Log-Diameter MST Verification and Sensitivity in MPC

We consider two natural variants of the problem of minimum spanning tree (\(\text {MST}\)) of a graph in the parallel setting: MST verification (verifying if a given tree is an \(\text {MST}\)) and the sensitivity analysis of an MST (finding the lowest cost replacement edge for each edge of the \(\text {MST}\)). These two problems have been studied extensively for sequential algorithms and for parallel algorithms in the \(\textrm{PRAM}\) model of computation. In this paper, we extend the study to the standard model of Massive Parallel Computation (\(\textrm{MPC}\)). It is known that for graphs of diameter D, the connectivity problem can be solved in \(O(\log D + \log \log n)\) rounds on an \(\textrm{MPC}\) with low local memory (each machine can store only \(O(n^{\delta })\) words for an arbitrary constant \(\delta > 0\)) and with linear global memory, that is, with optimal utilization. However, for the related task of finding an \(\text {MST}\), we need \(\Omega (\log D_{\text {MST}})\) rounds, where \(D_{\text {MST}}\) denotes the diameter of the minimum spanning tree. The state of the art upper bound for \(\text {MST}\) is \(O(\log n)\) rounds; the result follows by simulating existing \(\textrm{PRAM}\) algorithms. While this bound may be optimal for general graphs, the benchmark of connectivity and lower bound for \(\text {MST}\) suggest the target bound of \(O(\log D_\text {MST})\) rounds, or possibly \(O(\log D_\text {MST} + \log \log n)\) rounds. As for now, we do not know if this bound is achievable for the \(\text {MST}\) problem on an \(\textrm{MPC}\) with low local memory and linear global memory. In this paper, we show that two natural variants of the \(\text {MST}\) problem: \(\text {MST}\) verification and sensitivity analysis of an \(\text {MST}\), can be completed in \(O(\log D_T)\) rounds on an \(\textrm{MPC}\) with low local memory and with linear global memory, that is, with optimal utilization; here \(D_T\) is the diameter of the input “candidate \(\text {MST}\) ” T. The algorithms asymptotically match our lower bound, conditioned on the 1-vs-2-cycle conjecture.