Abstract
In the k-set agreement task each process proposes a value, and it is required that each correct process has to decide a value which was proposed and at most k distinct values must be decided. Using topological arguments it has been proved that k-set agreement is unsolvable in the asynchronous wait-free read/write shared memory model, when k < n, the number of processes.
This paper presents a simple, non-topological impossibility proof of k-set agreement. The proof depends on two simple properties of the immediate snapshot executions, a subset of all possible executions, and on the well known graph theory result stating that every graph has an even number of vertices with odd degree (the handshaking lemma).
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References
Attiya, H., Rajsbaum, S.: The Combinatorial Structure of Wait-Free Solvable Tasks. SIAM Journal on Computing 31(4), 1286–1313 (2002)
Attiya, H., Welch, J.: Distributed Computing: Fundamentals, Simulations and Advanced Topics. McGraw-Hill, New York (1998)
Borowsky, E., Gafni, E.: Generalized FLP Impossibility Result for t-Resilient Asynchronous Computations. In: Proc. 25th ACM Symposium on Theory of Computing (STOC 1993), pp. 91–100. ACM Press, New York (1993)
Borowsky, E., Gafni, E.: Immediate Atomic Snapshots and Fast Renaming. In: Proc. 12th ACM Symposium on Principles of Distributed Computing (PODC 1993), pp. 41-51 (1993)
Castañeda, A., Rajsbaum, S.: New Combinatorial Topology Upper and Lower Bounds for Renaming. In: Proc. 27th ACM Symposium on Principles of Distributed Computing (PODC 2008), pp. 295–304. ACM Press, New York (2008)
Chaudhuri, S.: More Choices Allow More Faults: Set Consensus Problems in Totally Asynchronous Systems. Information and Computation 105(1), 132–158 (1993)
Fischer, M.J., Lynch, N.A., Paterson, M.S.: Impossibility of Distributed Consensus with One Faulty Process. Journal of the ACM 32(2), 374–382 (1985)
Gafni, E., Koutsoupias, E.: Three-Processor Tasks Are Undecidable. SIAM Journal on Computing 28(3), 970–983 (1999)
Gafni, E., Rajsbaum, S., Herlihy, M.P.: Subconsensus Tasks: Renaming is Weaker Than Set Agreement. In: Dolev, S. (ed.) DISC 2006. LNCS, vol. 4167, pp. 329–338. Springer, Heidelberg (2006)
Henle, M.: A Combinatorial Introduction to Topology. Dover, New York (1994)
Herlihy, M.: Wait-free synchronization. Transactions on Programming Languages and Systems 13(1), 124–149 (1991)
Herlihy, M.P., Rajsbaum, S.: Set Consensus Using Arbitrary Objects (Preliminary Version). In: Proc. 13th Annual ACM Symposium on Principles on Distributed Computing (PODC 1994), pp. 324–333. ACM Press, New York (1994)
Herlihy, M.P., Rajsbaum, S.: Algebraic Spans. Mathematical Structures in Computer Science 10(4), 549–573 (2000)
Herlihy, M.P., Rajsbaum, S.: A Classification of Wait-free Loop Agreement Tasks. Theoretical Computer Science 291(1), 55–77 (2003)
Herlihy, M.P., Shavit, N.: The Topological Structure of Asynchronous Computability. Journal of the ACM 46(6), 858–923 (1999)
Hoest, G., Shavit, N.: Toward a Topological Characterization of Asynchronous Complexity. SIAM Journal on Computing 36(2), 457–497 (2006)
Loui, M., Abu-Amara, H.: Memory requirements for agreement among unreliable asynchronous processes. In: Preparata, F.P. (ed.) Advances in Computing Research, vol. 4, pp. 163–183. JAI Press, Greenwich (1987)
Saks, M., Zaharoglou, F.: Wait-Free k-Set Agreement Is Impossible: The Topology of Public Knowledge. SIAM Journal on Computing 29(5), 1449–1483 (2000)
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Attiya, H., Castañeda, A. (2011). A Non-topological Proof for the Impossibility of k-Set Agreement. In: Défago, X., Petit, F., Villain, V. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2011. Lecture Notes in Computer Science, vol 6976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24550-3_10
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DOI: https://doi.org/10.1007/978-3-642-24550-3_10
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