We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $Γ$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of $G$, is the spanning ratio of $Γ$.
First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio $1$, a proper straight-line drawing with spanning ratio $1$, and a planar straight-line drawing with spanning ratio $1$ are NP-complete, $\exists \mathbb R$-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex.
Second, we show that moving from spanning ratio $1$ to spanning ratio $1+ε$ allows us to draw every graph. Namely, we prove that, for every $ε>0$, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than $1+ε$.
Third, our drawings with spanning ratio smaller than $1+ε$ have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.
This tutorial review provides a guiding reference to researchers who want to have an overview of the large body of literature about graph spanners. It reviews the current literature covering various research streams about graph spanners, such as different formulations, sparsity and lightness results, computational complexity, dynamic algorithms, and applications. As an additional contribution, we offer a list of open problems on graph spanners.
Grigni and Hung~\cite{GH12} conjectured that H-minor-free graphs have $(1+ε)$-spanners that are light, that is, of weight $g(|H|,ε)$ times the weight of the minimum spanning tree for some function $g$. This conjecture implies the {\em efficient} polynomial-time approximation scheme (PTAS) of the traveling salesperson problem in $H$-minor free graphs; that is, a PTAS whose running time is of the form $2^{f(ε)}n^{O(1)}$ for some function $f$. The state of the art PTAS for TSP in H-minor-free-graphs has running time $n^{1/ε^c}$. We take a further step toward proving this conjecture by showing tha…
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 constructions can seamlessly handle edge weights, so long as error is measured \emph{multiplicatively}. In this work, we investigate whether one can similarly extend constructions of spanners with purely \emph{additive} error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for …
Tree spanners approximate distances within graphs; a subtree of a graph is a tree $t$-spanner of the graph if and only if for every pair of vertices their distance in the subtree is at most $t$ times their distance in the graph. When a graph contains a subtree of diameter at most $t$, then trivially admits a tree $t$-spanner. Now, determining whether a graph admits a tree $t$-spanner of diameter at most $t+1$ is an NP complete problem, when $t\geq 4$, and it is tractable, when $t\leq 3$. Although it is not known whether it is tractable to decide graphs that admit a tree 3-spanner of any diamet…
A tree t-spanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the tree t-spanner problem is to compute a tree t-spanner of G if one exists. This decision problem is known to be NP-complete even in the restricted class of unweighted planar graphs. We present a linear-time reduction from tree t-spanner in outerplanar graphs to the supply-demand tree partition problem. Based on this reduction, we obtain a linear-time algorithm to solve tree t…
We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8(29+23sqrt(2)). Enroute to this, we also show that the Yao graph Y4 in the Linf metric is a planar spanner with stretch factor 8.
It is an open problem whether Yao-Yao graphs $\mathsf{YY}_k$ (also known as sparse-Yao graphs) are all spanners when the integer parameter $k$ is large enough. In this paper we show that, for any integer $k\geq 42$, the Yao-Yao graph $\mathsf{YY}_{2k}$ is a $t_k$-spanner, with stretch factor $t_k=6.03+O(k^{-1})$ when $k$ tends to infinity. Our result generalizes the best known result which asserts that all $\mathsf{YY}_{6k}$ are spanners for $k$ large enough [Bauer and Damian, SODA'13]. Our proof is also somewhat simpler.
A tree $t$-spanner of a graph $G$ is a spanning tree of $G$ such that the distance between pairs of vertices in the tree is at most $t$ times their distance in $G$. Deciding tree $t$-spanner admissible graphs has been proved to be tractable for $t<3$ and NP-complete for $t>3$, while the complexity status of this problem is unresolved when $t=3$. For every $t>2$ and $b>0$, an efficient dynamic programming algorithm to decide tree $t$-spanner admissibility of graphs with vertex degrees less than $b$ is presented. Only for $t=3$, the algorithm remains efficient, when graphs $G$ with degrees less …
A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently. Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but als…
A roundtrip spanner of a directed graph $G$ is a subgraph of $G$ preserving roundtrip distances approximately for all pairs of vertices. Despite extensive research, there is still a small stretch gap between roundtrip spanners in directed graphs and undirected graphs. For a directed graph with real edge weights in $[1,W]$, we first propose a new deterministic algorithm that constructs a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}\log (nW))$ edges for every integer $k> 1$, then remove the dependence of size on $W$ to give a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/…
We introduce and investigate a new notion of resilience in graph spanners. Let $S$ be a spanner of a graph $G$. Roughly speaking, we say that a spanner $S$ is resilient if all its point-to-point distances are resilient to edge failures. Namely, whenever any edge in $G$ fails, then as a consequence of this failure all distances do not degrade in $S$ substantially more than in $G$ (i.e., the relative distance increases in $S$ are very close to those in the underlying graph $G$). In this paper we show that sparse resilient spanners exist, and that they can be computed efficiently.
