Abstract
The objective of this paper is to understand the effect of partitioning in distributed computing models. In spite of being quite similar agreement problems, (deterministic) consensus (1-set agreement) and k-set agreement (for \(k>1\)) require surprisingly different techniques for proving impossibilities. There is a widely applicable generic theorem, however, which allows to reduce the impossibility of k-set agreement to consensus in message-passing models that allow some partitioning. In this paper, we provide the topological representation of this theorem, which reveals how partitioning is reflected in the protocol complex: It turns out that this leads to a “color splitting” of the algorithm’s decision map, which separates the sub-complexes representing the partitioned processes. We also harvest a general advantage of topological results, which allowed us to carry over our findings to shared memory systems. We first demonstrate the utility of our reduction theorem by proving that d-set agreement cannot be solved in the d-solo asynchronous read-write model even when a single process may crash, not just in the wait-free case. Moreover, our new insights into the structure of protocol complexes gave us the idea for a simple proof of the fact that no partitioning argument can provide a valid impossibility proof for wait-free set agreement in the iterated immediate snapshot model: For any set of partition-compatible runs (which do not contain runs where all processes always have a complete view), we provide a way to construct a simple algorithm that solves set agreement.
This work has been supported by the PAPIIT-UNAM grant IN109917 and the Austrian Science Fund (FWF) projects RiSE/SHiNE (S11405) and ADynNet (P28182).
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Notes
- 1.
Lacking space forced us to relegate all proofs into the full version [22] of our paper.
- 2.
Note that the original BRS theorem actually assumed that every process starts with a unique value.
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Rincon Galeana, H., Winkler, K., Schmid, U., Rajsbaum, S. (2019). A Topological View of Partitioning Arguments: Reducing k-Set Agreement to Consensus. In: Ghaffari, M., Nesterenko, M., Tixeuil, S., Tucci, S., Yamauchi, Y. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2019. Lecture Notes in Computer Science(), vol 11914. Springer, Cham. https://doi.org/10.1007/978-3-030-34992-9_25
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