Streaming Complexity of Spanning Tree Computation

The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an $n$-node input graph to be read sequentially in $p$ passes using $\tilde{O}(n)$ space. In this model, some graph problems, such as spanning trees and $k$-connectivity, can be exactly solved in a single pass; while other graph problems, such as triangle detection and unweighted all-pairs shortest paths, are known to require $\tilde{\Omega}(n)$ passes to compute. For many fundamental graph problems, the tractability in these models is open. In this paper, we study the tractability of computing some standard spanning trees. Our results are: (1) Maximum-Leaf Spanning Trees. This problem is known to be APX-complete with inapproximability constant $\rho\in[245/244,2)$. By constructing an $\varepsilon$-MLST sparsifier, we show that for every constant $\varepsilon > 0$, MLST can be approximated in a single pass to within a factor of $1+\varepsilon$ w.h.p. (albeit in super-polynomial time for $\varepsilon \le \rho-1$ assuming $\mathrm{P} \ne \mathrm{NP}$). (2) BFS Trees. It is known that BFS trees require $\omega(1)$ passes to compute, but the na\"{i}ve approach needs $O(n)$ passes. We devise a new randomized algorithm that reduces the pass complexity to $O(\sqrt{n})$, and it offers a smooth tradeoff between pass complexity and space usage. (3) DFS Trees. The current best algorithm by Khan and Mehta {[}STACS 2019{]} takes $\tilde{O}(h)$ passes, where $h$ is the height of computed DFS trees. Our contribution is twofold. First, we provide a simple alternative proof of this result, via a new connection to sparse certificates for $k$-node-connectivity. Second, we present a randomized algorithm that reduces the pass complexity to $O(\sqrt{n})$, and it also offers a smooth tradeoff between pass complexity and space usage.