Communication-efficient randomized consensus

We consider the problem of consensus in the challenging classic model. In this model, the adversary is adaptive; it can choose which processors crash at any point during the course of the algorithm. Further, communication is via asynchronous message passing: there is no known upper bound on the time to send a message from one processor to another, and all messages and coin flips are seen by the adversary. We describe a new randomized consensus protocol with expected message complexity O(n2log2n) when fewer than n / 2 processes may fail by crashing. This is an almost-linear improvement over the best previously known protocol, and within logarithmic factors of a known Ω(n2) message lower bound. The protocol further ensures that no process sends more than O(nlog3n) messages in expectation, which is again within logarithmic factors of optimal. We also present a generalization of the algorithm to an arbitrary number of failures t, which uses expected O(nt+t2log2t) total messages. Our approach is to build a message-efficient, resilient mechanism for aggregating individual processor votes, implementing the message-passing equivalent of a weak shared coin. Roughly, in our protocol, a processor first announces its votes to small groups, then propagates them to increasingly larger groups as it generates more and more votes. To bound the number of messages that an individual process might have to send or receive, the protocol progressively increases the weight of generated votes. The main technical challenge is bounding the impact of votes that are still “in flight” (generated, but not fully propagated) on the final outcome of the shared coin, especially since such votes might have different weights. We achieve this by leveraging the structure of the algorithm, and a technical argument based on martingale concentration bounds. Overall, we show that it is possible to build an efficient message-passing implementation of a shared coin, and in the process (almost-optimally) solve the classic consensus problem in the asynchronous message-passing model

How to Elect a Leader Faster than a Tournament

The problem of electing a leader from among $n$ contenders is one of the fundamental questions in distributed computing. In its simplest formulation, the task is as follows: given $n$ processors, all participants must eventually return a win or lose indication, such that a single contender may win. Despite a considerable amount of work on leader election, the following question is still open: can we elect a leader in an asynchronous fault-prone system faster than just running a $\Theta(\log n)$-time tournament, against a strong adaptive adversary? In this paper, we answer this question in the affirmative, improving on a decades-old upper bound. We introduce two new algorithmic ideas to reduce the time complexity of electing a leader to $O(\log^* n)$, using $O(n^2)$ point-to-point messages. A non-trivial application of our algorithm is a new upper bound for the tight renaming problem, assigning $n$ items to the $n$ participants in expected $O(\log^2 n)$ time and $O(n^2)$ messages. We complement our results with lower bound of $\Omega(n^2)$ messages for solving these two problems, closing the question of their message complexity

Never Say Never Probabilistic & Temporal Failure Detectors (Extended)

The failure detector approach for solving distributed computing problems has been celebrated for its modularity. This approach allows the construction of algorithms using abstract failure detection mechanisms, defined by axiomatic properties, as building blocks. The minimal synchrony assumptions on communication, which enable to implement the failure detection mechanism, are studied separately. Such synchrony assumptions are typically expressed as eventual guarantees that need to hold, after some point in time, forever and deterministically. But in practice, they never do. Synchrony assumptions may hold only probabilistically and temporarily. In this paper, we study failure detectors in a realistic distributed system N, with asynchrony inflicted by probabilistic synchronous communication. We address the following paradox about the weakest failure detector to solve the consensus problem (and many equivalent problems), i.e., S: an implementation of “consensus with probability 1” is possible in N without using randomness in the algorithm itself, while an implementation of “S with probability 1” is impossible to achieve in N. We circumvent this paradox by introducing a new failure detector S*, a variant of S with probabilistic and temporal accuracy. We prove that S* is implementable in N and we provide an optimal S* implementation. Interestingly, we show that S* can replace S , in several existing deterministic consensus algorithms using S, to yield an algorithm that solves “consensus with probability 1”. In fact, we show that such result holds for all decisive problems (not only consensus) and also for failure detector P (not only S). The resulting algorithms combine the modularity of distributed computing practices with the practicality of networking ones