proposed efficient protocols for such fundamental problems with limited communication graphs. More concretely, Angluin et al proposed a protocol that constructs a spanning tree with regular graphs [6].…”

“…Alistarh et al showed that protocols for complete graphs (including the leader election protocol, the majority protocol, etc.) can be simulated efficiently in regular graphs [2].…”

“…Case ( 2) is that an agent having an L t l token (resp., an L t r token) interacts with an agent that does not have the L t se token (resp., the L t se ′ token). Figure 3(A-2) and (B-2) shows an example of case (2). By the interaction, agents make the L t l token (resp., the L t r token) transition to an L l token (resp., an L r token) by the behavior of lines 13-14 or 31-37, and thus the condition in line 25 or 28 is never satisfied in the trial.…”

“…In case (2), if star of an agent transitions to never, star of each agent converges to never similarly to case (1); otherwise, since initially star of each agent is no, star of each agent converges to no. Therefore, the lemma holds.…”