New Separations and Reductions for Directed Preservers and Hopsets

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) $\Omega(n^{2/3}p^{2/3})$ 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 $O(n^{\lambda}p^{\mu} + n)$ for constants $\lambda, \mu > 0$, then directed distance preservers have size $O\left( n^{\frac{1}{2-\lambda}}p^{\frac{1+\mu-\lambda}{2-\lambda}} + n^{1/2}p + n\right).$ 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) $\Omega(n^{2/9})$ for $O(m)$-size shortcut sets, improving the previous $\Omega(n^{1/5})$ bound [Vassilevska Williams, Xu and Xu, SODA 2024]; 2) $\Omega(n^{2/7})$ for $O(m)$-size exact hopsets in unweighted graphs, improving the previous $\Omega(n^{1/4})$ bound [Bodwin and Hoppenworth, FOCS 2023]; and 3) $\Omega(n^{1/2})$ 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].