There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
An optimal synchronizer for the hypercube
SIAM Journal on Computing
High probability parallel transitive-closure algorithms
SIAM Journal on Computing
New sparseness results on graph spanners
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
On sparse spanners of weighted graphs
Discrete & Computational Geometry
Polylog-time and near-linear work approximation scheme for undirected shortest paths
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Clique partitions, graph compression and speeding-up algorithms
Journal of Computer and System Sciences
Near-Linear Time Construction of Sparse Neighborhood Covers
SIAM Journal on Computing
All-Pairs Almost Shortest Paths
SIAM Journal on Computing
Distributed computing: a locality-sensitive approach
Distributed computing: a locality-sensitive approach
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
(1 + &egr;&Bgr;)-spanner constructions for general graphs
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Computing almost shortest paths
Proceedings of the twentieth annual ACM symposium on Principles of distributed computing
Compact Oracles for Reachability and Approximate Distances in Planar Digraphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Proximity-preserving labeling schemes
Journal of Graph Theory
Sparse source-wise and pair-wise distance preservers
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
New constructions of (α, β)-spanners and purely additive spanners
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Computing almost shortest paths
ACM Transactions on Algorithms (TALG)
Spanners and emulators with sublinear distance errors
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs
Random Structures & Algorithms
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
On the locality of distributed sparse spanner construction
Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing
Approximating Shortest Paths in Graphs
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
Fault-tolerant spanners for general graphs
Proceedings of the forty-first annual ACM symposium on Theory of computing
Additive spanners and (α, β)-spanners
ACM Transactions on Algorithms (TALG)
Fault Tolerant Spanners for General Graphs
SIAM Journal on Computing
Sparse spanners vs. compact routing
Proceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures
Fully dynamic randomized algorithms for graph spanners
ACM Transactions on Algorithms (TALG)
Multipath spanners via fault-tolerant spanners
MedAlg'12 Proceedings of the First Mediterranean conference on Design and Analysis of Algorithms
Small stretch pairwise spanners
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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For an unweighted graph G = (V, E), G′ = (V, E′) is a subgraph if E′ ⊆ E, and G″ = (V″, E″, ω) is a Steiner graph if V ⊆ V″, and for any pair of vertices u, w ∈ V, the distance bet-ween them in G″(denoted dG″, (u, w)) is at least the distance between them in G (denoted da(u, w)).In this paperwe introduce the notion of distance preserver. A subgraph (resp., Steiner graph) G′ of a graph G is a subgraph (resp., Steiner) D-preserver of G if for every pair of vertices u, w ∈ V with dG(u, w) ≥ D, dG′, (u, w) = dG(u, w). We show that anygraph (resp., digraph) has a subgraph D-preserver with at most O(n2/D) edges (resp., arcs), and there are graphs and digraphs for which any undirected Steiner D-preserver contains Ω(n2/D) edges. However, we show that if one allows a directed Steiner (or, shortly, diS-teiner) D-preserver, then these bounds can be improved. Specifically, we show that for any graph or digraph there exists a diSteiner D-preserver with O(n2.log D/D.log n arcs, and that this result is tight up to a constant factor.We also study D-preserving distance labeling schemes, that are labeling schemes that guarantee precise calculation ofdistances between pairs of vertices that are at distance at least D one from another. We show that there exists a D-preserving labeling scheme with labels of size O(n/Dlog2 n), and that labels of size Ω(n/D log D) are required for any D-preserving labeling scheme.Finally, we study additive spanners. A subgraph G′ of an undirected graph G = (V, E) is its additive β-spanner if for any pair of vertices u, w ∈ V, dG′, (u, w) ≤ dG(u, w)+β. It is known that for any n-vertex graph there exists an additive 2-spanner with O(n3/2) edges, and an additive Steiner 4-spanner with O(n4/3) edges. However, no construction of additive spanners with o(n3/2) edges or Steiner additive spanners with o(n4/3) edges are known so far. We devise a construction of additive O(21/δn(1-δ)[1/δ]--2/[1/δ]--1)-spanner with O(n1+δ) edges for any graph and any δ