Given an undirected graph $G=(V,E)$, an {\em $(α,β)$-spanner} $H=(V,E')$ is a subgraph that approximately preserves distances; for every $u,v\in V$, $d_H(u,v)\le α\cdot d_G(u,v)+β$. An $(α,β)$-hopset is a graph $H=(V,E")$, so that adding its edges to $G$ guarantees every pair has an $α$-approximate shortest path that has at most $β$ edges (hops), that is, $d_G(u,v)\le d_{G\cup H}^{(β)}(u,v)\le α\cdot d_G(u,v)$. Given the usefulness of spanners and hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter $α$.
In this work we develop a single algorithm that can attain all state-of-the-art spanners and hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm.
In \cite{BP20}, given a parameter $k$, a $(O(k^ε),O(k^{1-ε}))$-hopset of size $\tilde{O}(n^{1+1/k})$ was shown for any $n$-vertex graph and parameter $0<ε<1$, and they asked whether this result is best possible. We resolve this open problem, showing that any $(α,β)$-hopset of size $O(n^{1+1/k})$ must have $α\cdot β\geΩ(k)$.
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 added to the graph $G$ reduces the number of hops (edges) of near-exact shortest paths. Spanners and distance preservers, on the other hand, ask for removing many edges from the graph while approximately preserving shortest path distances. We provide a general reduction scheme from graph hopsets to the known metric compression schemes of spanners, emulators and…
We use exponential start time clustering to design faster and more work-efficient parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of the overall process, giving: * Linear work parallel algorithms that construct spanners with $O(k)$ stretch and size $O(n^{1+1/k})$ in unweighted graphs, and size $O(n^{1+1/k} \log k)$ in weighted graphs. * Hopsets that lead to…
Given an {\em unweighted} undirected graph $G = (V,E)$, and a pair of parameters $ε> 0$, $β= 1,2,\ldots$, a subgraph $G' =(V,H)$, $H \subseteq E$, of $G$ is a {\em $(1+ε,β)$-spanner} (aka, a {\em near-additive spanner}) of $G$ if for every $u,v \in V$, $$d_{G'}(u,v) \le (1+ε)d_G(u,v) + β~.$$ It was shown in \cite{EP01} that for any $n$-vertex $G$ as above, and any $ε> 0$ and $κ= 1,2,\ldots$, there exists a $(1+ε,β)$-spanner $G'$ with $O_{ε,κ}(n^{1+1/κ})$ edges, with $$β= β_{EP} = \left({{\log κ} \over ε}\right)^{\log κ- 2}~.$$ This bound remains state-of-the-art, and its dependence on $ε$ (for…
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 $O(\lceil k /d \rceil)$ for any distance value $d \in [1,k^{1-o(1)}]$, and $d \geq k^{1+o(1)}$. We show the following: 1. There exists an $f(d)$-spanner $H \subseteq G$ with $f(d)\leq 7k$ for any $d \in [1,\sqrt{k}/2]$ with expected size $O_{k}(n^{1+1/k})$. This in particular gives $(α,β)$ spanners with $α=O(\sqrt{k})$ and $β=O(k)$. 2. For any $ε\in (0,1/2]…
To our knowledge, there are only two known algorithms for constructing sparse and light spanners for general graphs. One of them is the greedy algorithm of Alth$\ddot{o}$fer et al. \cite{ADDJS93}, analyzed by Chandra et al. in SoCG'92. The greedy algorithm consructs, for every \emph{weighted} undirected $n$-vertex $m$-edge graph $G = (V,E)$ and any integer $k \ge 1$, a $(2k-1)$-spanner with $O(n^{1 + 1/k})$ edges and weight $O(k \cdot n^{(1+\eps)/k}) \cdot ω(MST(G))$, for any $\eps > 0$. The drawback of the greedy algorithm is that it requires $O(m \cdot (n^{1 + 1/k} + n \cdot \log n))$ time. …
We present the first polynomial time algorithm for the f vertex fault tolerant spanner problem, which achieves almost optimal spanner size. Our algorithm for constructing f vertex fault tolerant spanner takes $O(k\cdot n\cdot m^2 \cdot W)$ time, where W is the maximum edge weight, and constructs a spanner of size $O(n^{1+1/k}f^{1-1/k}\cdot (\log n)^{1-1/k})$. Our spanner has almost optimal size and is at most a $\log n$ factor away from the upper bound on the worst-case size. Prior to this work, no other polynomial time algorithm was known for constructing f vertex fault tolerant spanner with …
Consider an undirected weighted graph $G = (V,E,w)$. We study the problem of computing $(1+ε)$-approximate shortest paths for $S \times V$, for a subset $S \subseteq V$ of $|S| = n^r$ sources, for some $0 < r \le 1$. We devise a significantly improved algorithm for this problem in the entire range of parameter $r$, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of $r$ in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time $\tilde{O}(|E| \cdot n^{o(1)} + n^{ω(r)})$, where $n^{ω(r)}…
