Fault-Tolerant Graph Realizations in the Congested Clique

In this paper, we study the graph realization problem in the Congested Clique model of distributed computing under crash faults. We consider {\em degree-sequence realization}, in which each node $v$ is associated with a degree value $d(v)$, and the resulting degree sequence is realizable if it is possible to construct an overlay network with the given degrees. Our main result is a $O(f)$-round deterministic algorithm for the degree-sequence realization problem in a $n$-node Congested Clique, of which $f$ nodes could be faulty ($f<n$). The algorithm uses $O(n^2)$ messages. We complement the result with lower bounds to show that the algorithm is tight w.r.t the number of rounds and the messages simultaneously. We also extend our result to the Node Capacitated Clique (NCC) model, where each node is restricted to sending and receiving at-most $O(\log n)$ messages per round. In the NCC model, our algorithm solves degree-sequence realization in $O(nf/\log n)$ rounds and $O(n^2)$ messages. For both settings, our algorithms work without the knowledge of $f$, the number of faults. To the best of our knowledge, these are the first results for the graph realization problem in the crash-fault distributed network.