Graph Spanners Research Articles

Blackout-tolerant temporal spanners

We introduce the notions of blackout-tolerant temporal α-spanner of a temporal graph G which is a subgraph of G that preserves the distances between pairs of vertices of interest in G up to a multiplicative factor of α, even when the graph edges at a single time-instant become unavailable. In particular, we consider the single-source, single-pair, and all-pairs cases and, for each case we look at three quality requirements: exact distances (i.e., α=1), almost-exact distances (i.e., α=1+ε for an arbitrarily small constant ε>0), and connectivity (i.e., unbounded α). We provide almost tight bounds on the size of such spanners for general temporal graphs and for temporal cliques, showing that they are either very sparse (i.e., they have O˜(n) edges) or they must have size Ω(n2) in the worst case, where n is the number of vertices of G. We also investigate multiple blackouts and k-edge fault-tolerant temporal spanners.

Faster Edge‐Path Bundling through Graph Spanners

AbstractEdge‐Path bundling is a recent edge bundling approach that does not incur ambiguities caused by bundling disconnected edges together. Although the approach produces less ambiguous bundlings, it suffers from high computational cost. In this paper, we present a new Edge‐Path bundling approach that increases the computational speed of the algorithm without reducing the quality of the bundling. First, we demonstrate that biconnected components can be processed separately in an Edge‐Path bundling of a graph without changing the result. Then, we present a new edge bundling algorithm that is based on observing and exploiting a strong relationship between Edge‐Path bundling and graph spanners. Although the worst case complexity of the approach is the same as of the original Edge‐Path bundling algorithm, we conduct experiments to demonstrate that the new approach is 5–256 times faster than Edge‐Path bundling depending on the dataset, which brings its practical running time more in line with traditional edge bundling algorithms.

Reliable Spanners for Metric Spaces

A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes can only cause a small damage in the remaining graph in terms of the dilation. In other words, the spanner property is maintained for almost all nodes in the residual graph. Constructions of reliable spanners of near linear size are known in the low-dimensional Euclidean settings. Here, we present new constructions of reliable spanners for planar graphs, trees, and (general) metric spaces.

Quantum Speedup for Graph Sparsification, Cut Approximation, and Laplacian Solving

Graph underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, sparsification reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. In this work we demonstrate a polynomial quantum speedup for spectral and many of its applications. In particular, we give a quantum algorithm that, given a weighted graph with $n$ nodes and $m$ edges, outputs a classical description of an $\epsilon$-spectral sparsifier in sublinear time $\tilde{O}(\sqrt{mn}/\epsilon)$. This contrasts with the optimal classical complexity $\tilde{O}(m)$. We also prove that our quantum algorithm is optimal up to polylog-factors. The algorithm builds on a string of existing results on sparsification, graph spanners, quantum algorithms for shortest paths, and efficient constructions for $k$-wise independent random strings. Our algorithm implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut.

The statistics of the Debye–Hückel limiting law

The Debye–Hückel Limiting Law (DHLL) correctly predicts the thermodynamic behavior of dilute electrolyte solutions. Most articles and books explain this law using Peter Debye and Erich Hückel’s original formalism of linearizing the Poisson–Boltzmann equation for a simple electrolyte model. Brilliant in its own right, this approach does not fully explain which microstates contribute in the range of the Debye–Hückel theory. Notably, the original formalism does not establish the Energy Multiplicity Distribution (EMD), which is the energy distribution of a system’s microstates. This work establishes an analytical expression for the EMD that satisfies the DHLL. Specifically, an EMD that is proportional to exp(aUel3) satisfies the DHLL for a monovalent electrolyte solution. Here, Uel is the effective electrostatic energy due to ion–ion interactions. The proposed proportionality shows quantitative agreement with the simulated EMDs of a Coulomb lattice gas that corresponds to an aqueous sodium chloride solution at a concentration of 3.559 × 10−4 M. The lattice gas that is used does not incorporate solvent molecules, but the Coulomb interactions are scaled through a permittivity that emulates the solvent—similar to the Debye–Hückel theory. Moreover, this work explains the proportionality by partitioning Uel into a set of energy contributions using minimal spanning graphs. This discussion on the EMD is new in the field. It widens the scope of the Debye–Hückel theory and could lead to a new parameterization option for developing equations of state.

