Adaptive Massively Parallel Connectivity in Optimal Space

We study the problem of finding connected components in the Adaptive Massively Parallel Computation (AMPC) model. We show that when we require the total space to be linear in the size of the input graph the problem can be solved in O( log*n) rounds in forests (with high probability) and 2 (O( log*n)) expected rounds in general graphs. This improves upon an existing O(log log(m/n) n) round algorithm. For the case when the desired number of rounds is constant we show that both problems can be solved using Theta(m +n log((k)) n) total space in expectation (in each round), where k is an arbitrarily large constant and log((k)) is the k-th iterate of the log(2) function. This improves upon existing algorithms requiring Omega(m +n log n) total space.