A trade-off between space and efficiency for routing tables
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An optimal synchronizer for the hypercube
SIAM Journal on Computing
On sparse spanners of weighted graphs
Discrete & Computational Geometry
Journal of Algorithms
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Parallel shortcutting of rooted trees
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Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Fast Algorithms for Constructing t-Spanners and Paths with Stretch t
SIAM Journal on Computing
Design networks with bounded pairwise distance
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Polylog-time and near-linear work approximation scheme for undirected shortest paths
Journal of the ACM (JACM)
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Distributed computing: a locality-sensitive approach
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Proceedings of the twentieth annual ACM symposium on Principles of distributed computing
Monotonicity testing over general poset domains
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Roundtrip spanners and roundtrip routing in directed graphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
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SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Improved Testing Algorithms for Monotonicity
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WG '92 Proceedings of the 18th International Workshop on Graph-Theoretic Concepts in Computer Science
Exact and Approximate Distances in Graphs - A Survey
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
Space-time tradeoff for answering range queries (Extended Abstract)
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Compact roundtrip routing in directed networks
Journal of Algorithms
On the strength of comparisons in property testing
Information and Computation
Compact oracles for reachability and approximate distances in planar digraphs
Journal of the ACM (JACM)
Journal of the ACM (JACM)
Approximating k-spanner problems for k 2
Theoretical Computer Science
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Proceedings of the 12th ACM conference on Computer and communications security
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Proceedings of the eleventh ACM symposium on Access control models and technologies
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Lower Bounds for Additive Spanners, Emulators, and More
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Approximate distance oracles for unweighted graphs in expected O(n2) time
ACM Transactions on Algorithms (TALG)
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Information and Computation
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Theory of Computing Systems
Set connectivity problems in undirected graphs and the directed Steiner network problem
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Efficient provably-secure hierarchical key assignment schemes
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
Extensions and limits to vertex sparsification
Proceedings of the forty-second ACM symposium on Theory of computing
Lower bounds for local monotonicity reconstruction from transitive-closure spanners
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
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New results on the complexity of the max- and min-rep problems
SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
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SIAM Journal on Computing
Property testing of massively parametrized problems – a survey
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Property testing of massively parametrized problems – a survey
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ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
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An optimal-time construction of sparse Euclidean spanners with tiny diameter
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SIAM Journal on Discrete Mathematics
Label cover instances with large girth and the hardness of approximating basic k-spanner
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Approximation algorithms for spanner problems and Directed Steiner Forest
Information and Computation
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ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
Optimal bounds for monotonicity and lipschitz testing over hypercubes and hypergrids
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We define the notion of a transitive-closure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were studied implicitly in access control, property testing, and data structures, and properties of these spanners have been rediscovered over the span of 20 years. We bring these areas under the unifying framework of TC-spanners. We abstract the common task implicitly tackled in these diverse applications as the problem of constructing sparse TC-spanners. We study the approximability of the size of the sparsest k-TC-spanner for a given digraph. Our technical contributions fall into three categories: algorithms for general digraphs, inapproximability results, and structural bounds for a specific graph family which imply an efficient algorithm with a good approximation ratio for that family. Algorithms. We present two efficient deterministic algorithms that find k-TC-spanners of near optimal size. The first algorithm gives an Õ(n1-1/k)-approximation for k 2. Our method, based on a combination of convex programming and sampling, yields the first sublinear approximation ratios for (1) Directed k-Spanner, a well-studied generalization of k-TC-Spanner, and (2) its variants Client/Server Directed k-Spanner, and the k-Diameter Spanning Subgraph. This resolves the main open question of Elkin and Peleg (IPCO, 2001). The second algorithm, specific to the k-TC-spanner problem, gives an Õ(n/k2)-approximation. It shows that for k = Ω(√n), our problem has a provably better approximation ratio than Directed k-Spanner and its variants. This algorithm also resolves an open question of Hesse (SODA, 2003). Inapproximability. Our main technical contribution is a pair of strong inapproximability results. We resolve the approximability of 2-TC-spanners, showing that it is θ(log n) unless P = NP. For constant k ≥ 3, we prove that the size of the sparsest k-TC-spanner is hard to approximate within 2log1-ε n, for any ε 0, unless NP ⊆ DTIME (npolylog n). Our hardness result helps explain the difficulty in designing general efficient solutions for the applications above, and it cannot be improved without resolving a long-standing open question in complexity theory. It uses an involved application of generalized butterfly and broom graphs, as well as noise-resilient transformations of hard problems, which may be of independent interest. Structural bounds. Finally, we study the size of the sparsest TC-spanner for H-minor-free digraphs, which include planar, bounded genus, and bounded tree-width graphs, explicitly investigated in applications above. We show that every H-minor-free digraph has an efficiently con-structible k-TC-spanner of size Õ(n). This implies an Õ(1)-approximation algorithm for this family. Furthermore, using our insight that 2-TC-spanners yield property testers, we obtain a monotonicity tester with O(log2 n/ε) queries for any poset whose transitive reduction is an H-minor free digraph. This improves and generalizes the previous θ(√n log n/ε)-query tester of Fischer et al (STOC, 2002).