In this paper, we focus on the uniform bipartition problem in the population protocol model. This problem aims to divide a population into two groups of equal size. In particular, we consider the problem in the context of \emph{arbitrary} communication graphs. As a result, we clarify the solvability of the uniform bipartition problem with arbitrary communication graphs when agents in the population have designated initial states, under various assumptions such as the existence of a base station, symmetry of the protocol, and fairness of the execution. When the problem is solvable, we present protocols for uniform bipartition. When global fairness is assumed, the space complexity of our solutions is tight.
We focus on a uniform partition problem in a population protocol model. The uniform partition problem aims to divide a population into k groups of the same size, where k is a given positive integer. In the case of k=2 (called uniform bipartition), a previous work clarified space complexity under various assumptions: 1) an initialized base station (BS) or no BS, 2) weak or global fairness, 3) designated or arbitrary initial states of agents, and 4) symmetric or asymmetric protocols, except for the setting that agents execute a protocol from arbitrary initial states under weak fairness in the mo…
We study population protocols: networks of anonymous agents that interact under a scheduler that picks pairs of agents uniformly at random. The _size counting problem_ is that of calculating the exact number $n$ of agents in the population, assuming no leader (each agent starts in the same state). We give the first protocol that solves this problem in sublinear time. The protocol converges in $O(\log n \log \log n)$ time and uses $O(n^{60})$ states ($O(1) + 60 \log n$ bits of memory per agent) with probability $1-O(\frac{\log \log n}{n})$. The time complexity is also $O(\log n \log \log n)$ in…
We study uniform population protocols: networks of anonymous agents whose pairwise interactions are chosen at random, where each agent uses an identical transition algorithm that does not depend on the population size $n$. Many existing polylog$(n)$ time protocols for leader election and majority computation are nonuniform: to operate correctly, they require all agents to be initialized with an approximate estimate of $n$ (specifically, the exact value $\lfloor \log n \rfloor$). Our first main result is a uniform protocol for calculating $\log(n) \pm O(1)$ with high probability in $O(\log^2 n)…
Population protocols are a model for distributed computing that is focused on simplicity and robustness. A system of $n$ identical agents (finite state machines) performs a global task like electing a unique leader or determining the majority opinion when each agent has one of two opinions. Agents communicate in pairwise interactions with randomly assigned communication partners. Quality is measured in two ways: the number of interactions to complete the task and the number of states per agent. We present protocols for the majority problem that allow for a trade-off between these two measures.…
In this work, we initiate the study of \emph{smoothed analysis} of population protocols. We consider a population protocol model where an adaptive adversary dictates the interactions between agents, but with probability $p$ every such interaction may change into an interaction between two agents chosen uniformly at random. That is, $p$-fraction of the interactions are random, while $(1-p)$-fraction are adversarial. The aim of our model is to bridge the gap between a uniformly random scheduler (which is too idealistic) and an adversarial scheduler (which is too strict). We focus on the fundamen…
We consider the problem of counting the population size in the population model. In this model, we are given a distributed system of $n$ identical agents which interact in pairs with the goal to solve a common task. In each time step, the two interacting agents are selected uniformly at random. In this paper, we consider so-called uniform protocols, where the actions of two agents upon an interaction may not depend on the population size $n$. We present two population protocols to count the size of the population: protocol Approximate, which computes with high probability either $\lfloor\log n…
