412 research outputs found
Locally Solvable Tasks and the Limitations of Valency Arguments
An elegant strategy for proving impossibility results in distributed computing was introduced in the celebrated FLP consensus impossibility proof. This strategy is local in nature as at each stage, one configuration of a hypothetical protocol for consensus is considered, together with future valencies of possible extensions. This proof strategy has been used in numerous situations related to consensus, leading one to wonder why it has not been used in impossibility results of two other well-known tasks: set agreement and renaming. This paper provides an explanation of why impossibility proofs of these tasks have been of a global nature. It shows that a protocol can always solve such tasks locally, in the following sense. Given a configuration and all its future valencies, if a single successor configuration is selected, then the protocol can reveal all decisions in this branch of executions, satisfying the task specification. This result is shown for both set agreement and renaming, implying that there are no local impossibility proofs for these tasks
Wait-Free Solvability of Equality Negation Tasks
We introduce a family of tasks for n processes, as a generalization of the two process equality negation task of Lo and Hadzilacos (SICOMP 2000). Each process starts the computation with a private input value taken from a finite set of possible inputs. After communicating with the other processes using immediate snapshots, the process must decide on a binary output value, 0 or 1. The specification of the task is the following: in an execution, if the set of input values is large enough, the processes should agree on the same output; if the set of inputs is small enough, the processes should disagree; and in-between these two cases, any output is allowed. Formally, this specification depends on two threshold parameters k and l, with k<l, indicating when the cardinality of the set of inputs becomes "small" or "large", respectively. We study the solvability of this task depending on those two parameters. First, we show that the task is solvable whenever k+2 <= l. For the remaining cases (l = k+1), we use various combinatorial topology techniques to obtain two impossibility results: the task is unsolvable if either k <= n/2 or n-k is odd. The remaining cases are still open
Why Extension-Based Proofs Fail
We introduce extension-based proofs, a class of impossibility proofs that
includes valency arguments. They are modelled as an interaction between a
prover and a protocol. Using proofs based on combinatorial topology, it has
been shown that it is impossible to deterministically solve k-set agreement
among n > k > 1 processes in a wait-free manner in certain asynchronous models.
However, it was unknown whether proofs based on simpler techniques were
possible. We show that this impossibility result cannot be obtained for one of
these models by an extension-based proof and, hence, extension-based proofs are
limited in power.Comment: This version of the paper is for the NIS model. Previous versions of
the paper are for the NIIS mode
How to Elect a Leader Faster than a Tournament
The problem of electing a leader from among contenders is one of the
fundamental questions in distributed computing. In its simplest formulation,
the task is as follows: given 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 -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 , using
point-to-point messages. A non-trivial application of our algorithm is a new
upper bound for the tight renaming problem, assigning items to the
participants in expected time and messages. We
complement our results with lower bound of messages for solving
these two problems, closing the question of their message complexity
General Tasks and Extension-Based Proofs
The concept of extension-based proofs models the idea of a valency argument
which is widely used in distributed computing. Extension-based proofs are
limited in power: it has been shown that there is no extension-based proof of
the impossibility of a wait-free protocol for -set agreement among processes. A discussion of a restricted type of reduction has shown
that there are no extension-based proofs of the impossibility of wait-free
protocols for some other distributed computing problems.
We extend the previous result to general reductions that allow multiple
instances of tasks. The techniques used in the previous work are designed for
certain tasks, such as the -set agreement task. We give a necessary and
sufficient condition for general colorless tasks to have no extension-based
proofs of the impossibility of wait-free protocols, and show that different
types of extension-based proof are equivalent in power for colorless tasks.
Using this necessary and sufficient condition, the result about reductions can
be understood from a topological perspective
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