A $t$-spanner of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality -- in this work we focus on the latter. Specifically, it is shown that for any parameters $k\ge 1$ and $ε>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+ε)$-stretch spanner of weight at most $w(MST(G))\cdot O_ε(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a mi…
Graph spanners are fundamental graph structures with a wide range of applications in distributed networks. We consider a standard synchronous message passing model where in each round $O(\log n)$ bits can be transmitted over every edge (the CONGEST model). The state of the art of deterministic distributed spanner constructions suffers from large messages. The only exception is the work of Derbel et al. '10, which computes an optimal-sized $(2k-1)$-spanner but uses $O(n^{1-1/k})$ rounds. In this paper, we significantly improve this bound. We present a deterministic distributed algorithm that gi…
Given a weighted undirected graph $G=(V,E,w)$, a hopset $H$ of hopbound $β$ and stretch $(1+ε)$ 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 $β$ hops, whose length is within a $(1+ε)$ factor from the distance between $u$ and $v$ in $G$. We show the first efficient decremental algorithm for maintaining hopsets with a polylogarithmic hopbound. The update time of our algorithm matches the best known static algorithm up to polylogarithmic factors. All the previous decremental hopset constructions had a superpolylogarithmic (but subpolynom…
A multiplicative $α$-spanner $H$ is a subgraph of $G=(V,E)$ with the same vertices and fewer edges that preserves distances up to the factor $α$, i.e., $d_H(u,v)\leqα\cdot d_G(u,v)$ for all vertices $u$, $v$. While many algorithms have been developed to find good spanners in terms of approximation guarantees, no experimental studies comparing different approaches exist. We implemented a rich selection of those algorithms and evaluate them on a variety of instances regarding, e.g., their running time, sparseness, lightness, and effective stretch.
A $t$-{\em spanner} $H$ of a weighted graph $G=(V,E,w)$ is a subgraph that approximates all pairwise distances up to a factor of $t$. The {\em lightness} of $H$ is defined as the ratio between the weight of $H$ to that of the minimum spanning tree. An $(α,β)$-{\em Shallow Light Tree} (SLT) is a tree of lightness $β$, that approximates all distances from a designated root vertex up to a factor of $α$. A long line of works resulted in efficient algorithms that produce (nearly) optimal light spanners and SLTs. Some of the most notable algorithmic applications of light spanners and SLTs are in dis…
An $(α,β)$-spanner of an $n$-vertex graph $G=(V,E)$ is a subgraph $H$ of $G$ satisfying that $dist(u, v, H) \leq α\cdot dist(u, v, G)+β$ for every pair $(u, v)\in V \times V$, where $dist(u,v,G')$ denotes the distance between $u$ and $v$ in $G' \subseteq G$. It is known that for every integer $k \geq 1$, every graph $G$ has a polynomially constructible $(2k-1,0)$-spanner of size $O(n^{1+1/k})$. This size-stretch bound is essentially optimal by the girth conjecture. It is therefore intriguing to ask if one can "bypass" the conjecture by settling for a multiplicative stretch of $2k-1$ only for \…
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. Only very recently [ABSKS20] extended the classical +2 (respectively, +4) spanners for unweighted graphs of size $O(n^{3/2})$ (resp., $O(n^{7/5})$) to the weighted setting, where the additive error is $+2W$ (resp., $+4W$). This means that for every pair $u,v$, the additive stretch is at most $+2W_{u,v}$, where $W_{u,v}$ is the maximal edge weight on the shortes…
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 schemes and routing to solving linear systems and spectral sparsification. A $k$-spanner maintains pairwise distances up to multiplicative factor of $k$. It is a folklore that for every $n$-vertex graph $G$, one can construct a $(2k-1)$ spanner with $O(n^{1+1/k})$ edges. In a distributed setting, such spanners can be constructed in the standard CONGEST model u…
