Massively parallel algorithms for finding well-connected components in sparse graphs

Massively parallel computation (MPC) algorithms for graph problems have witnessed a resurgence of interest in recent years. Despite major progress for numerous graph problems however, the complexity of the sparse graph connectivity problem in this model has remained elusive: While classical logarithmic-round PRAM algorithms for finding connected components in any n-vertex graph have been known for more than three decades (and imply the same bounds for MPC model), no o(log n)-round MPC algorithms are known for this task with truly sublinear in n memory per machine (which is the only interesting regime for sparse graphs with O(n) edges). It is conjectured that an o(log n)-round algorithm for connectivity on general sparse graphs with n^{1- (1)} per-machine memory may not exist, a conjecture that also forms the basis for multiple conditional hardness results on the round complexity of other problems in the MPC model. We take an opportunistic approach towards the sparse graph connectivity problem by designing an algorithm with improved performance in terms of the connectivity structure of the input graph. Formally, we design an MPC algorithm that finds all connected components with spectral gap at least in a graph in O(log log n + log(1/)) MPC rounds and n memory per machine for any constant (0,1). While this algorithm still requires (log n) rounds in the worst-case, it achieves an exponential round reduction on "well-connected'' components with 1/polylog(n) using only n memory per machine and (n) total memory, and still operates in o(log n)l rounds even when = 1/n^o(1). En-route to our main result, we design a new distributed data structure for performing independent random walks from all vertices simultaneously, as well as a new leader-election algorithm with exponentially faster round complexity on random graphs.