A modular approach to shared-memory consensus, with applications to the probabilistic-write model

Abstract

We show that consensus can be solved by an alternating sequence of adopt-commit objects (Gafni in Proceedings of the seventeenth annual ACM symposium on principles of distributed computing, pp 143-152, 1998; Alistarh et al. in ISAAC, Lecture notes in computer science, vol 5878. Springer, Berlin, pp 943-953, 2009), which detect agreement, and conciliators, which ensure agreement with some probability. We observe that most known randomized consensus algorithms have this structure. We give a deterministic implementation of an m-valued adopt-commit object for an unbounded number of processes that uses lg m + (log log m) space and individual work. We also give a randomized conciliator for any number of values in the probabilistic-write model with n processes that guarantees agreement with constant probability while using one multiwriter register, O(log n) expected individual work, and (n) expected total work. Combining these objects gives a consensus protocol for the probabilistic-write model that uses O(log m +log n) individual work and O(n log m) total work. No previous protocol in this model uses sublinear individual work or linear total work for constant m.

Tight Bounds for Adopt-Commit Objects

“…In this model, a protocol of Aspnes [3], based on combining adopt-commit objects and a class of randomized objects called conciliators, gives an anonymous protocol for m-valued consensus with expected O(log m + log n) individual step complexity, where O(log m) is the cost of the adopt-commit and O(log n) is the cost of the conciliator using implementations given in [3].…”

“…The difference is that agreement (which requires that all outputs are the same) is replaced by the weaker requirements of coherence and convergence. As observed in [3], this means that consensus objects satisfy the requirements of adopt-commit objects, if each process returns the decision bit commit together with its output value. It follows that lower bounds on adopt-commit objects immediately give lower bounds on consensus objects.…”

“…Until now, the best implementations of m-valued adopt-commit objects had Θ(n) individual step complexity, for n processes [15], or Θ(log m) individual step complexity, for any number of processes [3]. Both these implementations are deterministic, but the latter is also anonymous.…”

“…We use agreement instead for the stronger unconditional agreement property of consensus objects. The term coherence is from [3].…”

An $O(\sqrt n)$ Space Bound for Obstruction-Free Leader Election

“…Randomized algorithms can solve consensus and guarantee randomized wait-freedom even if only registers are available. The randomized step complexity of the consensus problem has been studied thoroughly and is well understood for most of the common adversary models [8][9][10][11][12][13].…”