New Separations and Reductions for Directed Preservers and Hopsets

TL;DR

This work advances the theory of distance preservers and hopsets in directed graphs by (i) establishing new sparsity bounds for exact and approximate distance preservers, (ii) proving a novel directed-to-undirected reduction that transfers undirected preserver bounds to directed settings, and (iii) deriving new lower bounds for directed hopsets and shortcut sets that separate finite-stretch preservers from reachability preservers and reveal aspect-ratio dependent separations. The key techniques include the Dense Low-Diameter Cluster Lemma, a directed analogue of sourcewise preservers, and the Over-Under Weighting Scheme to convert undirected lower bounds into directed ones, enabling tight bounds like $\tilde{O}(n^{5/6}p^{2/3}+n)$ for exact directed preservers and $\Omega(n^{2/7})$-type hopset lower bounds. The results yield structural insights into shortest-path systems in directed graphs and provide algorithmic implications for APSP reductions and network design. Overall, the paper enhances understanding of sparsification and augmentation in directed graphs, with implications for efficiently computing near-shortest paths and preserving essential distance information.

Abstract

We study distance preservers, hopsets, and shortcut sets in $n$-node, $m$-edge directed graphs, and show improved bounds and new reductions for various settings of these problems. Our first set of results is about exact and approximate distance preservers. We give the following bounds on the size of directed distance preservers with $p$ demand pairs: 1) $\tilde{O}(n^{5/6}p^{2/3} + n)$ edges for exact distance preservers in unweighted graphs; and 2) $Ω(n^{2/3}p^{2/3})$ edges for approximate distance preservers with any given finite stretch, in graphs with arbitrary aspect ratio. Additionally, we establish a new directed-to-undirected reduction for exact distance preservers. We show that if undirected distance preservers have size $O(n^λp^μ + n)$ for constants $λ, μ> 0$, then directed distance preservers have size $O\left( n^{\frac{1}{2-λ}}p^{\frac{1+μ-λ}{2-λ}} + n^{1/2}p + n\right).$ As a consequence of the reduction, if current upper bounds for undirected preservers can be improved for some $p \leq n$, then so can current upper bounds for directed preservers. Our second set of results is about directed hopsets and shortcut sets. For hopsets in directed graphs, we prove that the hopbound is: 1) $Ω(n^{2/9})$ for $O(m)$-size shortcut sets, improving the previous $Ω(n^{1/5})$ bound [Vassilevska Williams, Xu and Xu, SODA 2024]; 2) $Ω(n^{2/7})$ for $O(m)$-size exact hopsets in unweighted graphs, improving the previous $Ω(n^{1/4})$ bound [Bodwin and Hoppenworth, FOCS 2023]; and 3) $Ω(n^{1/2})$ for $O(n)$-size approximate hopsets with any given finite stretch, in graphs with arbitrary aspect ratio. This result establishes a separation between this setting and $O(n)$-size approximate hopsets for graphs with polynomial aspect ratio, which have hopbound $\widetilde{O}(n^{1/3})$ [Bernstein and Wein, SODA 2023].

Figures (2)

• Figure 1: This figure depicts an example of a path system $\Pi = \{\pi, \pi', \pi", \pi"'\}$. The paths in $\Pi \setminus \{\pi\}$, restricted to the nodes in path $\pi$, form a DAG.
• Figure 2: This figure depicts an example graph constructed by a point-line incidence system. All edges on lines with a larger slope than the line containing $x_1$ and $x_4$ are marked as red, wider lines. Except the path $(x_1, x_2, x_3, x_4)$, all other paths from $x_1$ to $x_4$ use red edges, which have weight much larger than the edges on $(x_1, x_2, x_3, x_4)$.

Theorems & Definitions (86)

• Definition 1.1: Approximate Preservers
• Theorem 1.2: Consistent Tiebreaking Schemes
• Theorem 1.3: Main Result 1
• Theorem 1.4: Main Result 2
• Corollary 1.5
• Corollary 1.6
• Definition 1.7: Approximate Hopsets
• Theorem 1.8: Folklore Sampling UY91BRR10
• Theorem 1.9: Main Result 3
• Lemma 2.2: Dense Low-Diameter Cluster Lemma (cf. Lemma \ref{['lem:low-diam-scc']})