Constructing light spanners deterministically in near-linear time

Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [10] improved the state-of-the-art for light spanners by constructing a (2k−1)(1+ε)-spanner with O(n1+1k) edges and Oε(n1k) lightness. Soon after, Filtser and Solomon [18] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of O(mn1+1k) (which is faster than [10]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Ωε(kn1k), even when randomization is used.The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an Oε(n2+1k+ε′) time spanner construction which achieves the state-of-the-art bounds. Our second result is an Oε(m+nlog⁡n) time construction of a spanner with (2k−1)(1+ε) stretch, O(log⁡k⋅n1+1k) edges and Oε(log⁡k⋅n1k) lightness. This is an exponential improvement in the dependence on k compared to the previous result with such running time. Finally, for the important special case where k=log⁡n, for every constant ε>0, we provide an O(m+n1+ε) time construction that produces an O(log⁡n)-spanner with O(n) edges and O(1) lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any k=ω(1).To achieve our constructions, we show a novel deterministic incremental approximate distance oracle. Our new oracle is crucial in our construction, as known randomized dynamic oracles require the assumption of a non-adaptive adversary. This is a strong assumption, which has seen recent attention in prolific venues. Our new oracle allows the order of the edge insertions to not be fixed in advance, which is critical as our spanner algorithm chooses which edges to insert based on the answers to distance queries. We believe our new oracle is of independent interest.

The threshold bias of the clique-factor game

Let r≥4 be an integer and consider the following game on the complete graph Kn for n∈rZ: Two players, Maker and Breaker, alternately claim previously unclaimed edges of Kn such that in each turn Maker claims one and Breaker claims b∈N edges. Maker wins if her graph contains a Kr-factor, that is a collection of n/r vertex-disjoint copies of Kr, and Breaker wins otherwise. In other words, we consider a b-biased Kr-factor Maker–Breaker game. We show that the threshold bias for this game is of order n2/(r+2). This makes a step towards determining the threshold bias for making bounded-degree spanning graphs and extends a result of Allen et al. who resolved the case r∈{3,4} up to a logarithmic factor.

Affine invariant triangulations

We study affine invariant 2D triangulation methods. That is, methods that produce the same triangulation for a point set S for any (unknown) affine transformation of S. Our work is based on a method by Nielson (1993) that uses the inverse of the covariance matrix of S to define an affine invariant norm, denoted AS, and an affine invariant triangulation, denoted DTAS[S]. We revisit the AS-norm from a geometric perspective, and show that DTAS[S] can be seen as a standard Delaunay triangulation of a transformed point set based on S. We prove that it retains all of its well-known properties such as being 1-tough, containing a perfect matching, and being a constant spanner of the complete geometric graph of S. We show that the AS-norm extends to a hierarchy of related geometric structures such as the minimum spanning tree, nearest neighbor graph, Gabriel graph, relative neighborhood graph, and higher order versions of these graphs. In addition, we provide different affine invariant sorting methods of a point set S and of the vertices of a polygon P that can be combined with known algorithms to obtain other affine invariant triangulation methods of S and of P. Our sorting methods provide a novel alternative to computing affine invariant geometric structures that are computationally simpler than using the AS-norm.In this work we focus on the theoretical part of affine invariant methods and as such do not provide experimental results.

Expected Path Degradation when Searching over a Sparse Grid Hierarchy

The traditional focus of combinatorial search research is to speed up the search algorithm. An alternative, however, is to create a sparser representation of the search space. This relates to the idea of spanners from graph theory. These are subgraphs which retain paths between any two vertices of the original graph while guaranteeing a maximum stretch in path length. In practice, the path degradation of graph spanners is significantly smaller than the theoretical bound. Even so, expected path degradation of graph spanners is typically not studied. This work focuses on grid path-finding to propose an algorithm that constructs a grid spanner, where analysis for the obstacle-free case shows that significant performance gains can be achieved with a small decrease in expected path quality. This is an important first step towards studying the expected performance of spanners. Experiments on game maps show that expected path quality with obstacles is only sometimes marginally lower than that in the obstacle-free case and that a significant reduction in the size of the search space can be achieved.

New Results on Linear Size Distance Preservers

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 3 January 2019Accepted: 30 December 2020Published online: 12 April 2021Keywordsshortest paths, graph spanners, extremal graph theory, distance preserversAMS Subject Headings68, 05Publication DataISSN (print): 0097-5397ISSN (online): 1095-7111Publisher: Society for Industrial and Applied MathematicsCODEN: smjcat

Very fast construction of bounded‐degree spanning graphs via the semi‐random graph process

In this paper, we study the following recently proposed semi‐random graph process: starting with an empty graph on n vertices, the process proceeds in rounds, where in each round we are given a uniformly random vertex v, and must immediately (in an online manner) add to our graph an edge incident with v. The end goal is to make the constructed graph satisfy some predetermined monotone graph property. Alon asked whether every given bounded‐degree spanning graph can be constructed with high probability in O(n) rounds. We answer this question positively in a strong sense, showing that any n‐vertex graph with maximum degree can be constructed with high probability in rounds. This is tight up to a multiplicative factor of . We also obtain tight bounds for the number of rounds necessary to embed bounded‐degree spanning trees, and consider a nonadaptive variant of this setting.

