Renaming in an asynchronous environment
Journal of the ACM (JACM)
Generalized FLP impossibility result for t-resilient asynchronous computations
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Three-Processor Tasks Are Undecidable
SIAM Journal on Computing
The topological structure of asynchronous computability
Journal of the ACM (JACM)
Wait-Free k-Set Agreement is Impossible: The Topology of Public Knowledge
SIAM Journal on Computing
The Combinatorial Structure of Wait-Free Solvable Tasks
SIAM Journal on Computing
Mathematical Structures in Computer Science
Toward a Topological Characterization of Asynchronous Complexity
SIAM Journal on Computing
New combinatorial topology upper and lower bounds for renaming
Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing
Subconsensus tasks: renaming is weaker than set agreement
DISC'06 Proceedings of the 20th international conference on Distributed Computing
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In the weak symmetry breaking (WSB) task, each of n+1 processes has to decide either 0 or 1 such that not all processes decide the same value. A WSB algorithm should be wait-free, namely, processes are asynchronous (there is no bound on their relative speeds), and potentially faulty (any proper subset may halt without warning). Also, it should be comparison based, namely, processes can only use comparison operations (,=) on the values they read from the memory. Symmetric chromatic subdivisions of an n-simplex have been used to represent the executions of a distributed algorithm solving the WSB task. Informally, each simplex of such a subdivision corresponds to an execution of the algorithm. Each vertex is labeled with the local state of a process in the execution; it is colored with the process ID, and also with the binary value the process decides in the execution. The symmetry properties of such a complex come from the comparison based requirement of a WSB algorithm. Let C denote the number of monochromatic n-simplexes of such a subdivision, counted by orientation. Previous work has shown that C=1+@?"i"="0^n^-^1(n+1i+1)k"i for some coefficients k"i@?Z. This characterization of C implies that the WSB task on n+1 processes is solvable if and only if n is such that the binomial coefficients are relatively prime, or equivalently, if and only if n is not a primer power. This paper presents an inductive style procedure that yields an alternative proof of the characterization of C. Roughly speaking, the proof consists in a procedure for modifying gradually the binary coloring of a symmetric chromatic subdivision, and computing the degree of the maps produced during the procedure.