Being Fast Means Being Chatty: The Local Information Cost of Graph Spanners

We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to jointly solve a graph problem. We show that the local information cost presents a natural lower bound on the communication complexity of distributed algorithms. We demonstrate the application of local information cost by deriving a lower bound on the communication complexity of computing a $(2t-1)$-spanner that consists of at most $O(n^{1+\frac{1}{t} + \epsilon})$ edges, where $\epsilon = \Theta \left( {1}/{t^2} \right)$. Our main result is that any $O(\text{poly}(n))$-time algorithm must send at least $\tilde\Omega\left(\tfrac{1}{t^2} n^{1+{1}/{2t}}\right)$ bits in the CONGEST model under the KT1 assumption, where each node has knowledge of its neighbors' IDs initially. Previously, only a trivial lower bound of $\tilde \Omega(n)$ bits was known for this problem; in fact, our result is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem under the KT1 assumption. A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing spanner algorithms for this setting. In light of the work of King, Kutten, and Thorup (PODC 2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with $\tilde O(n)$ communication complexity. Our result also implies time complexity lower bounds for constructing a spanner in the node-congested clique of Augustine et al. (2019) and in the push-pull gossip model with limited bandwidth.