A Topological Treatment of Early-Deciding Set-Agreement

This paper considers the k-set-agreement problem in a synchronous message passing distributed system where up to t processes can fail by crashing. We determine the number of communication rounds needed for all correct processes to reach a decision in a given run, as a function of k, the degree of coordination, and f <= t the number of processes that actually fail in the run. We prove a lower bound of min(\floor{f/k}+2,\floor{t/k}+1) rounds. Our proof uses simple topological tools to reason about runs of a full information set-agreement protocol. In particular, we introduce a new topological operator, which we call the early deciding operator, to capture rounds where k processes fail but correct processes see only k-1 failures

A Topological Treatment of Early-Deciding Set-Agreement

The k-set-agreement problem consists for a set of n processes to agree on less than k among n possibly different values, each initially known to only one process. The problem is at the heart of distributed computing and generalizes the celebrated consensus problem. This paper considers the k-set-agreement problem in a synchronous message passing distributed system where up to t processes can fail by crashing. We determine the number of communication rounds needed for all correct processes to reach a decision in a given run, as a function of the degree of coordination k and the number of processes that actually fail in the run, f ≤ t. We prove that, for any integer 1 ≤ k &lt; n, for any set-agreement protocol, for any integer 0 ≤ f ≤ t, not all correct processes can decide within ⌊f/k ⌋ + 1 rounds, in any run with at most f process crashes. More specifically, we prove a lower bound of min(⌊f/k ⌋ + 2, ⌊t/k ⌋ + 1) rounds for early-deciding set-agreement. This bound is tight because there is a set-agreement protocol that matches it, and the bound generalizes both the min(f + 2, t + 1) bound previously obtained for early-deciding consensus and the t + 1 bound previously obtained for the worst-case complexity of set-agreement