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An Axiomatic Approach to Computing the Connectivity of Synchronous and Asynchronous Systems
"... We present a unified, axiomatic approach to proving lower bounds for the kset agreement problem in both synchronous and asynchronous messagepassing models. The proof involves constructing the set of reachable states, proving that these states are highly connected, and then appealing to a wellknow ..."
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We present a unified, axiomatic approach to proving lower bounds for the kset agreement problem in both synchronous and asynchronous messagepassing models. The proof involves constructing the set of reachable states, proving that these states are highly connected, and then appealing to a wellknown topological result that high connectivity implies that set agreement is impossible. We construct the set of reachable states in an iterative fashion using a round operator that we define, and our proof of connectivity is an inductive proof based on this iterative construction and simple properties of the round operator. 1
A Topological Treatment of EarlyDeciding SetAgreement
, 2008
"... The ksetagreement problem consists for a set of n processes to agree on less than k among n possibly different values, each initially known to only one process. The problem is at the heart of distributed computing and generalizes the celebrated consensus problem. This paper considers the ksetagr ..."
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The ksetagreement problem consists for a set of n processes to agree on less than k among n possibly different values, each initially known to only one process. The problem is at the heart of distributed computing and generalizes the celebrated consensus problem. This paper considers the ksetagreement problem in a synchronous message passing distributed system where up to t processes can fail by crashing. We determine the number of communication rounds needed for all correct processes to reach a decision in a given run, as a function of the degree of coordination k and the number of processes that actually fail in the run, f ≤ t. We prove that, for any integer 1 ≤ k < n, for any setagreement protocol, for any integer 0 ≤ f ≤ t, not all correct processes can decide within ⌊f/k ⌋ + 1 rounds, in any run with at most f process crashes. More specifically, we prove a lower bound of min(⌊f/k ⌋ + 2, ⌊t/k ⌋ + 1) rounds for earlydeciding setagreement. This bound is tight because there is a setagreement protocol that matches it, and the bound generalizes both the min(f + 2, t + 1) bound previously obtained for earlydeciding consensus and the t + 1 bound previously obtained for the worstcase complexity of setagreement.
Structured Derivation of SemiSynchronous Algorithms
"... Abstract. The semisynchronous model is an important middle ground between the synchronous and the asynchronous models of distributed computing. In this model, processes can detect (timeout) when other processes fail. However, since detection is done by timing out, it incurs a cost much higher than ..."
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Abstract. The semisynchronous model is an important middle ground between the synchronous and the asynchronous models of distributed computing. In this model, processes can detect (timeout) when other processes fail. However, since detection is done by timing out, it incurs a cost much higher than the typical delay of messages. The paper presents a new communication primitive, Timely Announced Broadcast (TAB), and uses it in algorithms for consensus and set consensus in the semisynchronous model. Separate implementations of TAB, withstanding different types of failures, allow to derive algorithms for consensus and set consensus under crash and omission failures. The time bounds obtained by our algorithms asymptotically match, or improve, the previously known bounds.
Transforming Worstcase Optimal Solutions for Simultaneous Tasks into Allcase Optimal Solutions
"... Decision tasks require that nonfaulty processes make decisions based on their input values. Simultaneous decision tasks require that nonfaulty processes decide in the same round. Most decision tasks have known worstcase lower bounds. Most also have known worstcase optimal protocols that halt in th ..."
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Decision tasks require that nonfaulty processes make decisions based on their input values. Simultaneous decision tasks require that nonfaulty processes decide in the same round. Most decision tasks have known worstcase lower bounds. Most also have known worstcase optimal protocols that halt in the number of rounds given by the worstcase lower bound, and some have earlystopping protocols that can halt earlier than the worstcase lower bound (sometimes in as early as two rounds). We consider what might be called earliestpossible protocols for simultaneous decision tasks. We present a new technique that converts worstcase optimal decision protocols into allcase optimal simultaneous decision protocols: For every behavior of the adversary, the allcase optimal protocol decides as soon as any protocol can decide in a run with the same adversarial behavior. Examples to which this can be applied include set consensus, conditionbased consensus, renaming and orderpreserving renaming. Some of these tasks can be solved significantly faster than the classical simultaneous consensus task. A byproduct of the analysis is a proof that improving on the worstcase bound for any simultaneous task by even a single round is as hard as reaching simultaneous consensus.
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"... Abstract. In the kset agreement problem, each processor starts with a private input value and eventually decides on an output value. At most k distinct output values may be chosen, and every processor’s output value must be one of the proposed values. We consider a synchronous message passing syste ..."
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Abstract. In the kset agreement problem, each processor starts with a private input value and eventually decides on an output value. At most k distinct output values may be chosen, and every processor’s output value must be one of the proposed values. We consider a synchronous message passing system, and we prove a tight bound of f/k+2 rounds of communication for all processors to decide in every run in which at most f processors fail. The lower bound proof proceeds through a simulation of a synchronous solution to kset agreement in message passing, in an asynchronous shared memory system in which k − 1 processors may fail, and which was proven to be impossible using topological approaches. In contrast to past complexity results on set agreement, our lower bound proof is purely algorithmic. It does not use any direct topological argument but uses instead the impossibility of asynchronous set agreement to encapsulate the needed topology. We thus derive an adaptive complexity lower bound for a message passing system from a static impossibility in a shared memory system.
Abstract GETCO 2004 Preliminary Version The Complexity of Early Deciding Set Agreement: How can Topology help?
"... The aim of this paper is to pose a challenge to the experts of (algebraic) topology techniques. We present an early deciding algorithm that solves the set agreement problem, i.e., the problem which triggered research on applying topology techniques to distributed computing. We conjecture the algorit ..."
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The aim of this paper is to pose a challenge to the experts of (algebraic) topology techniques. We present an early deciding algorithm that solves the set agreement problem, i.e., the problem which triggered research on applying topology techniques to distributed computing. We conjecture the algorithm to be optimal, and we discuss the need and challenges of applying topology techniques to prove the lower bound.