Optimal lower bounds for distributed and streaming spanning forest computation

We show optimal lower bounds for spanning forest computation in two different models: • One wants a data structure for fully dynamic spanning forest in which updates can insert or delete edges amongst a base set of n vertices. The sole allowed query asks for a spanning forest, which the data structure should successfully answer with some given (potentially small) constant probability > 0. We prove that any such data structure must use Ω(nlog³ n) bits of memory. • There is a referee and n vertices in a network sharing public randomness, and each vertex knows only its neighborhood; the referee receives no input. The vertices each send a message to the referee who then computes a spanning forest of the graph with constant probability > 0. We prove the average message length must be Ω(log³ n) bits. Both our lower bounds are optimal, with matching upper bounds provided by the AGM sketch [AGM12] (which even succeeds with probability 1 − 1/poly(n)). Furthermore, for the first setting we show optimal lower bounds even for low failure proba-1− bility δ, as long as δ > 2⁻n .