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We study the consensus problem in a synchronous distributed system of $n$nodes under an adaptive adversary that has a slightly outdated view of thesystem and can block all incoming and outgoing communication of a constantfraction of the nodes in each round. Motivated by a result of Ben-Or andBar-Joseph (1998), showing that any consensus algorithm that is resilientagainst a linear number of crash faults requires $\tilde \Omega(\sqrt{n})$rounds in an $n$-node network against an adaptive adversary, we consider a lateadaptive adversary, who has full knowledge of the network state at thebeginning of the previous round and unlimited computational power, but isoblivious to the current state of the nodes.  Our main contributions are randomized distributed algorithms that achievealmost-everywhere consensus w.h.p. against a late adaptive adversary who canblock up to $\epsilon n$ nodes in each round, for a small constant $\epsilon>0$. Our first protocol achieves binary and plurality consensus, and the secondone achieves multi-value consensus. Both of our algorithms succeed in $O(\logn)$ rounds with high probability, thus showing an exponential gap to theaforementioned lower bound for strongly adaptive crash-failure adversaries,which can be strengthened to $\Omega(n)$ when allowing the adversary to blocknodes instead of permanently crashing them. In our algorithms each nodecontacts only an (amortized) constant number of peers in each communicationround. We show that our algorithms are optimal up to constant (resp.sub-logarithmic) factors by proving that every almost-everywhere consensusprotocol takes $\Omega(\log_d n)$ rounds in the worst case, where $d$ is anupper bound on the number of communication requests initiated per node in eachround. We complement our theoretical results with an experimental evaluation ofthe first protocol revealing a short convergence time.

Breaking the $\tilde\Omega(\sqrt{n})$ Barrier: Fast Consensus under a Late Adversary