Multi-colored spanning graphs

We study a problem motivated by sparse set visualization. Given n points in the plane, each labeled with one or more primary colors, a colored spanning graph (for short, CSG) is a graph in which the vertices of each primary color induce a connected subgraph. The Min-CSG problem asks for the minimum sum of edge lengths in a colored spanning graph. We show that the problem is NP-hard for k primary colors when k≥3 and provide a (2−13+2ϱ)-approximation algorithm for k=3 that runs in polynomial time, where ϱ is the Steiner ratio. Further, we give an O(n) time algorithm in the special case that the given points are collinear and k is constant.

Plane hop spanners for unit disk graphs: Simpler and better

The unit disk graph (UDG) is a widely employed model for the study of wireless networks. In this model, wireless nodes are represented by points in the plane and there is an edge between two points if and only if their Euclidean distance is at most one. A hop spanner for the UDG is a spanning subgraph H such that for every edge (p,q) in the UDG the topological shortest path between p and q in H has a constant number of edges. The hop stretch factor of H is the maximum number of edges of these paths. A hop spanner is plane (i.e. embedded planar) if its edges do not cross each other.The problem of constructing hop spanners for the UDG has received considerable attention in both computational geometry and wireless ad hoc networks. Despite this attention, there has not been significant progress on getting hop spanners that (i) are plane, and (ii) have low hop stretch factor. Previous constructions either do not ensure the planarity or have high hop stretch factor. The only construction that satisfies both conditions is due to Catusse, Chepoi, and Vaxès (2010); their plane hop spanner has hop stretch factor at most 449.Our main result is a simple algorithm that constructs a plane hop spanner for the UDG. In addition to the simplicity, the hop stretch factor of the constructed spanner is at most 341. Even though the algorithm itself is simple, its analysis is rather involved. Several results on the plane geometry are established in the course of the proof. These results are of independent interest.

Local Algorithms for Sparse Spanning Graphs

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning subgraph containing at most \((1+\epsilon )n\) edges (where n is the number of vertices and \(\epsilon \) is a given approximation/sparsity parameter). In the local setting, the goal is to quickly determine whether a given edge e belongs to such a subgraph, without constructing the whole subgraph, but rather by inspecting (querying) the local neighborhood of e. The challenge is to maintain consistency. That is, to provide answers concerning different edges according to the same spanning subgraph. We first show that for general bounded-degree graphs, the query complexity of any such algorithm must be \(\Omega (\sqrt{n})\). This lower bound holds for constant-degree graphs that have high expansion. Next we design an algorithm for (bounded-degree) graphs with high expansion, obtaining a result that roughly matches the lower bound. We then turn to study graphs that exclude a fixed minor (and are hence non-expanding). We design an algorithm for such graphs, which may have an unbounded maximum degree. The query complexity of this algorithm is \(\mathrm{poly}(1/\epsilon , h)\) (independent of n and the maximum degree), where h is the number of vertices in the excluded minor. Though our two algorithms are designed for very different types of graphs (and have very different complexities), on a high-level there are several similarities, and we highlight both the similarities and the differences.

Colored spanning graphs for set visualization

We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected.We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast (12ρ+1)-approximation algorithm, where ρ is the Steiner ratio.

Source-wise round-trip spanners

In this paper, we study a new type of graph spanners, called source-wise round-trip spanners. Given any source vertex set S⊆V in a directed graph G(V,E), such a spanner (with stretch k) has the property that the round-trip shortest path distance from any vertex s∈S to any vertex v∈V is at most k times of their round-trip distance in G. We present an algorithm to construct the source-wise round-trip spanners with stretch (2k+ϵ) and size O((k2/ϵ)ns1/klog⁡(nw)) where n=|V|,s=|S| and w is the maximum edge weight. The result out-performs the state-of-the-art traditional round-trip spanners when the source vertex set S has small cardinality, and at the same time, it matches the traditional spanners when S is the whole vertex set V.

