Related papers: Better Distance Preservers and Additive Spanners
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…
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…
For a given graph $G$ and a subset of vertices $S$, a \emph{distance preserver} is a subgraph of $G$ that preserves shortest paths between the vertices of $S$. We distinguish between a \emph{subsetwise} distance preserver, which preserves…
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…
A spanner is a sparse subgraph of a given graph $G$ which preserves distances, measured w.r.t.\ some distance metric, up to a multiplicative stretch factor. This paper addresses the problem of constructing graph spanners w.r.t.\ the group…
Designing sparse directed spanners, which are subgraphs that approximately maintain distance constraints, has attracted sustained interest in TCS, especially due to their wide applicability, as well as the difficulty to obtain tight…
We give a simple algorithm for maintaining a $n^{o(1)}$-approximate spanner $H$ of a graph $G$ with $n$ vertices as $G$ receives edge updates by reduction to the dynamic All-Pairs Shortest Paths (APSP) problem. Given an initially empty…
In the pairwise weighted spanner problem, the input consists of an $n$-vertex-directed graph, where each edge is assigned a cost and a length. Given $k$ vertex pairs and a distance constraint for each pair, the goal is to find a…
Let $P \subset \mathbb{R}^2$ be a planar $n$-point set such that each point $p \in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that for any $p, q \in P$, there is…
A \emph{spanner} of a graph $G$ is a subgraph $H$ that approximately preserves shortest path distances in $G$. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner…
It is $\mathsf{NP}$-hard to determine the minimum number of branching vertices needed in a single-source distance-preserving subgraph of an undirected graph. We show that this problem can be solved in polynomial time if the input graph is…
In this paper we address the problem of computing a sparse subgraph of a weighted directed graph such that the exact distances from a designated source vertex to all other vertices are preserved under bounded weight increment. Finding a…
A $k$-additive spanner of a graph is a subgraph that preserves the distance between any two nodes up to a total additive error of $+k$. Efficient algorithms have been devised for constructing 2 [Aingworth et al. SIAM '99], 6 [Baswana et al.…
Let $G$ be an unweighted, undirected graph. An additive $k$-spanner of $G$ is a subgraph $H$ that approximates all distances between pairs of nodes up to an additive error of $+k$, that is, it satisfies $d_H(u,v) \le d_G(u,v)+k$ for all…
Given an undirected unweighted graph $G = (V, E)$ on $n$ vertices and $m$ edges, a subgraph $H\subseteq G$ is a spanner of $G$ with stretch function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, if for every pair $s, t$ of vertices in $V$,…
Graph spanners are sparse subgraphs that faithfully preserve the distances in the original graph up to small stretch. Spanner have been studied extensively as they have a wide range of applications ranging from distance oracles, labeling…
Fault-tolerant spanners are fundamental objects that preserve distances in graphs even under edge failures. A long line of work culminating in Bodwin, Dinitz, Robelle (SODA 2022) gives $(2k-1)$-stretch, $f$-fault-tolerant spanners with…
The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [Dist. Comp. '02] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two…
Fault tolerant distance preservers (spanners) are sparse subgraphs that preserve (approximate) distances between given pairs of vertices under edge or vertex failures. So-far, these structures have been studied mainly from a centralized…
An \emph{additive +$\beta W$ spanner} of an edge weighted graph $G=(V,E)$ is a subgraph $H$ of $G$ such that for every pair of vertices $u$ and $v$, $d_{H}(u,v) \le d_G(u,v) + \beta W$, where $d_G(u,v)$ is the shortest path length from $u$…