In this paper we devise a novel \emph{unified} construction of Euclidean spanners that trades between the maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1+eps)-spanner with maximum degree O(k), diameter O(log_k n + alpha(k)), weight O(k \cdot log_k n \cdot log n) \cdot w(MST(S)), and O(n) edges. Note that for k= n^{1/alpha(n)} this gives rise to diameter O(alpha(n)), weight O(n^{1/alpha(n)} \cdot log n \cdot alpha(n)) \cdot w(MST(S)) and maximum degree O(n^{1/alpha(n)}), which improves upon a classical result of Arya et al. \cite{ADMSS95}…
This paper presents significantly improved deterministic algorithms for some of the key problems in the area of distributed graph algorithms, including network decomposition, hitting sets, and spanners. As the main ingredient in these results, we develop novel randomized distributed algorithms that we can analyze using only pairwise independence, and we can thus derandomize efficiently. As our most prominent end-result, we obtain a deterministic construction for $O(\log n)$-color $O(\log n \cdot \log\log\log n)$-strong diameter network decomposition in $\tilde{O}(\log^3 n)$ rounds. This is the…
A tree $t$-spanner $T$ of a graph $G$ is a spanning tree of $G$ such that the distance in $T$ between every pair of verices is at most $t$ times the distance in $G$ between them. There are efficient algorithms that find a tree $t\cdot O(\log n)$-spanner of a graph $G$, when $G$ admits a tree $t$-spanner. In this paper, the search space is narrowed to $v$-concentrated spanning trees, a simple family that includes all the breadth first search trees starting from vertex $v$. In this case, it is not easy to find approximate tree spanners within factor almost $o(\log n)$. Specifically, let $m$ and …
In STOC'95 Arya et al. (1995) conjectured that for any constant dimensional $n$-point Euclidean space, a $(1+\eps)$-spanner with constant degree, hop-diameter $O(\log n)$ and weight $O(\log n) \cdot ω(MST)$ can be built in $O(n \log n)$ time. Recently Elkin and Solomon (technical report, April 2012) proved this conjecture of Arya et al. in the affirmative. In fact, the proof of Elkin and Solomon is more general in two ways. First, it applies to arbitrary doubling metrics. Second, it provides a complete tradeoff between the three involved parameters that is tight (up to constant factors) in the…
We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor $C_{del\approx 2.42$; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor $ρ=1+2π(k\cos{\frac{π{k)^{-1$ and degree bounded by $k$, for any integer parameter $k\geq 14$. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean gra…
Spanners have been shown to be a powerful tool in graph algorithms. Many spanner constructions use a certain type of clustering at their core, where each cluster has small diameter and there are relatively few spanner edges between clusters. In this paper, we provide a clustering algorithm that, given $k\geq 2$, can be used to compute a spanner of stretch $2k-1$ and expected size $O(n^{1+1/k})$ in $k$ rounds in the CONGEST model. This improves upon the state of the art (by Elkin, and Neiman [TALG'19]) by making the bounds on both running time and stretch independent of the random choices of th…
We show that for every $n$-point metric space $M$ there exists a spanning tree $T$ with unweighted diameter $O(\log n)$ and weight $ω(T) = O(\log n) \cdot ω(MST(M))$. Moreover, there is a designated point $rt$ such that for every point $v$, $dist_T(rt,v) \le (1+ε) \cdot dist_M(rt,v)$, for an arbitrarily small constant $ε> 0$. We extend this result, and provide a tradeoff between unweighted diameter and weight, and prove that this tradeoff is \emph{tight up to constant factors} in the entire range of parameters. These results enable us to settle a long-standing open question in Computational Ge…
Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied during the last decade. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of $2κ-1$ versus $n^{1+1/κ}$, for an integer parameter $κ$. (In fact, existing solutions also incur an extra factor of $1+ε$ in the stretch for weighted graphs, and an…