Given a weighted graph $G(V,E)$ and $t \ge 1$, a subgraph $H$ is a \emph{$t$--spanner} of $G$ if the lengths of shortest paths in $G$ are preserved in $H$ up to a multiplicative factor of $t$. The \emph{subsetwise spanner} problem aims to preserve distances in $G$ for only a subset of the vertices. We generalize the minimum-cost subsetwise spanner problem to one where vertices appear on multiple levels, which we call the \emph{multi-level graph spanner} (MLGS) problem, and describe two simple heuristics. Applications of this problem include road/network building and multi-level graph visualiza…
Given an edge-weighted graph $G$ and $ε>0$, a $(1+ε)$-spanner is a spanning subgraph $G'$ whose shortest path distances approximate those of $G$ within a $(1+ε)$ factor. If $G$ is from certain minor-closed graph families (at least bounded genus graphs and apex graphs), then we know that light spanners exist. That is, we can compute a $(1+ε)$-spanner $G'$ with total edge weight at most a constant times the weight of a minimum spanning tree. This constant may depend on $ε$ and the graph family, but not on the particular graph $G$ nor on its edge weighting. For weighted graphs from several minor-…
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…
Seminal works on light spanners over the years have provided spanners with optimal lightness in various graph classes, such as general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (i) The runtimes of these constructions are almost always sub-optimal and usually far from optimal. (ii) These constructions are optimal in the standard and crude sense but not in a refined sense that takes into account a wider range of involved parameters. (iii) The techniques are ad hoc per graph class and thus can't be applied broadly. This work aim…
In approximation algorithm design, light spanners has applications in graph-metric problems such as metric TSP (the traveling salesman problem). We have developed an efficient algorithm for light spanners in bounded pathwidth graphs, based on an intermediate data structure called monotone tree. In this paper, we extended the results to include bounded catwidth graphs.
Given a directed graph G and an integer k >= 1, a k-transitive-closure-spanner (k-TCspanner) of G is a directed graph H that has (1) the same transitive-closure as G and (2) diameter at most k. In some applications, the shortcut paths added to the graph in order to obtain small diameter can use Steiner vertices, that is, vertices not in the original graph G. The resulting spanner is called a Steiner transitive-closure spanner (Steiner TC-spanner). Motivated by applications to property reconstruction and access control hierarchies, we concentrate on Steiner TC-spanners of directed acyclic graph…
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…
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 which approximately preserve all pairwise shortest-path distances in an input graph. The notion of approximation can be additive, multiplicative, or both, and many variants of this problem have been extensively studied. We study the problem of computing a graph spanner when the edges of the input graph are distributed across two or more sites in an arbitrary, possibly worst-case partition, and the goal is for the sites to minimize the communication used to output a spanner. We assume the message-passing model of communication, for which there is a point-to-p…
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 consider additive spanners of unweighted undirected graphs. Let $G$ be a graph and $H$ a subgraph of $G$. The most naïve way to construct an additive $k$-spanner of $G$ is the following: As long as $H$ is not an additive $k$-spanner repeat: Find a pair $(u,v) \in H$ that violates the spanner-condition and a shortest path from $u$ to $v$ in $G$. Add the edges of this path to $H$. We show that, with a very simple initial graph $H$, this naïve method gives additive $6$- and $2$-spanners of sizes matching the best known upper bounds. For additive $2$-spanners we start with $H=\emptyset$ and end…
We show that, for any integer k > 5, the Sparse-Yao graph YY_{6k} (also known as Yao-Yao) is a spanner with stretch factor 11.67. The stretch factor drops down to 4.75 for k > 7.
For a positive integer $t$ and a graph $G$, an additive $t$-spanner of $G$ is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus $t$. Minimum Additive $t$-Spanner Problem is to find an additive $t$-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive $t$-spanners, Minimum Additive $t$-Spanner Problem is hard to handle, and hence only few results are known for it. In this paper, we study Minimum Additive $t$-Spanner Pr…
The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a natural question to ask how much can we sparsify a graph and still guarantee that its diameter remains preserved within an approximation $t$. This property is captured by the notion of extremal-distance spanners. Given a graph $G=(V,E)$, a subgraph $H=(V,E_H)$ is defined to be a $t$-diameter spanner if the diameter of $H$ is at most $t$ times the diameter of $G$.…
There has been significant recent progress on algorithms for approximating graph spanners, i.e., algorithms which approximate the best spanner for a given input graph. Essentially all of these algorithms use the same basic LP relaxation, so a variety of papers have studied the limitations of this approach and proved integrality gaps for this LP in a variety of settings. We extend these results by showing that even the strongest lift-and-project methods cannot help significantly, by proving polynomial integrality gaps even for $n^{Ω(ε)}$ levels of the Lasserre hierarchy, for both the directed 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 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…
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…
We prove that the size of the sparsest directed k-spanner of a graph can be approximated in polynomial time to within a factor of $\tilde{O}(\sqrt{n})$, for all k >= 3. This improves the $\tilde{O}(n^{2/3})$-approximation recently shown by Dinitz and Krauthgamer.
