Synchronization and Symmetry Breaking in Distributed Systems

Abstract

An emerging characteristic of modern computer systems is that it is becoming ever more frequent that the amount of communication involved in a solution to a given problem is the determining cost factor. In other words, the convenient abstraction of a random access memory machine performing sequential operations does not adequately reflect reality anymore. Rather, a multitude of spatially separated agents cooperates in solving a problem, where at any time each individual agent has a limited view of the entire system’s state. As a result, coordinating these agents’ efforts in a way making best possible use of the system’s resources becomes a fascinating and challenging task. This dissertation treats of several such coordination problems arising in distributed systems. In the clock synchronization problem, devices carry clocks whose times should agree to the best possible degree. As these clocks are not perfect, the devices need to perpetually resynchronize by exchanging messages repeatedly. We consider two different varieties of this problem. First, we examine the problem in sensor networks, where for the purpose of energy conservation it is mandatory to reduce communication to a minimum. We give an algorithm that achieves an asymptotically optimal maximal clock difference throughout the network using a minimal number of transmissions. Subsequently, we explore a worst-case model allowing for arbitrary network dynamics, i.e., network links may fail and (re)appear at arbitrary times. For this model, we devise an algorithm achieving an asymptotically optimal gradient property. That is, if two devices in a larger network have access to precise estimates of each other’s clock values, their clock difference is much smaller than the maximal one. Naturally, this property can only hold for devices that had such estimates for a sufficiently long period of time. We prove that the time span necessary for our algorithm to fully establish the gradient property when better estimates become available is also asymptotically optimal. Many load balancing tasks can be abstracted as distributing n balls as evenly as possible into n bins. In a distributed setting, we assume the balls and bins to act as independent entities that seek to coordinate at a minimal communication complexity. We show that under this constraint, a natural class of algorithms requires a small, but non-constant number of communication rounds to achieve a constant maximum bin load. We complement the respective bounds by demonstrating that if any of the preconditions of the lower bound is dropped, a constant-time solution is possible. Finally, we consider two basic combinatorial structures, maximal independent sets and dominating sets. A maximal independent set is a subset of the agents containing no pair of agents that can communicate directly, while there is no agent that can be added to the set without destroying this property. A dominating set is a subset of the agents that—as a whole—can contact all agents by direct communication. For several families of graphs, we shed new light on the distributed complexity of computing dominating sets of approximatively minimal size or maximal independent sets, respectively.