Better computing on the anonymous ring

We consider a bidirectional ring of n processors, where processors are anonymous, i.e., are indistinguishable. In this model it is known that “most” functions (in particular XOR and orientation) have worst case message complexity Θ(n2) for asynchronous computations, and Θ(n log n) for synchronous computations. The average case behavior is different; an algorithm that computes XOR asynchronously with O(nn messages on the average is known. In this paper we give tight bounds on the average complexity of various problems. We show the following:
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•An asynchronous deterministic algorithm that computes any computable function with O(n log n) messages, on the average (improving the O(nn algorithm). A matching lower bound is proven for functions such as XOR and orientation.

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•An asynchronous probabilistic algorithm that computes any computable function with O(n log n) expected messages on any input, using one random bit per processor. A matching lower bound is proven.

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•A Monte-Carlo asynchronous algorithm that computes any computable function with O(n) expected messages on any input, using one random bit per processor, with fixed error probability e > 0.

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•A synchronous algorithm that computes any computable function optimally in O(n) messages, on the average.

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•A synchronous probabilistic algorithm that computes any computable function optimally in O(n) expected messages on any input, using one random bit per processor

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•Lower bounds on the complexity of Monte-Carlo algorithms that always terminate.

Better Computing on the Anonymous Ring

Leader election is a fundamental problem in distributed computing where a unique node from a set of nodes declares itself as the leader (a distinguished state). We propose a notion of fairness in the leader election problem; a leader election algorithm is said to be fair if each node has the same probability of getting elected as leader. We show that existing deterministic algorithms based on comparisons of identifiers fail to achieve fairness. We demonstrate how fairness can be achieved through randomization and propose new leader election algorithms based on randomized voting. We separate the task of fair leader election on unidirectional rings into two subtasks: attrition and solitude detection following [9]. We show that tight interleaving of these two procedures as in [7], results in fair leader election on asynchronous, anonymous, unidirectional rings using expected O(n log n) bits of communication, in expected O(log n) rounds. (Here n is the number of nodes in the ring.) This matches the performance of the optimal algorithm of Abrahamson et al. [1]. Similar algorithms are presented for electing a fair leader in other models.

Fair leader election by randomized voting

LetR be a unidirectional asynchronous ring ofn identical processors each with a single input bit. Letf be any cyclic nonconstant function ofn boolean variables. Moran and Warmuth (1986) prove that anydeterministic algorithm that evaluatesf onR has communication complexity Ω (n logn) bits. They also construct a family of cyclic nonconstant boolean functions that can be evaluated inO(n logn) bits by a deterministic algorithm. This contrasts with the following new results:

Randomized function evaluation on a ring

Let $R$ be a unidirectional asynchronous ring of $n$ processors each with a single input bit. Let $f$ be any cyclic non-constant function of $n$ boolean variables. Moran and Warmuth [8] prove that any deterministic algorithm for $R$ that evaluates $f$ has communication complexity $\Omega (n \log n)$ bits. They also construct a cyclic non-constant boolean function that can be evaluated in $O(n \log n)$ bits by a deterministic algorithm. This contrasts with the following new results: \begin{enumerate} \item There exists a cyclic non-constant boolean function which can be evaluated with expected complexity $O (n \sqrt{\log n})$ bits by a randomized algorithm for $R$. \item Any nondeterministic algorithm for $R$ which evaluates any cyclic non-constant function has communication complexity $\Omega (n \sqrt{\log n})$ bits. \end{enumerate

Randomized Function Evaluation on a Ring

The communication complexity of fundamental problems in distributed computing on an asynchornous ring are examined from both the algorithmic and lower bound perspective. A detailed study is made of the effect on complexity of a number of assumptions about the algorithms. Randomization is shown to influence both the computability and complexity of several problems. Communication complexity is also shown to exhibit varying degrees of sensitivity to additional parameters including admissibility of error, kinds of error, knowledge of ring size, termination requirements, and the existence of identifiers.
A unified collection of formal models of distributed computation on asynchronous rings is developed which captures the essential characteristics of a spectrum of distributed algorithms--those that are error free (deterministic, Las Vegas, and nondeterministic), and those that err with small probability (Monte Carlo and nondeterministic/probabilistic). The nondeterministic and nondeterministic/probabilistic models are introduced as natural generalizations of the Las Vegas and Monte Carlo models respectively, and prove useful in deriving lower bounds. The unification helps to clarify the essential differences between the progressively more general notions of a distributed algorithm. In addition, the models reveal the sensitivity of various problems to the parameters listed above.
Complexity bounds derived using these models typically vary depending on the type of algorithm being investigated. The lower bounds are complemented by algorithms with matching complexity while frequently the lower bounds hold on even more powerful models than those required by the algorithms.
Among the algorithms and lower bounds presented are two specific results which stand out because of their relative significance. (1) If g is any nonconstant cyclic function of n variables, then any nondeterministic algorithm for computing g on an anonymous ring of size n has complexity $\Omega(n\sqrt{log\ n})$ bits of communication; and, there is a nonconstant cyclic boolean function f, such that f can be computed by a Las Vegas algorithm in O($n\sqrt{log\ n}$) expected bits of communication on a ring of size n. (2) The expected complexity of computing AND (and a number of other natural functions) on a ring of fixed size n in the Monte Carlo model is $\Theta(n\min\{$log n, log log$(1/\epsilon)\})$ messages and bits where $\epsilon$ is the allowable probability of error.

Lisa Higham

Papers

Randomized distributed computing on rings