Spanners for Geodesic Graphs and Visibility Graphs

Let \(\mathcal{P}\) be a set of n points inside a polygonal domain \(\mathcal{D}\). A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set \(\mathcal{P}\) with respect to the geodesic distance function \(\pi \) where for any two points p and q, \(\pi (p,q)\) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., \(h=0\)), we construct a (\(\sqrt{10}+\epsilon \))-spanner that has \(O(n \log ^2 n)\) edges. For a case where there are h holes, our construction gives a (\(5+\epsilon \))-spanner with the size of \(O(n\sqrt{h}\log ^2 n)\). Moreover, we study t-spanners for the visibility graph of \(\mathcal{P}\) (\(VG(\mathcal{P})\), for short) with respect to a hole-free polygonal domain \(\mathcal{D}\). The graph \(VG(\mathcal{P})\) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (\(3+\epsilon \))-spanner of size \(O(n^{4/3+\delta })\) for some \(\delta >0\). In addition, we show that there is a set \(\mathcal{P}\) of n points such that any \((3-\epsilon )\)-spanner of \(VG(\mathcal{P})\) must contain \(\varOmega (n^2)\) edges.

One-Class Classifiers Based on Entropic Spanning Graphs.

One-class classifiers offer valuable tools to assess the presence of outliers in data. In this paper, we propose a design methodology for one-class classifiers based on entropic spanning graphs. Our approach also takes into account the possibility to process nonnumeric data by means of an embedding procedure. The spanning graph is learned on the embedded input data, and the outcoming partition of vertices defines the classifier. The final partition is derived by exploiting a criterion based on mutual information minimization. Here, we compute the mutual information by using a convenient formulation provided in terms of the -Jensen difference. Once training is completed, in order to associate a confidence level with the classifier decision, a graph-based fuzzy model is constructed. The fuzzification process is based only on topological information of the vertices of the entropic spanning graph. As such, the proposed one-class classifier is suitable also for data characterized by complex geometric structures. We provide experiments on well-known benchmarks containing both feature vectors and labeled graphs. In addition, we apply the method to the protein solubility recognition problem by considering several representations for the input samples. Experimental results demonstrate the effectiveness and versatility of the proposed method with respect to other state-of-the-art approaches.

Fast Constructions of Lightweight Spanners for General Graphs

It is long known that for every weighted undirected n -vertex m -edge graph G = ( V , E , ω), and every integer k ⩾ 1, there exists a ((2 k − 1) · (1 + ϵ))-spanner with O ( n 1 + 1/ k ) edges and weight O ( k · n 1/ k · ω( MST ( G )), for an arbitrarily small constant ϵ > 0. (Here ω( MST ( G )) stands for the weight of the minimum spanning tree of G .) To our knowledge, the only algorithms for constructing sparse and lightweight spanners for general graphs admit high running times. Most notable in this context is the greedy algorithm of Althöfer et al. [1993], analyzed by Chandra et al. [1992], which requires O ( m · ( n 1 + 1/ k + n · log n )) time. In this article, we devise an efficient algorithm for constructing sparse and lightweight spanners. Specifically, our algorithm constructs ((2 k − 1) · (1 + ϵ))-spanners with O ( k · n 1 + 1/ k ) edges and weight O ( k · n 1/ k ) · ω( MST ( G )), where ϵ > 0 is an arbitrarily small constant. The running time of our algorithm is O ( k · m + min { n · log n , m · α( n )}). Moreover, by slightly increasing the running time we can reduce the other parameters. These results address an open problem by Roditty and Zwick [2004].

Label Cover Instances with Large Girth and the Hardness of Approximating Basic k -Spanner

We study the well-known Label Cover problem under the additional requirement that problem instances have large girth. We show that if the girth is some k , the problem is roughly 2 log 1-ϵ n)/k hard to approximate for all constant ϵ > 0. A similar theorem was claimed by Elkin and Peleg [2000] as part of an attempt to prove hardness for the basic k -spanner problem, but their proof was later found to have a fundamental error. Thus, we give both the first nontrivial lower bound for the problem of Label Cover with large girth as well as the first full proof of strong hardness for the basic k -spanner problem, which is both the simplest problem in graph spanners and one of the few for which super-logarithmic hardness was not known. Assuming NP ⊆ BPTIME (2 polylog(n) , we show (roughly) that for every k ⩾ 3 and every constant ϵ > 0, it is hard to approximate the basic k -spanner problem within a factor better than 2 log 1-ϵ n)/k . This improves over the previous best lower bound of only Ω(log n )/ k from Kortsarz [2001]. Our main technique is subsampling the edges of 2-query probabilistically checkable proofs (PCPs), which allows us to reduce the degree of a PCP to be essentially equal to the soundness desired. This turns out to be enough to basically guarantee large girth.