相关论文: Near-Additive Spanners and Near-Exact Hopsets, A U…

Given an undirected graph $G=(V,E)$, an {\em $(\alpha,\beta)$-spanner} $H=(V,E')$ is a subgraph that approximately preserves distances; for every $u,v\in V$, $d_H(u,v)\le \alpha\cdot d_G(u,v)+\beta$. An $(\alpha,\beta)$-hopset is a graph…

Given an undirected possibly weighted $n$-vertex graph $G=(V,E)$ and a set $\mathcal{P}\subseteq V^2$ of pairs, a subgraph $S=(V,E')$ is called a ${\cal P}$-pairwise $\alpha$-spanner of $G$, if for every pair $(u,v)\in\mathcal{P}$ we have…

An $(\alpha,\beta)$-spanner of a weighted graph $G=(V,E)$, is a subgraph $H$ such that for every $u,v\in V$, $d_G(u,v) \le d_H(u,v)\le\alpha\cdot d_G(u,v)+\beta$. The main parameters of interest for spanners are their size (number of edges)…

Given parameters $\alpha\geq 1,\beta\geq 0$, a subgraph $G'=(V,H)$ of an $n$-vertex unweighted undirected graph $G=(V,E)$ is called an $(\alpha,\beta)$-spanner if for every pair $u,v\in V$ of vertices, $d_{G'}(u,v)\leq \alpha…

An $f(d)$-spanner of an unweighted $n$-vertex graph $G=(V,E)$ is a subgraph $H$ satisfying that $dist_H(u, v)$ is at most $f(dist_G(u, v))$ for every $u,v \in V$. We present new spanner constructions that achieve a nearly optimal stretch of…

For a positive parameter $\beta$, the $\beta$-bounded distance between a pair of vertices $u,v$ in a weighted undirected graph $G = (V,E,\omega)$ is the length of the shortest $u-v$ path in $G$ with at most $\beta$ edges, aka {\em hops}.…

Given a graph $G = (V,E)$, a subgraph $H$ is an \emph{additive $+\beta$ spanner} if $\dist_H(u,v) \le \dist_G(u,v) + \beta$ for all $u, v \in V$. A \emph{pairwise spanner} is a spanner for which the above inequality only must hold for…

A $(\beta,\epsilon)$-hopset for a weighted undirected $n$-vertex graph $G=(V,E)$ is a set of edges, whose addition to the graph guarantees that every pair of vertices has a path between them that contains at most $\beta$ edges, whose length…

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…

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 $V$,…

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…

Hopsets and spanners are fundamental graph structures, playing a key role in shortest path computation, distributed communication, and more. A (near-exact) hopset for a given graph $G$ is a (small) subset of weighted edges $H$ that when…

For any constants $d\ge 1$, $\epsilon >0$, $t>1$, and any $n$-point set $P\subset\mathbb{R}^d$, we show that there is a geometric graph $G=(P,E)$ having $O(n\log^2 n\log\log n)$ edges with the following property: For any $F\subseteq P$,…

An \emph{additive $+\beta$ spanner} of a graph $G$ is a subgraph which preserves distances up to an additive $+\beta$ error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted…

A $t$-spanner of a graph $G=(V,E)$ is a subgraph $H=(V,E')$ that contains a $uv$-path of length at most $t$ for every $uv\in E$. It is known that every $n$-vertex graph admits a $(2k-1)$-spanner with $O(n^{1+1/k})$ edges for $k\geq 1$. This…

Given a weighted graph $G$, a $(\beta,\varepsilon)$-hopset $H$ is an edge set such that for any $s,t \in V(G)$, where $s$ can reach $t$ in $G$, there is a path from $s$ to $t$ in $G \cup H$ which uses at most $\beta$ hops whose length is in…

Given a weighted undirected graph $G=(V,E,w)$, a hopset $H$ of hopbound $\beta$ and stretch $(1+\epsilon)$ is a set of edges such that for any pair of nodes $u, v \in V$, there is a path in $G \cup H$ of at most $\beta$ hops, whose length…

A $(\beta,\epsilon)$-$\textit{hopset}$ is, informally, a weighted edge set that, when added to a graph, allows one to get from point $a$ to point $b$ using a path with at most $\beta$ edges ("hops") and length…

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 $u$…

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…