On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
Approximating k-Spanner Problems for k2
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
MFCS '02 Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science
A Variant of the Arrow Distributed Directory with Low Average Complexity
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
Tree Spanners on Chordal Graphs: Complexity, Algorithms, Open Problems
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
Tree Spanners for Subgraphs and Related Tree Covering Problems
WG '00 Proceedings of the 26th International Workshop on Graph-Theoretic Concepts in Computer Science
Tree spanners on chordal graphs: complexity and algorithms
Theoretical Computer Science
The non-approximability of bicriteria network design problems
Journal of Discrete Algorithms
Spanners and message distribution in networks
Discrete Applied Mathematics - Special issue on international workshop on algorithms, combinatorics, and optimization in interconnection networks (IWACOIN '99)
Approximating Minimum Max-Stretch spanning Trees on unweighted graphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Minimum spanning tree with hop restrictions
Journal of Algorithms - Special issue: Twelfth annual ACM-SIAM symposium on discrete algorithms
Approximating k-spanner problems for k 2
Theoretical Computer Science
Additive sparse spanners for graphs with bounded length of largest induced cycle
Theoretical Computer Science
Scalable data aggregation for dynamic events in sensor networks
Proceedings of the 4th international conference on Embedded networked sensor systems
Approximation algorithms for embedding general metrics into trees
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Spanners for bounded tree-length graphs
Theoretical Computer Science
The zoo of tree spanner problems
Discrete Applied Mathematics
A distance approximating trees
Discrete Applied Mathematics
A PTAS for the Sparsest Spanners Problem on Apex-Minor-Free Graphs
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Additive Spanners for Circle Graphs and Polygonal Graphs
Graph-Theoretic Concepts in Computer Science
Tree 3-spanners in 2-sep directed path graphs: Characterization, recognition, and construction
Discrete Applied Mathematics
Additive spanners for k-chordal graphs
CIAC'03 Proceedings of the 5th Italian conference on Algorithms and complexity
Computing geometric minimum-dilation graphs is NP-hard
GD'06 Proceedings of the 14th international conference on Graph drawing
Collective additive tree spanners of homogeneously orderable graphs
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Tree 3-spanners in 2-sep chordal graphs: Characterization and algorithms
Discrete Applied Mathematics
Spanners of bounded degree graphs
Information Processing Letters
Approximation of minimum weight spanners for sparse graphs
Theoretical Computer Science
Complexity results for the spanning tree congestion problem
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Journal of Computer and System Sciences
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Navigating in a Graph by Aid of Its Spanning Tree Metric
SIAM Journal on Discrete Mathematics
On spanners of geometric graphs
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Collective tree spanners in graphs with bounded genus, chordality, tree-width, or clique-width
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
A linear time algorithm for constructing tree 4-spanner in 2-trees
CIT'04 Proceedings of the 7th international conference on Intelligent Information Technology
Distance approximating trees: complexity and algorithms
CIAC'06 Proceedings of the 6th Italian conference on Algorithms and Complexity
Max-stretch reduction for tree spanners
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
Collective additive tree spanners for circle graphs and polygonal graphs
Discrete Applied Mathematics
Minimum weight Euclidean t-spanner is NP-hard
Journal of Discrete Algorithms
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A tree $t$-spanner $T$ of a graph $G$ is a spanning tree in which the distance between every pair of vertices is at most $t$ times their distance in $G$. This notion is motivated by applications in communication networks, distributed systems, and network design. This paper studies graph-theoretic, algorithmic, and complexity issues about tree spanners. It is shown that a tree 1-spanner, if it exists, in a weighted graph with $m$ edges and $n$ vertices is a minimum spanning tree and can be found in $O(m \log \beta(m, n))$ time, where $\beta(m, n) = \min\{i\mid\log^{(i)}n \leq m/n\}$. On the other hand, for any fixed $t 1$, the problem of determining the existence of a tree $t$-spanner in a weighted graph is proven to be NP-complete. For unweighted graphs, it is shown that constructing a tree 2-spanner takes linear time, whereas determining the existence of a tree $t$-spanner is NP-complete for any fixed $t \geq 4$. A theorem that captures the structure of tree 2-spanners is presented for unweighted graphs. For digraphs, an $O((m + n)\alpha(m, n))$ algorithm is provided for finding a tree $t$-spanner with $t$ as small as possible, where $\alpha(m, n)$ is a functional inverse of Ackerman's function. The results for tree spanners on undirected graphs are extended to "quasi-tree spanners" on digraphs. Furthermore, linear-time algorithms are derived for verifying tree spanners and quasi-tree spanners.