The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive. In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is \emph{existentially optimal} (or existentially near-optimal) for several important graph families, in terms of both the size and weight. Roughly speaking, the greedy spanner is said to be existen…
We show that every $H$-minor-free graph has a light $(1+ε)$-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung. Our lightness bound is \[O\left(\frac{σ_H}{ε^3}\log \frac{1}ε\right)\] where $σ_H = |V(H)|\sqrt{\log |V(H)|}$ is the sparsity coefficient of $H$-minor-free graphs. That is, it has a practical dependency on the size of the minor $H$. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in $H$-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient …
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…
Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (1) The techniques are ad hoc per graph class, and thus can't be applied broadly. (2) The runtimes of these constructions are almost always sub-optimal, and usually far from optimal. (3) These constructions are optimal in the standard and crude sense, but not in a refined sense that takes into account a wider range of involved parameters. This work aims 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. ACM '10, Woodruff ICALP '13], and 8-additive spanners [Knudsen '17], but no efficient algorithms for 4-additive spanners have yet been discovered. In this paper we present a modification of Chechik's 4-additive spanner construction [Chechik SODA '13] that produces a 4-additive spanner on $\Oish(n^{7/5})$ edges, with an improved runtime of $\Oish(mn^{3/5})$ …
Resolving an open question from 2006, we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple $\mathcal{O}(\log^*n)$-round distributed algorithm in the LOCAL model of computation, that given a unit ball graph $G$ with $n$ vertices and a positive constant $ε< 1$ finds a $(1+ε)$-spanner with constant bounds on its maximum degree and its lightness using only 2-hop neighborhood information. This immediately improves the best prior lightness bound, the algorithm of Damian, Pandit, and Pemmaraju, which r…
A $t$-spanner of a graph $G$ is a subgraph $H$ in which all distances are preserved up to a multiplicative $t$ factor. A classical result of Althöfer et al. is that for every integer $k$ and every graph $G$, there is a $(2k-1)$-spanner of $G$ with at most $O(n^{1+1/k})$ edges. But for some settings the more interesting notion is not the number of edges, but the degrees of the nodes. This spurred interest in and study of spanners with small maximum degree. However, this is not necessarily a robust enough objective: we would like spanners that not only have small maximum degree, but also have "f…
Given a directed graph G = (V,E) and an integer k>=1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, E_H) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were implicitly studied in access control, data structures, and property testing, and properties of these spanners have been rediscovered over the span of 20 years. The main goal in each of these applications is to obtain the sparsest k-TC-spanners. We bring these diverse areas under the unifying framework of TC-spanners. We initiate the study of approximability of the s…
Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing very sparse small-stretch subgraphs. Remarkably, it was then shown that the known (+6)-spanner constructions are essentially the sparsest possible, that is, a larger additive stretch cannot guarantee a sparser spanner, which brought the stretch-sparsity trade-off to its limit. Distributed constructions of spanners are also abundant. However, for additive spanners, while there wer…
We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in $H$-minor-free graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in minor-closed families initiated by Grigni, Koutsoupias and Papadimitriou in FOCS'95. The main technical ingredient in our PTAS is a construction of a nearly light subset $(1+ε)$-spanner for any given edge-weighted $H$-minor-free graph. This construction is based on a necessary and sufficient condition given by \emph{sparse spanner oracles}: light subset spanners exist if and only if sparse spanner or…
Given an undirected graph $G=(V,E)$ on $n$ vertices, $m$ edges, and an integer $t\ge 1$, a subgraph $(V,E_S)$, $E_S\subseteq E$ is called a $t$-spanner if for any pair of vertices $u,v \in V$, the distance between them in the subgraph is at most $t$ times the actual distance. We present streaming algorithms for computing a $t$-spanner of essentially optimal size-stretch trade offs for any undirected graph. Our first algorithm is for the classical streaming model and works for unweighted graphs only. The algorithm performs a single pass on the stream of edges and requires $O(m)$ time to process…
Graph sketching is a powerful technique introduced by the seminal work of Ahn, Guha and McGregor'12 on connectivity in dynamic graph streams that has enjoyed considerable attention in the literature since then, and has led to near optimal dynamic streaming algorithms for many fundamental problems such as connectivity, cut and spectral sparsifiers and matchings. Interestingly, however, the sketching and dynamic streaming complexity of approximating the shortest path metric of a graph is still far from well-understood. Besides a direct $k$-pass implementation of classical spanner constructions (…
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…
A spanner graph on a set of points in $R^d$ contains a shortest path between any pair of points with length at most a constant factor of their Euclidean distance. In this paper we investigate new models and aim to interpret why good spanners 'emerge' in reality, when they are clearly built in pieces by agents with their own interests and the construction is not coordinated. Our main result is to show that if edges are built in an arbitrary order but an edge is built if and only if its endpoints are not 'close' to the endpoints of an existing edge, the graph is a $(1 + \eps)$-spanner with a lin…
Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA'12) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes.…
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 …
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. …
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…
