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 solve a graph problem. We show that the local information cost (LIC) presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST-KT₁ model, where each node has initial knowledge of its neighbors’ IDs, we prove that Ω (LIC_γ (P )/log τ log n) bits are required for solving a graph problem P with a τ-round algorithm that errs with probability at most γ. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST-KT₁ model.
We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a multiplicative spanner with stretch 2t − 1 that consists of at most O (n^{1 + 1/t + ε}) edges, where ε = O (1/t²). Our main result is that any O(poly(n))-time algorithm must send at least Ω (1/t² n^1+1/2t ) bits in the CONGEST model under the KT₁ assumption. Previously, only a trivial lower bound of Ω(n)  bits was known for this problem; in fact, this is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem in this setting.
A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing a distributed spanner algorithm. In light of the work of King, Kutten, and Thorup (2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with 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 bandwidt.