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Approximation algorithms and an integer program for multi-level graph spanners [article]

Reyan Ahmed, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, Faryad Darabi Sahneh, Richard Spence
2019 arXiv   pre-print
Given a weighted graph G(V,E) and t > 1, a subgraph H is a t--spanner of G if the lengths of shortest paths in G are preserved in H up to a multiplicative factor of t.  ...  The subsetwise spanner problem aims to preserve distances in G for only a subset of the vertices.  ...  In the pairwise spanner problem [11] , distances only need to be preserved for a subset P ⊆ V × V of pairs of vertices.  ... 
arXiv:1904.01135v1 fatcat:vcadmu7k2faa7jlvcejbeut7f4

On Pairwise Spanners

Marek Cygan, Fabrizio Grandoni, Telikepalli Kavitha, Marc Herbstritt
2013 Symposium on Theoretical Aspects of Computer Science  
Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same P) and of the best known spanners (with the same f (·)).  ...  We show that there exists an (S × S)-spanner of size O(n |S|) with f (d) = d + 2, and an (S × V )-spanner of size O(n |S| log n) with f (d) = d + 2 log n.  ...  in [8] are at the same time more accurate and sparser than our spanners for small enough values of |P|.  ... 
doi:10.4230/lipics.stacs.2013.209 dblp:conf/stacs/CyganGK13 fatcat:qmfqkwfkhndyxgdsufqvohxehe

On Pairwise Spanners [article]

Marek Cygan and Fabrizio Grandoni and Telikepalli Kavitha
2013 arXiv   pre-print
Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same ) and of the best known spanners (with the same f(·)).  ...  We show that there exists an (S× S)-spanner of size O(n√(|S|)) with f(d)=d+2, and an (S× V)-spanner of size O(n√(|S| n)) with f(d)=d+2 n.  ...  An (α, β) spanner is a spanner with stretch functions f (d) = α · d + β. (α and β are the multiplicative stretch and additive stretch of the spanner, respectively).  ... 
arXiv:1301.1999v1 fatcat:y5pdjxbifze2fn25ki46sakguu

Approximating Spanners and Directed Steiner Forest: Upper and Lower Bounds [article]

Eden Chlamtáč, Michael Dinitz, Guy Kortsarz, Bundit Laekhanukit
2016 arXiv   pre-print
We give an O(n^3/5 + ϵ)-approximation for distance preservers and pairwise spanners (for arbitrary constant ϵ > 0).  ...  in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Godwin  ...  In this paper we bring together these lines of work, by studying the approximability of distance preservers, pairwise spanners, and additive spanners.  ... 
arXiv:1607.06068v1 fatcat:q5owvfqnrjcozmglkx3qb5tngy

New Pairwise Spanners

Telikepalli Kavitha
2016 Theory of Computing Systems  
We show sparse pairwise spanners with additive stretch 4 and with additive stretch 6.  ...  A subgraph H of G is called an (all-pairs) purely additive spanner with stretch β if for every (u, v) The problem of computing sparse spanners with small stretch β is well-studied.  ...  [9] were the first to study a variant of pairwise preservers -they studied D-preservers where the problem was to compute a sparse pairwise preserver for the set P = {(u, v) : δ G (u, v) ≥ D}.  ... 
doi:10.1007/s00224-016-9736-7 fatcat:n3e7nn2c4jauvh3wi6nymnrapa

Graph Spanners: A Tutorial Review [article]

Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, Richard Spence
2020 arXiv   pre-print
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  ...  As an additional contribution, we offer a list of open problems on graph spanners.  ...  Small stretch pairwise spanners. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 601-612. Springer, 2013 [116] Telikepalli Kavitha. New pairwise spanners.  ... 
arXiv:1909.03152v2 fatcat:vqme4ewuvbbd5ac5erinyykwsy

Improved Purely Additive Fault-Tolerant Spanners [chapter]

Davide Bilò, Fabrizio Grandoni, Luciano Gualà, Stefano Leucci, Guido Proietti
2015 Lecture Notes in Computer Science  
one such spanner that preserves all pairwise distances.  ...  On one hand, we show how to augment (by adding a small number of edges) a fault-tolerant additive sourcewise spanner (that approximately preserves distances only from a given set of source nodes) into  ...  We show that, by carefully choosing S and by augmenting A S with a conveniently selected small subset of edges, it is possible to construct a fault-tolerant spanner (approximately preserving all pairwise  ... 
doi:10.1007/978-3-662-48350-3_15 fatcat:lqwrqjtwfvbtfehw7hjnsttu7i

Distributed Construction of Purely Additive Spanners [article]

Keren Censor-Hillel, Telikepalli Kavitha, Ami Paz, Amir Yehudayoff
2016 arXiv   pre-print
We complement our algorithms with a lower bound on the number of rounds required for computing pairwise spanners.  ...  Instead, to obtain our lower bound, we define a new communication complexity problem that reduces to computing a sparse spanner, and prove a lower bound on its communication complexity using information  ...  Acknowledgements We thank Merav Parter for a helpful discussion on the lower bound, and the anonymous referees of DISC 2016 for valuable comments.  ... 
arXiv:1607.05597v1 fatcat:rwrhqal3gbfvhddnjvnmponbee

