Related papers: Lower Bounds on Sparse Spanners, Emulators, and Di…
We obtain improved lower bounds for additive spanners, additive emulators, and diameter-reducing shortcut sets. Spanners and emulators are sparse graphs that approximately preserve the distances of a given graph. A shortcut set is a set of…
We provide a variety of lower bounds for the well-known shortcut set problem: how much can one decrease the diameter of a directed graph on $n$ vertices and $m$ edges by adding $O(n)$ or $O(m)$ of shortcuts from the transitive closure of…
We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+\Omega(n^{3/17})$, improving on the previous best lower bound of $\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes the…
Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say $\tilde{O}(n^{1+\delta})$ bits. There is an inherent tradeoff between the…
For an input graph $G$, an additive spanner is a sparse subgraph $H$ whose shortest paths match those of $G$ up to small additive error. We prove two new lower bounds in the area of additive spanners: 1) We construct $n$-node graphs $G$ for…
For an $n$-vertex digraph $G=(V,E)$, a \emph{shortcut set} is a (small) subset of edges $H$ taken from the transitive closure of $G$ that, when added to $G$ guarantees that the diameter of $G \cup H$ is small. Shortcut sets, introduced by…
We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers, which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work…
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…
Near-additive (aka $(1+\epsilon,\beta)$-) emulators and spanners are a fundamental graph-algorithmic construct, with numerous applications for computing approximate shortest paths and related problems in distributed, streaming and dynamic…
For $\alpha \ge 1$, $\beta \ge 0$, and a graph $G$, a spanning subgraph $H$ of $G$ is said to be an $(\alpha, \beta)$-spanner if $\dist(u, v, H) \le \alpha \cdot \dist(u, v, G) + \beta$ holds for any pair of vertices $u$ and $v$. These type…
We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers, which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work…
Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs.…
Let $G$ be an unweighted $n$-node undirected graph. A \emph{$\beta$-additive spanner} of $G$ is a spanning subgraph $H$ of $G$ such that distances in $H$ are stretched at most by an additive term $\beta$ w.r.t. the corresponding distances…
Maintaining and updating shortest paths information in a graph is a fundamental problem with many applications. As computations on dense graphs can be prohibitively expensive, and it is preferable to perform the computations on a sparse…
For an n-vertex directed graph $G = (V,E)$, a $\beta$-\emph{shortcut set} $H$ is a set of additional edges $H \subseteq V \times V$ such that $G \cup H$ has the same transitive closure as $G$, and for every pair $u,v \in V$, there is a…
We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an $\widetilde{\Omega}(n)$ lower bound for…
Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures,…
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…
We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+\epsilon$ error must…
Given an input graph $G = (V, E)$, an additive emulator $H = (V, E', w)$ is a sparse weighted graph that preserves all distances in $G$ with small additive error. A recent line of inquiry has sought to determine the best additive error…