Distributed MST: A Smoothed Analysis

We study smoothed analysis of distributed graph algorithms, focusing on the fundamental minimum spanning tree (MST) problem. With the goal of studying the time complexity of distributed MST as a function of the "perturbation" of the input graph, we posit a {\em smoothing model} that is parameterized by a smoothing parameter $0 \leq \epsilon(n) \leq 1$ which controls the amount of {\em random} edges that can be added to an input graph $G$ per round. Informally, $\epsilon(n)$ is the probability (typically a small function of $n$, e.g., $n^{{-\frac{1}{4}}$)} that a random edge can be added to a node per round. The added random edges, once they are added, can be used (only) for communication. We show upper and lower bounds on the time complexity of distributed MST in the above smoothing model. We present a distributed algorithm that, with high probability,\footnote{Throughout, with high probability (whp) means with probability at least $1 - n^{-c}$, for some fixed, positive constant $c$.} computes an MST and runs in $\tilde{O}(\min{\frac{1}{\sqrt{\epsilon(n)}} 2^{{O(\sqrt{\log} n})}, D + \sqrt{n}})$ rounds\footnote{The notation $\tilde{O}$ hides a $\polylog(n)$ factor and $\tilde{\Omega}$ hides a $\frac{1}{\polylog{(n)}}$ factor, where $n$ is the number of nodes of the graph.} where $\epsilon$ is the smoothing parameter, $D$ is the network diameter and $n$ is the network size. To complement our upper bound, we also show a lower bound of $\tilde{\Omega}(\min{\frac{1}{\sqrt{\epsilon(n)}}, D+\sqrt{n}})$. We note that the upper and lower bounds essentially match except for a multiplicative $2^{{O(\sqrt{\log} n})} \polylog(n)$ factor. Our work can be considered as a first step in understanding the smoothed complexity of distributed graph algorithms.