Tree Covers for Directed Graphs: Distance Oracles, Routing Schemes and Spanners

Abstract

In a seminal work [STOC'01; JACM'05], Thorup and Zwick showed that every weighted undirected graph admits a tree cover where each vertex belongs to $O(kn^{1/k})$ trees in expectation, and every pair has a path of stretch at most $2k-1$ in some tree. In this paper, we extend this structural property to directed graphs with nonnegative edge weights. Let $\mathrm{dist}_G(u \leftrightarrows v) = \mathrm{dist}_G(u \to v) + \mathrm{dist}_G(v \to u)$ denote the roundtrip distance. For every integer $k$, we construct a collection $\mathcal{T}$ of tree-pairs, each consisting of an in-tree and an out-tree rooted at the same vertex. Each vertex belongs to $O(kn^{1/k})$ tree-pairs in expectation, and for every ordered pair $(u, v)$ some tree-pair $T_{in}\cup T_{out} \in \mathcal{T}$ satisfies $\mathrm{dist}_{T_{in}\cup T_{out}}(u \to v) \le \mathrm{dist}_G(u \to v) + (k-1)\mathrm{dist}_G(u \leftrightarrows v).$ We call this a \emph{$k$-strong (roundtrip) tree cover}. While prior techniques partition distances into geometric scales and build separate structures for each scale, our construction handles all distances simultaneously within a single, unified hierarchy. This yields the first directed tree cover for general $k$ matching the classic Thorup-Zwick stretch-size trade-off. Our cover immediately implies improved constructions of roundtrip spanners, path-reporting distance oracles, and compact routing schemes, as well as their source-wise and prioritized variants. We also provide faster algorithms for these structures: for general $k \ge 4$, all the above objects can be constructed in $\widetilde{O}(\min\{mn, n^{2.5}\})$ expected time, breaking the $mn$ barrier in dense directed graphs.