Graph spanners: A tutorial review

Ahmed, Reyan; Bodwin, Greg; Sahneh, Faryad Darabi; Hamm, Keaton; Jebelli, Mohammad Javad Latifi; Kobourov, Stephen G.; Spence, Richard

doi:10.1016/j.cosrev.2020.100253

Cited by 53 publications

(50 citation statements)

References 67 publications

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“…We show that Directed Multiplicative Spanner admits a kernel of size O(k 4 t 5 ). We complement this result by observing that the problem can be solved in (4t) k • n O (1) time by a Monte Carlo algorithm with false negatives. We leave open the question whether the problem is FPT when parameterized by k only.…”

Section: Introductionmentioning

confidence: 76%

“…Originally, the concept was Previous work. We refer to [1] for the comprehensive survey of the known results and mention here only those that directly concern our work. First, we point that the considered graph spanner problems are computationally hard.…”

Section: Introductionmentioning

confidence: 99%

“…In [10], it was proved that Multiplicative Spanner admits a polynomial kernel of size O(k 2 t 2 ). For Additive Spanner, it was shown in [11] that the problem can be solved in time 2 O((k 2 +kt) log t) • n O (1) , that is, the problem is fixed-parameter tractable (FPT) when parameterized by k and t.…”

Section: Introductionmentioning

confidence: 99%

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Parameterized Complexity of Directed Spanner Problems

et al. 2021

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We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most $$m-k$$ m - k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most $$m-k$$ m - k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size $$\mathcal {O}(k^4t^5)$$ O ( k 4 t 5 ) and can be solved in randomized $$(4t)^k\cdot n^{\mathcal {O}(1)}$$ ( 4 t ) k · n O ( 1 ) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in $$k^{2k}\cdot n^{\mathcal {O}(1)}$$ k 2 k · n O ( 1 ) time on directed acyclic graphs, (iii) Directed Additive Spanner is $${{\,\mathrm{\mathsf{W}}\,}}[1]$$ W [ 1 ] -hard when parameterized by k for every fixed $$t\ge 1$$ t ≥ 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is $${{\,\mathrm{\mathsf{FPT}}\,}}$$ FPT when parameterized by t and k.

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Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

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Parameterized Complexity of Directed Spanner Problems

et al. 2021

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“…At least one of these halves has length greater than 2 5 , let us assume without loss of generality that it is the half where we change the zeros in x to ones. During these at least 2 5 steps, the sum of coordinates increases by +1 each time. However, we have no control over the steps that we take in H 2 , they can both increase and decrease the coordinate sum by 1.…”

Section: Casementioning

confidence: 99%

“…Sparse subgraphs of the hypercube with strong distance-preserving properties have been studied extensively in the literature, and have found many practical applications in distributed computing and communication networks. We refer the reader to the recent survey [2].…”

Section: Introductionmentioning

confidence: 99%

Bounded Degree Spanners of the Hypercube

Nenadov¹,

Sawhney²,

Sudakov

et al. 2020

Electron. J. Combin.

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In this short note we study two questions about the existence of subgraphs of the hypercube $Q_n$ with certain properties. The first question, due to Erdős–Hamburger–Pippert–Weakley, asks whether there exists a bounded degree subgraph of $Q_n$ which has diameter $n$. We answer this question by giving an explicit construction of such a subgraph with maximum degree at most 120. The second problem concerns properties of $k$-additive spanners of the hypercube, that is, subgraphs of $Q_n$ in which the distance between any two vertices is at most $k$ larger than in $Q_n$. Denoting by $\Delta_{k,\infty}(n)$ the minimum possible maximum degree of a $k$-additive spanner of $Q_n$, Arizumi–Hamburger–Kostochka showed that $$\frac{n}{\ln n}e^{-4k}\leq \Delta_{2k,\infty}(n)\leq 20\frac{n}{\ln n}\ln \ln n.$$ We improve their upper bound by showing that $$\Delta_{2k,\infty}(n)\leq 10^{4k} \frac{n}{\ln n}\ln^{(k+1)}n,$$where the last term denotes a $k+1$-fold iterated logarithm.

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Spanners in randomly weighted graphs: Euclidean case

Frieze

Pegden

2023

Journal of Graph Theory

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Given a connected graph G=(V,E) $G=(V,E)$ and a length function ℓ:E→double-struckR $\ell :E\to {\mathbb{R}}$ we let dv,w ${d}_{v,w}$ denote the shortest distance between vertex v $v$ and vertex w $w$. A t $t$‐spanner is a subset Efalse′⊆E $E^{\prime} \subseteq E$ such that if dv,w′ ${d}_{v,w}^{^{\prime} }$ denotes shortest distances in the subgraph Gfalse′=(V,Efalse′) $G^{\prime} =(V,E^{\prime} )$ then dv,w′≤tdv,w ${d}_{v,w}^{^{\prime} }\le t{d}_{v,w}$ for all v,w∈V $v,w\in V$. We study the size of spanners in the following scenario: we consider a random embedding Xp ${{\mathscr{X}}}_{p}$ of Gn,p ${G}_{n,p}$ into the unit square with Euclidean edge lengths. For ϵ>0 $\epsilon \gt 0$ constant, we prove the existence w.h.p. of (1+ϵ) $(1+\epsilon )$‐spanners for Xp ${{\mathscr{X}}}_{p}$ that have Oϵ(n) ${O}_{\epsilon }(n)$ edges. These spanners can be constructed in Oϵ(n2logn) ${O}_{\epsilon }({n}^{2}\mathrm{log}n)$ time. (We will use Oϵ ${O}_{\epsilon }$ to indicate that the hidden constant depends on ε $\varepsilon $). There are constraints on p $p$ preventing it going to zero too quickly.

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Graph spanners: A tutorial review

Cited by 53 publications

References 67 publications

Parameterized Complexity of Directed Spanner Problems

Parameterized Complexity of Directed Spanner Problems

Bounded Degree Spanners of the Hypercube

Spanners in randomly weighted graphs: Euclidean case

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