Having Hope in Missing Spanners: New Distance Preservers and Light Hopsets

An r-missing spanner for a graph G is a sparse subgraph H ⊆ G satisfying that for any u, v pair there is a (possibly approximate) u-v shortest path P in G such that |P \ H| ≤ r. That is, H misses at most r edges from every u-v (approximate) shortest path. [Kogan and Parter, FOCS’22] introduced the notion of missing spanners as an intermediate step for translating hopset constructions into spanners and distance preservers. In this work, we provide new constructions of missing spanners that lead to improved distance preservers. We also present a reduction in the reverse direction to that of [KP’22] by translating a special class of distance preservers into hopsets. Missing spanners serve as key structures in our constructions, leading to: Any p = Ω(n²/pRS(n)) vertex pairs in a directed unweighted graph have a distance preserver on O(p) edges, where RS(n) is the Ruzsa-Szemerédi function from combinatorial graph theory. This resolves the (second) open problem of [Bodwin, SODA’16]. In light of the classical quadratic lower bound for undirected weighted graphs, this provides the first separation between weighted and unweighted directed distance preservers. Any p ∈ [n, n²/RS(n)] vertex pairs in unweighted DAGs have a distance preserver with o(n√p) edges. This improves over the two-decade old bound by [Coppersmith and Elkin, SODA’05] for the family of DAGs. Any consistent distance preservers (where the shortest path’s tiebreaking scheme is consistent) for p pairs in n-vertex graphs of size g(n, p) can be transformed into r-missing (exact) spanners of the same size for r = g(n, p)/p. Missing spanners can provide new light hopsets: Any n-vertex weighted DAG (or weighted undirected graph) G = (V, E, ω) with aspect ratio M admits exact √n-hopsets with O^e(n) edges and a total edge weight, denoted as the lightness, of O^e(Mn), hence near-linear for unweighted graphs. This should be contrasted with all prior constructions that yield a lightness of Ω(Mn³/²), already for unweighted undirected graphs. Our results provide new links between graph augmentation and reduction structures, which for many years have been studied in isolation. We believe that these connections should have further combinatorial and algorithmic applications.