Being Fast Means Being Chatty: The Local Information Cost of Graph Spanners
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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+1/t + ϵ) edges, where ϵ = Θ( 1/t^2). Our main result is that any O(poly(n))-time algorithm must send at least Ω̃(1t^2 n^1+1/2t) 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 Ω̃(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 Õ(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.
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