Optimal Lower Bounds for Distributed and Streaming Spanning Forest Computation

TL;DR

This paper establishes optimal lower bounds on memory and communication for distributed and streaming algorithms computing spanning forests, matching existing upper bounds and advancing understanding of their fundamental complexity.

Contribution

The paper proves tight lower bounds for dynamic spanning forest data structures and distributed algorithms, extending known results to low failure probabilities.

Findings

Memory lower bound of (n n) bits for dynamic data structures.

Message length lower bound of ( n) bits in distributed models.

Lower bounds are tight, matching existing upper bounds like AGM sketch.

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] (which even succeeds with probability $1 - 1/\mathrm{poly}(n)$). Furthermore, for the first setting we show optimal lower bounds even for low failure probability $\delta$, as long as $\delta > 2^{-n^{1-\epsilon}}$.