Abstract
We address the problem of solving a task T=(T 1,...T m ) (called (m,1)-BG), in which a processor returns in an arbitrary one of m simultaneous consensus subtasks T 1,...T m . Processor p i submits to T an input vector of proposals (prop i,1,...,prop i,m ), one entry per subtask, and outputs, from just one subtask ℓ, a pair (ℓ, prop j,l ) for some j. All processors that output at ℓ output the same proposal.
Let d be a bound on the number of distinct input vectors that may be submitted to T. For example, d=3 if Democrats always vote Democrats across the board, and similarly for Republicans and Libertarians. A wait-free algorithm that immaterial of the number of processors solves T provided m ≥d is presented. In addition, if in each T j we allow k-set consensus rather than consensus, i.e., for each ℓ, the outputs satisfy |{j | prop j , ℓ}| ≤k, then the same algorithm solves T if m ≥⌈d/k ⌉.
What is the power of T=(T 1,...,T m ) when given as a subroutine, to be used by any number of processors with any number of input vectors? Obviously, T solves m-set consensus since each processor p i can submit the vector (id i ,id i ,...id i ), but can m-set consensus solve T? We show it does, and thus simultaneous consensus is a new characterization of set-consensus.
Finally, what if each T j is just a binary-consensus rather than consensus? Then we get the novel problem that was recently introduced of the Committee-Decision. It was shown that for 3 processors and m=2, the simultaneous binary-consensus is equivalent to (3,2)-set consensus. Here, using a variation of our wait-free algorithms mentioned above, we show that a task, in which a processor is required to return in one of m simultaneous binary-consensus subtasks, when used by n processors, is equivalent to (n,m)-set consensus. Thus, while set-consensus unlike consensus, has no binary version, now that we characterize m-set consensus through simultaneous consensus, the notion of binary-set-consensus is well defined. We have then showed that binary-set-consensus is equivalent to set consensus as it was with consensus.
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Afek, Y., Gafni, E., Rajsbaum, S., Raynal, M., Travers, C. (2006). Simultaneous Consensus Tasks: A Tighter Characterization of Set-Consensus. In: Chaudhuri, S., Das, S.R., Paul, H.S., Tirthapura, S. (eds) Distributed Computing and Networking. ICDCN 2006. Lecture Notes in Computer Science, vol 4308. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11947950_36
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DOI: https://doi.org/10.1007/11947950_36
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