A non-topological proof for the impossibility of $k$-set agreement

Abstract

In the $k$-set agreement task, each process proposes a value and each correct process has to decide a value which was proposed, so that at most $k$ distinct values are 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 an elementary, 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 handshaking lemma stating that every graph has an even number of vertices with odd degree.