Transformations in distributed computations and applications to set agreement

In a distributed application, high-availability of a critical online service is ensured despite failures by duplicating the vital components of the server. Whilst guaranteeing the access to the server at all times, duplication requires particular care, so as to maintain a consistent state amongst all components. A distributed computation plays a key-role in preserving the consistency of the state in this case. Indeed, one of the fundamental forms of distributed computation is distributed agreement. In a distributed agreement problem, the components of the system, called processes, are supposed to all propose a value and eventually decide on a common value that depends on the proposed values. Designing correct distributed agreement algorithms relies on a mathematical model abstracting the details of the underlying physical model. In practice, there are many models, inspired from various types of physical failures. Coping with one type of failures rather than another one, for a distributed agreement algorithm, is not of equal difficulty. Indeed some types are intrinsically harder to cope with than other ones. The thesis studies how distributed algorithms can be transformed from one model to another one. The idea is to design distributed agreement algorithms in models in which the task is made as easiest and least error-prone as possible, and execute the transformed algorithms in models imposed by the underlying physical constraints. We investigate another aspect of transforming distributed computations, the notion of reducibility amongst distributed computations. Reducibility offers an elegant way for deducing lower or upper bounds on the complexity of a distributed algorithm from complexity bounds that have been proved elsewhere, about a different distributed computation or in a different model. For investigating reducibility, the thesis considers two very general classes of models: synchronous models, in which the communication delay and the process execution speed are known and bounded, and asynchronous models, in which the communication delay and the process execution speed are unknown and potentially unbounded. The thesis investigates reducibility of a distributed agreement problem called set agreement. Precisely, we consider the early-deciding variant of set agreement, in which processes must decide as fast as they can in any execution. We prove a lower bound on the complexity of early-deciding set agreement in the synchronous model. The proof works by reducing the impossibility of set agreement in the asynchronous model to the desired lower bound. In some cases, complexity bounds must still be derived with ad-hoc arguments, for example when there is no apparent connection with existing results. The thesis presents an alternative approach to reducibility, namely the connection of distributed computing with algebraic topology. Within this framework, we derive a lower bound for the complexity of early-deciding set agreement in the synchronous model. For all the lower bounds that we derive in the thesis, we show that these bounds are tight, by presenting for each lower bound an algorithm matching the complexity of the lower bound.

The Complexity of Renaming

Dan Alistarh, Seth Gilbert, Rachid Guerraoui

We study the complexity of renaming, a fundamental problem in distributed computing in which a set of processes need to pick distinct names from a given namespace. We prove an individual lower bound of Omega(k) process steps for deterministic renaming into any namespace of size sub-exponential in k, where k is the number of participants. This bound is tight: it draws an exponential separation between deterministic and randomized solutions, and implies new tight bounds for deterministic fetch-and-increment registers, queues and stacks. The proof of the bound is interesting in its own right, for it relies on the first reduction from renaming to another fundamental problem in distributed computing: mutual exclusion. We complement our individual bound with a global lower bound of Omega(k log(k/c)) on the total step complexity of renaming into a namespace of size c k, for any c >= 1. This applies to randomized algorithms against a strong adversary, and helps derive new global lower bounds for randomized approximate counter and fetch-and-increment implementations, all tight within logarithmic factors.

Ieee Computer Soc Press, Customer Service Center, Po Box 3014, 10662 Los Vaqueros Circle, Los Alamitos, Ca 90720-1264 Usa2011

Tight Bounds for Asynchronous Renaming

Dan Alistarh, Seth Gilbert, Rachid Guerraoui

This article presents the first tight bounds on the time complexity of shared-memory renaming, a fundamental problem in distributed computing in which a set of processes need to pick distinct identifiers from a small namespace. We first prove an individual lower bound of P(h) process steps for deterministic renaming into any namespace of size subexponential in k, where k is the number of participants. The bound is tight: it draws an exponential separation between deterministic and randomized solutions, and implies new tight bounds for deterministic concurrent fetch-and-increment counters, queues, and stacks. The proof is based on a new reduction from renaming to another fundamental problem in distributed computing: mutual exclusion. We complement this individual bound with a global lower bound of Omega(k log(k/c)) on the total step complexity of renaming into a namespace of size ck, for any c >= 1. This result applies to randomized algorithms against a strong adversary, and helps derive new global lower bounds for randomized approximate counter implementations, that are tight within logarithmic factors. On the algorithmic side, we give a protocol that transforms any sorting network into a randomized strong adaptive renaming algorithm, with expected cost equal to the depth of the sorting network. This gives a tight adaptive renaming algorithm with expected step complexity O(log k), where k is the contention in the current execution. This algorithm is the first to achieve sublinear time, and it is time-optimal as per our randomized lower bound. Finally, we use this renaming protocol to build monotone-consistent counters with logarithmic step complexity and linearizable fetch-and-increment registers with polylogarithmic cost.

Association for Computing Machinery2014

The Time-Complexity of Local Decision in Distributed Agreement

Partha Dutta, Rachid Guerraoui, Bastian Pochon

Agreement is at the heart of distributed computing. In its simple form, it requires a set of processes to decide on a common value out of the values they propose. The time-complexity of distributed agreement problems is generally measured in terms of the number of communication rounds needed to achieve a global decision; i.e., for all non-faulty (correct) processes to reach a decision. This paper studies the time-complexity of local decisions in agreement problems, which we de¯ne as the number of communication rounds needed for at least one correct process to decide. We explore bounds for early local decision, that depend on the number f of actual failures (that occur in a given run of an algorithm), out of the maximum number t of failures tolerated (by the algorithm). We ¯rst consider the synchronous message-passing model where we give tight local decision bounds for three variants of agreement: consensus, uniform consensus and (non-blocking) atomic commit. We use these results to (1) show that, for consensus, local decision bounds are not compatible with global decision bounds (roughly speaking, they cannot be reached by the same algorithm), and (2) draw the ¯rst sharp line between the time-complexity of uniform consensus and atomic commit. Then we consider the eventually synchronous model where we give tight local decision bounds for synchronous runs of uniform consensus. (In this model, consensus and uniform consensus are similar, atomic commit is impossible, and one cannot bound the number of rounds to reach a decision in non-synchronous runs of consensus algorithms.) We prove a counter-intuitive result that the early local decision bound is the same as the early global decision bound. We also give a matching early deciding consensus algorithm that is signi¯cantly better than previous eventually synchronous consensus algorithms.

2007