Log Diameter Rounds Algorithms for $2$-Vertex and $2$-Edge Connectivity

Many modern parallel systems, such as MapReduce, Hadoop and Spark, can be modeled well by the MPC model. The MPC model captures well coarse-grained computation on large data --- data is distributed to processors, each of which has a sublinear (in the input data) amount of memory and we alternate between rounds of computation and rounds of communication, where each machine can communicate an amount of data as large as the size of its memory. This model is stronger than the classical PRAM model, and it is an intriguing question to design algorithms whose running time is smaller than in the PRAM model. In this paper, we study two fundamental problems, $2$-edge connectivity and $2$-vertex connectivity (biconnectivity). PRAM algorithms which run in $O(\log n)$ time have been known for many years. We give algorithms using roughly log diameter rounds in the MPC model. Our main results are, for an $n$-vertex, $m$-edge graph of diameter $D$ and bi-diameter $D'$, 1) a $O(\log D\log\log_{m/n} n)$ parallel time $2$-edge connectivity algorithm, 2) a $O(\log D\log^2\log_{m/n}n+\log D'\log\log_{m/n}n)$ parallel time biconnectivity algorithm, where the bi-diameter $D'$ is the largest cycle length over all the vertex pairs in the same biconnected component. Our results are fully scalable, meaning that the memory per processor can be $O(n^δ)$ for arbitrary constant $δ>0$, and the total memory used is linear in the problem size. Our $2$-edge connectivity algorithm achieves the same parallel time as the connectivity algorithm of Andoni et al. (FOCS 2018). We also show an $Ω(\log D')$ conditional lower bound for the biconnectivity problem.