Let $(V,δ)$ be a finite metric space, where $V$ is a set of $n$ points and $δ$ is a distance function defined for these points. Assume that $(V,δ)$ has a constant doubling dimension $d$ and assume that each point $p\in V$ has a disk of radius $r(p)$ around it. The disk graph that corresponds to $V$ and $r(\cdot)$ is a \emph{directed} graph $I(V,E,r)$, whose vertices are the points of $V$ and whose edge set includes a directed edge from $p$ to $q$ if $δ(p,q)\leq r(p)$. In \cite{PeRo08} we presented an algorithm for constructing a $(1+\eps)$-spanner of size $O(n/\eps^d \log M)$, where $M$ is the…
Given a metric space $\mathcal{M}=(X,δ)$, a weighted graph $G$ over $X$ is a metric $t$-spanner of $\mathcal{M}$ if for every $u,v \in X$, $δ(u,v)\le d_G(u,v)\le t\cdot δ(u,v)$, where $d_G$ is the shortest path metric in $G$. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points $(s_1, \ldots, s_n)$, where the points are presented one at a time (i.e., after $i$ steps, we saw $S_i = \{s_1, \ldots , s_i\}$). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to re…
In the distributed setting, the only existing constructions of \textit{sparse skeletons}, (i.e., subgraphs with $O(n)$ edges) either use randomization or large messages, or require $Ω(D)$ time, where $D$ is the hop-diameter of the input graph $G$. We devise the first deterministic distributed algorithm in the CONGEST model (i.e., uses small messages) for constructing linear-size skeletons in time $2^{O(\sqrt{{\log n}\cdot{\log{\log n}}})}$. We can also compute a linear-size spanner with stretch $polylog(n)$ in low deterministic polynomial time, i.e., $O(n^ρ)$ for an arbitrarily small constant …
In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph $G$ {\em admits a system of $μ$ collective additive tree $r$-spanners} (resp., {\em multiplicative tree $t$-spanners}) if there is a system $\cT(G)$ of at most $μ$ spanning trees of $G$ such that for any two vertices $x,y$ of $G$ a spanning tree $T\in \cT(G)$ exists such that $d_T(x,y)\leq d_G(x,y)+r$ (resp., $d_T(x,y)\leq t\cdot d_G(x,y)$). When $μ=1$ one gets the notion of {…
We (nearly) settle the time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT spanners are sparse subgraphs that preserve distance information, up to a small multiplicative stretch, in the presence of vertex failures. These structures were introduced by [Chechik et al., STOC 2009] and have received a lot of attention since then. We provide algorithms for computing nearly optimal $f$-VFT spanners for any $n$-vertex $m$-edge graph, with near optimal running time in several computational models: - A randomized sequential algor…
This paper presents efficient distributed algorithms for a number of fundamental problems in the area of graph sparsification: We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in $\textrm{polylog}(n)$ rounds in weighted graphs. Concretely, our algorithm outputs a spanning subgraph with only $n+o(n)$ edges in which the pairwise distances are stretched by a factor of at most $O(\log n \;\cdot\; 2^{O(\log^* n)})$. We provide a $\textrm{polylog}(n)$-round deterministic distributed algorithm that computes a spanner with stretch $(2k-1)$ and $O(nk + n^{1…
A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in t…
A new \emph{spanner} construction algorithm is presented, working under the \emph{LOCAL} model with unique edge IDs. Given an $n$-node communication graph, a spanner with a constant stretch and $O (n^{1 + \varepsilon})$ edges (for an arbitrarily small constant $\varepsilon > 0$) is constructed in a constant number of rounds sending $O (n^{1 + \varepsilon})$ messages whp. Consequently, we conclude that every $t$-round LOCAL algorithm can be transformed into an $O (t)$-round LOCAL algorithm that sends $O (t \cdot n^{1 + \varepsilon})$ messages whp. This improves upon all previous message-reducti…
Let $G=(V,E,w)$ be a weighted undirected graph with $n$ vertices and $m$ edges, and fix a set of $s$ sources $S\subseteq V$. We study the problem of computing {\em almost shortest paths} (ASP) for all pairs in $S \times V$ in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of $1+ε$, for an arbitrarily small constant $ε> 0$ . In this regime existing centralized algorithms require $Ω(\min\{|E|s,n^ω\})$ time, where $ω< 2.372$ is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka tim…
