Optimal Lower Bounds for Distributed and Streaming Spanning Forest Computation

Abstract: 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 $\epsilon>0$. We prove that any such data structure must use $\Omega(n\log^3 n)$ 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 $\epsilon>0$. We prove the average message length must be $\Omega(\log^3 n)$ bits. Both our lower bounds are optimal, with matching upper bounds provided by the AGM sketch AGM12. Furthermore, for the first setting we show optimal lower bounds even for low failure probability $\delta$, as long as $\delta > 2^{-n^{1-\epsilon}}$.