Antti Roeyskoe

Bernhard Haeupler, Yaowei Long, Antti Roeyskoe et al.

A fault-tolerant distance labeling scheme assigns a label to each vertex and edge of an undirected weighted graph $G$ with $n$ vertices so that, for any edge set $F$ of size $|F| \leq f$, one can approximate the distance between $p$ and $q$ in $G \setminus F$ by reading only the labels of $F \cup \{p,q\}$. For any $k$, we present a deterministic polynomial-time scheme with $O(k^{4})$ approximation and $\tilde{O}(f^{4}n^{1/k})$ label size. This is the first scheme to achieve a constant approximation while handling any number of edge faults $f$, resolving the open problem posed by Dory and Parter [DP21]. All previous schemes provided only a linear-in-$f$ approximation [DP21, LPS25]. Our labeling scheme directly improves the state of the art in the simpler setting of distance sensitivity oracles. Even for just $f = Θ(\log n)$ faults, all previous oracles either have super-linear query time, linear-in-$f$ approximation [CLPR12], or exponentially worse $2^{{\rm poly}(k)}$ approximation dependency in $k$ [HLS24].

Bernhard Haeupler, Antti Roeyskoe, Zhijun Zhang

All parallel algorithms for directed reachability and shortest paths crucially rely on efficient shortcut constructions. These constructions find directed paths and shortcut them by adding edges, with the goal to reduce the diameter of the graph. A long sequence of works has studied (efficient) shortcut constructions as well as impossibility results on the best diameter and therefore the best parallelism that can be achieved via this approach. This paper introduces a new conceptual tool for this line of research in the form of a simple and natural structural criterion: A shortcut $H$ for a graph $G$ is certified if for any shortcut edge $(u, v) \in H$, there exists a vertex $w$ such that the edges $(u, w)$ and $(w, v)$ are also in $G \cup H$. We show that this criterion captures constructiveness in the following sense: A shortcut $H$ can be constructed in $t$ time by repeatedly spending $\ell$ time on shortcutting a path of length $\ell$, if and only if, there exists a certified shortcut $H' \supseteq H$ of size $\tilde{O}(t)$. Furthermore, all known shortcut constructions with efficient algorithms can be extended to produce certified shortcuts of size $\tilde{O}(m)$. On the other hand, for shortcut constructions for which attempts to find efficient implementations have failed, we can show that this is impossible. We also obtain stronger diameter lower bounds for certified shortcuts and hopsets. For example, no certified shortcut construction with almost-linear size can reduce a graph's diameter below $n^{1/4-o(1)}$. This seems to be the best bound one can hope for with current techniques.