Improved Purely Additive Fault-Tolerant Spanners [article]

Davide Bilò, Fabrizio Grandoni, Luciano Gualà, Stefano Leucci, Guido Proietti
2015 arXiv   pre-print
one such spanner that preserves all pairwise distances.  ...  On one hand, we show how to augment (by adding a small number of edges) a fault-tolerant additive sourcewise spanner (that approximately preserves distances only from a given set of source nodes) into  ...  We show that, by carefully choosing S and by augmenting A S with a conveniently selected small subset of edges, it is possible to construct a fault-tolerant spanner (approximately preserving all pairwise  ... 
arXiv:1507.00505v1 fatcat:4egrechb3ve4fhv6mez3p7zunm

A Note on Distance-Preserving Graph Sparsification [article]

Greg Bodwin
2021 arXiv   pre-print
Examples include pairwise spanners, distance preservers, reachability preservers, etc.  ...  This simplifies and unifies a few proofs in the area, and it improves the size of the +4 pairwise spanner from O(np^2/7) [Kavitha Th. Comp. Sys. '17] to O(np^2/7).  ...  Not pictured: there is also a +0 pairwise spanner (distance preserver) by Coppersmith and Elkin [11] on O(n+ n 1/2 p) edges, which improves on these pairwise spanners in some regime of small p. some  ... 
arXiv:2001.07741v3 fatcat:23eqk3j5jfdpnf64o73d5f4k64

Better Distance Preservers and Additive Spanners

Greg Bodwin, Virginia Vassilevska Williams
2015 Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms  
We then develop a new graph clustering technique, based on distance preservers, and we apply this technique to show new upper bounds for additive (standard) spanners, in which all pairwise distances must  ...  be preserved up to an additive error function, and for subset spanners, in which only distances within a given node subset must be preserved up to an error function.  ...  These are: a +2 pairwise spanner onÕ(n|P | 1/3 ) edges due to Kavitha & Varma [KV13] , a +4 pairwise spanner onÕ(n|P | 2/7 ) edges due to Kavitha [Kav15] , and a +6 pairwise spanner on O(n|P | 1/4 )  ... 
doi:10.1137/1.9781611974331.ch61 dblp:conf/soda/BodwinW16 fatcat:66i75jt6tjfd3iprey47bp4f4y

Locality-Sensitive Orderings and Applications to Reliable Spanners [article]

Arnold Filtser, Hung Le
2022 arXiv   pre-print
Afterwards, we use our new LSO's to construct reliable spanners with improved stretch and sparsity parameters.  ...  Most prominently, we construct Õ(n)-size reliable spanners for trees and planar graphs with the optimal stretch of 2.  ...  They used their LSO to design simple dynamic algorithms for approximate nearest neighbor search, approximate bichromatic closest pair, approximate MST, spanners, and fault-tolerant spanners.  ... 
arXiv:2101.07428v3 fatcat:6iwemadvkncu7hgfwd2ym6fk3q

Multi-Level Graph Sketches via Single-Level Solvers [article]

Reyan Ahmed, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, Faryad Darabi Sahneh, Richard Spence
2019 arXiv   pre-print
Examples include Steiner trees (preserving connectivity among T) and subsetwise spanners (preserving shortest path distances up to a stretch factor).  ...  This is the first algorithm and weight guarantee for a multiplicative subsetwise spanner for nonplanar graphs.  ...  The research for this paper was partially supported by NSF grants CCF-1740858 and DMS-1839274.  ... 
arXiv:1905.00536v2 fatcat:ynptxpg3engvdlsxww6kb66ar4

Very Sparse Additive Spanners and Emulators [article]

Greg Bodwin, Virginia Vassilevska Williams
2021 arXiv   pre-print
The second part of this paper leverages these distance preservers in a new construction of additive spanners, which are subgraphs that preserve all pairwise distances up to an additive error function.  ...  split apart, and then intersect again later.  ...  Much of the initial theoretical work on spanners studied multiplicative stretch spanners, that is, all pairwise distances are preserved up to a small multiplicative factor. Althofer et al.  ... 
arXiv:1505.05630v2 fatcat:eqcs5cposrhxxikyixpd6fbfvy

Online Directed Spanners and Steiner Forests [article]

Elena Grigorescu, Young-San Lin, Kent Quanrud
2021 arXiv   pre-print
We present online algorithms for directed spanners and Steiner forests.  ...  Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized Õ(n^4/5)-competitive algorithm for graphs  ...  Acknowledgements We thank the anonymous reviewers for comments and suggestions that helped improve the presentation.  ... 
arXiv:2103.04543v2 fatcat:r54ordqp7vc37f6kmsrhfvzjqe
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