Transitive-Closure Spanners

@article{Bhattacharyya2008TransitiveClosureS,
  title={Transitive-Closure Spanners},
  author={Arnab Bhattacharyya and Elena Grigorescu and Kyomin Jung and Sofya Raskhodnikova and David P. Woodruff},
  journal={SIAM J. Comput.},
  year={2008},
  volume={41},
  pages={1380-1425}
}
Given a directed graph $G = (V,E)$ and an integer $k \geq 1$, a $k$-transitive-closure-spanner ($k$-TC-spanner) of $G$ is a directed graph $H = (V, E_H)$ that has (1) the same transitive-closure as $G$ and (2) diameter at most $k$. These spanners were implicitly studied in the context of circuit complexity, data structures, property testing, and access control, and properties of these spanners have been rediscovered over the span of 20 years. We abstract the common task implicitly tackled in… 

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References

SHOWING 1-10 OF 81 REFERENCES

Finding Sparser Directed Spanners

This work studies the computational problem of finding the sparsest spanners of a given directed graph, which it is referred to as DIRECTED $k-SPANNER (resp., $k$-TC-spanner) and improves all known approximation algorithms for these problems for $k\geq 3$.

Steiner transitive-closure spanners of low-dimensional posets

The dimension of a poset G is the smallest d such that G can be embedded into a d-dimensional directed hypergrid via an order-preserving embedding and a nearly tight lower bound on the size of Steiner 2-TC-spanners of d- dimensional directed hypergrids is presented.

Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners

Tight upper and lower bounds on the size of the sparsest 2-TC-spanners of the directed hypercube and hypergrid are presented, implying tighter lower bounds for local monotonicity reconstructors that nearly match the known upper bounds.

Approximating k-spanner problems for kge2

Approximating k-Spanner Problems for k>2

The technique introduced in the paper enables the studied algorithmic questions of approximability of the k-spanner and k-DSS problems to be reduced to purely graph-theoretical questions concerning the existence of certain combinatorial objects in families of graphs.

Generating low-degree 2-spanners

It is shown that the problem of finding a 2-spanner in a given graph is at least as hard to approximate as set cover, and a randomized approximation algorithm is provided with approximation ratio of $\tilde O(\Delta^{1/4})$.

Improved Approximation for the Directed Spanner Problem

The approximation ratio of the algorithm is O(n1/3) which almost matches the lower bound shown by Dinitz and Krauthgamer for the integrality gap of a natural linear programming relaxation.

On the Hardness of Approximating Spanners

It is proved that for every fixed k, approximation of the spanner problem is at least as hard as approximating the set-cover problem.

The Transitive Reduction of a Directed Graph

It is shown that the time complexity of the best algorithm for finding the transitive reduction of a graph is the same as the time to compute the transitives closure of agraph or to perform Boolean matrix multiplication.

Lower Bounds for Additive Spanners, Emulators, and More

  • David P. Woodruff
  • Mathematics, Computer Science
    2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
  • 2006
The study of pair-wise and source-wise distance preservers defined by Coppersmith and Elkin by considering their approximate variants and their relaxation to emulators and proves the first lower bounds for such graphs.
...