Population stability: regulating size in the presence of an adversary

We introduce a new coordination problem in distributed computing that we call the population stability problem. A system of agents each with limited memory and communication, as well as the ability to replicate and self-destruct, is subjected to attacks by a worst-case adversary that can at a bounded rate (1) delete agents chosen arbitrarily and (2) insert additional agents with arbitrary initial state into the system. The goal is perpetually to maintain a population whose size is within a constant factor of the target size $N$. The problem is inspired by the ability of complex biological systems composed of a multitude of memory-limited individual cells to maintain a stable population size in an adverse environment. Such biological mechanisms allow organisms to heal after trauma or to recover from excessive cell proliferation caused by inflammation, disease, or normal development. We present a population stability protocol in a communication model that is a synchronous variant of the population model of Angluin et al. In each round, pairs of agents selected at random meet and exchange messages, where at least a constant fraction of agents is matched in each round. Our protocol uses three-bit messages and $\omega(\log^2 N)$ states per agent. We emphasize that our protocol can handle an adversary that can both insert and delete agents, a setting in which existing approximate counting techniques do not seem to apply. The protocol relies on a novel coloring strategy in which the population size is encoded in the variance of the distribution of colors. Individual agents can locally obtain a weak estimate of the population size by sampling from the distribution, and make individual decisions that robustly maintain a stable global population size

Simple and Efficient Local Codes for Distributed Stable Network Construction

In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of the their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks.Comment: 43 pages, 7 figure

Asynchronous Silent Programmable Matter Achieves Leader Election and Compaction

We study models and algorithms for Programmable Matter (PM), that is matter with the ability to change its physical properties (e.g., shape or optical properties) in a programmable fashion. PM can be implemented by assembling a system of weak self-organizing computational elements, called particles, that can be programmed via distributed algorithms to collectively achieve some global task. Recent advances in the production of nanotechnologies have rendered such systems increasingly possible in practice, thus triggering research interests from many areas of computer science. The most established models for PM assume that particles: are modeled as finite state automata; are all identical, executing the same algorithm based on local observation of the surroundings; live and operate in the cells of a hexagonal grid; can move from one cell to another by repeatedly alternating between a contracted state (a particle occupies one cell) and an expanded state (a particle occupies two neighboring cells). Given these elementary features, it is rather hard to design distributed algorithms even for basic tasks and, in fact, all existing solutions to solve fundamental problems via PM have resorted to endowing PM systems with various capabilities to overcome such hardness, thus assuming quite unrealistic features. In this paper, we move toward more realistic computational models for PM. Specifically, we first introduce, a new modeling approach that relaxes several assumptions used in previous ones. Second, we present a distributed algorithm to solve, in the model, a foundational primitive for PM, namely Leader Election. This algorithm works in O(n) rounds for all initial configurations of n particles that are both connected (i.e. particles induce a connected graph) and compact (i.e. without holes, that is no empty cells surrounded by particles occur). As usual in asynchronous contexts, a round is intended as the time within which all particles have been activated at least once. Third, we show that, if the initial configuration admits holes, it is impossible to achieve leader election while preserving connectivity. Finally, by slightly empowering the robots, we design an algorithm to handle initial configurations admitting holes that in O(n2) rounds solves the leader election problem while obtaining also compaction

Selecting a Leader in a Network of Finite State Machines

This paper studies a variant of the leader election problem under the stone age model (Emek and Wattenhofer, PODC 2013) that considers a network of n randomized finite automata with very weak communication capabilities (a multi-frequency asynchronous generalization of the beeping model\u27s communication scheme). Since solving the classic leader election problem is impossible even in more powerful models, we consider a relaxed variant, referred to as k-leader selection, in which a leader should be selected out of at most k initial candidates. Our main contribution is an algorithm that solves k-leader selection for bounded k in the aforementioned stone age model. On (general topology) graphs of diameter D, this algorithm runs in O~(D) time and succeeds with high probability. The assumption that k is bounded turns out to be unavoidable: we prove that if k = omega (1), then no algorithm in this model can solve k-leader selection with a (positive) constant probability