The model of population protocols refers to a large collection of simple indistinguishable entities, frequently called {\em agents}. The agents communicate and perform computation through pairwise interactions. We study fast and space efficient leader election in population of cardinality $n$ governed by a random scheduler, where during each time step the scheduler uniformly at random selects for interaction exactly one pair of agents. We propose the first $o(\log^2 n)$-time leader election protocol. Our solution operates in expected parallel time $O(\log n\log\log n)$ which is equivalent to $…
We consider the standard population protocol model, where (a priori) indistinguishable and anonymous agents interact in pairs according to uniformly random scheduling. The self-stabilizing leader election problem requires the protocol to converge on a single leader agent from any possible initial configuration. We initiate the study of time complexity of population protocols solving this problem in its original setting: with probability 1, in a complete communication graph. The only previously known protocol by Cai, Izumi, and Wada [Theor. Comput. Syst. 50] runs in expected parallel time $Θ(n^…
The model of population protocols refers to the growing in popularity theoretical framework suitable for studying pairwise interactions within a large collection of simple indistinguishable entities, frequently called agents. In this paper the emphasis is on the space complexity in fast leader election via population protocols governed by the random scheduler, which uniformly at random selects pairwise interactions within the population of n agents. The main result of this paper is a new fast and space optimal leader election protocol. The new protocol utilises O(log^2 n) parallel time (which …
A population protocol can be viewed as a sequence of pairwise interactions of $n$ agents (nodes). During one interaction, two agents selected uniformly at random update their states by applying a specified deterministic transition function. In a long run, the whole system should stabilize at the correct output property. The main performance objectives in designing population protocols are small number of states per agent and fast stabilization time. We present a fast population protocol for the exact-majority problem which uses $Θ(\log n)$ states (per agent) and stabilizes in $O(\log^{5/3} n)$…
In this paper we study population protocols governed by the {\em random scheduler}, which uniformly at random selects pairwise interactions between $n$ agents. The main result of this paper is the first time and space optimal {\em exact majority population protocol} which also works with high probability. The new protocol operates in the optimal {\em parallel time} $O(\log n),$ which is equivalent to $O(n\log n)$ sequential {\em pairwise interactions}, where each agent utilises the optimal number of $O(\log n)$ states. The time optimality of the new majority protocol is possible thanks to the …
Population protocols are a formal model of sensor networks consisting of identical mobile devices. Two devices can interact and thereby change their states. Computations are infinite sequences of interactions in which the interacting devices are chosen uniformly at random. In well designed population protocols, for every initial configuration of devices, and for every computation starting at this configuration, all devices eventually agree on a consensus value. We address the problem of automatically computing a parametric bound on the expected time the protocol needs to reach this consensus. …
Order Optimal Information Spreading Using Algebraic Gossip
在本文中，我们研究了基于流言的信息传播与有限的消息大小。我们使用代数流言传播$k$不同的消息到网络中的所有$n$节点。对于任意网络，我们给出了$O((k+\log n + D)Δ)$轮的高概率一致代数流言的一个新的上界，其中$D$和$Δ$分别是网络的直径和最大度.对于许多拓扑和$k$的选择，这个界改进了以前的结果，特别是，对于一个恒定的最大度的图，它意味着一致的八卦是\emph{order optimal}和停止时间是$Θ(k + D)$。为了消除$Δ$的因素从上限，我们提出了一个非均匀的八卦协议，TAG，它是基于代数八卦和任意生成树协议$§$。TAG的停止时间是$O(k+\log n +d(§)+t(§))$，其中$t(§)$是生成树协议的停止时间，$d(§)$是生成树的直径。我们提供了两个一般情况下，这个界限导致一个顺序最优协议。首先是$k=Ω(n)$，在那里，使用一个简单的八卦广播协议，创建一个生成树在最线性的时间，我们表明，TAG完成后，$Θ(n)$轮的任何图。第二个使用一个复杂的，最近的八卦协议，建立一个快速生成树的图与大弱电导。反过来，这导致了$k=Ω(\mathrm{polylog}(n))$在这些图上的TAG的最优性。在我们的证明中使用的技术依赖于排队论，这是一种有趣的方法，可以在未来的八卦分析中使用。
In population protocols, the underlying distributed network consists of $n$ nodes (or agents), denoted by $V$, and a scheduler that continuously selects uniformly random pairs of nodes to interact. When two nodes interact, their states are updated by applying a state transition function that depends only on the states of the two nodes prior to the interaction. The efficiency of a population protocol is measured in terms of both time (which is the number of interactions until the nodes collectively have a valid output) and the number of possible states of nodes used by the protocol. By conventi…
