# 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…

## 91 Citations

### Finding Sparser Directed Spanners

- Mathematics, Computer ScienceFSTTCS
- 2010

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 $k\geq 3$.

### The Norms of Graph Spanners

- Mathematics, Computer ScienceICALP
- 2019

The study of graph spanners with respect to the $\ell_p$-norm of their degree vector is initiated, thus simultaneously modeling the number of edges and the maximum degree, and precise upper bounds for all ranges of $p$ and stretch $t$ are given.

### Steiner transitive-closure spanners of low-dimensional posets

- Mathematics, Computer ScienceCombinatorica
- 2014

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.

### Steiner transitive-closure spanners of low-dimensional posets

- Mathematics, Computer ScienceComb.
- 2014

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.

### Approximation algorithms and an integer program for multi-level graph spanners

- MathematicsSEA²
- 2019

A 0--1 integer linear program (ILP) of size $O(|E||V|^2)$ is formulated for the more general minimum \emph{pairwise spanner problem}, which resolves an open question by Sigurd and Zachariasen on whether this problem admits a useful polynomial-size ILP.

### Congested Clique Algorithms for Graph Spanners

- Computer Science, MathematicsDISC
- 2018

This work considers spanner constructions in the congested clique model, and shows a randomized construction and a deterministic construction of a $O(k)-spanner with $\widetilde{O}(n^{1+1/k})$ edges in $O(\log k)$ rounds.

### Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2010

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.

### Reachability Preservers: New Extremal Bounds and Approximation Algorithms

- Computer Science, MathematicsSODA
- 2018

A new connection between extremal graph sparsification results and classical Steiner Network Design problems is made and an efficient algorithm is designed that can always compute a preserver of existentially optimal size is designed.

### Exact Distance Oracles Using Hopsets

- Computer Science, MathematicsArXiv
- 2018

It is shown that $3-hopsets require exponentially fewer shortcuts per node than any previously described distance oracle while incurring only a quadratic increase in the query decoding time, and actually offer a speedup when compared to simple oracles based on a direct application of $2-hopset.

### Õ ( √ n ) Approximation for Directed Spanners

- Mathematics, Computer Science
- 2010

An approximation algorithm with O( √ n log n) ratio is described, which improves all known approximation algorithms for k > 4 by Ω̃(n).

## References

SHOWING 1-10 OF 81 REFERENCES

### Finding Sparser Directed Spanners

- Mathematics, Computer ScienceFSTTCS
- 2010

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

- Mathematics, Computer ScienceComb.
- 2014

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

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2010

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 k>2

- Mathematics, Computer ScienceIPCO
- 2001

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

- Mathematics, Computer ScienceSODA '94
- 1994

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

- Mathematics, Computer ScienceICALP
- 2011

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

- MathematicsAlgorithmica
- 2001

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

- MathematicsSIAM J. Comput.
- 1972

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

- Mathematics, Computer Science2006 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.