Abstract
We give an \(O(\sqrt{n}\log n)\)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V,E) with nonnegative edge lengths d: E → ℝ ≥ 0 and a stretch k ≥ 1, a subgraph H = (V,E H ) is a k-spanner of G if for every edge (u,v) ∈ E, the graph H contains a path from u to v of length at most k ·d(u,v). The previous best approximation ratio was \(\tilde{O}(n^{2/3})\), due to Dinitz and Krauthgamer (STOC ’11).
We also present an improved algorithm for the important special case of directed 3-spanners with unit edge lengths. The approximation ratio of our algorithm is \(\tilde{O}(n^{1/3})\) which almost matches the lower bound shown by Dinitz and Krauthgamer for the integrality gap of a natural linear programming relaxation. The best previously known algorithms for this problem, due to Berman, Raskhodnikova and Ruan (FSTTCS ’10) and Dinitz and Krauthgamer, had approximation ratio \(\tilde{O}(\sqrt{n})\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete & Computational Geometry 9(1), 81–100 (1993)
Awerbuch, B.: Communication-time trade-offs in network synchronization. In: PODC, pp. 272–276 (1985)
Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in expected \(\tilde{O} (n^2)\) time. ACM Transactions on Algorithms 2(4), 557–577 (2006)
Berman, P., Raskhodnikova, S., Ruan, G.: Finding sparser directed spanners. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS. LIPIcs, vol. 8, pp. 424–435. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2010)
Bhattacharyya, A., Grigorescu, E., Jha, M., Jung, K., Raskhodnikova, S., Woodruff, D.P.: Lower bounds for local monotonicity reconstruction from transitive-closure spanners. In: Serna, M.J., Shaltiel, R., Jansen, K., Rolim, J.D.P. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 448–461. Springer, Heidelberg (2010)
Bhattacharyya, A., Grigorescu, E., Jung, K., Raskhodnikova, S., Woodruff, D.: Transitive-closure spanners. In: SODA, pp. 932–941 (2009)
Cohen, E.: Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput. 28(1), 210–236 (1998)
Cohen, E.: Polylog-time and near-linear work approximation scheme for undirected shortest paths. JACM 47(1), 132–166 (2000)
Cowen, L.: Compact routing with minimum stretch. J. Algorithms 38(1), 170–183 (2001)
Cowen, L., Wagner, C.G.: Compact roundtrip routing in directed networks. J. Algorithms 50(1), 79–95 (2004)
Dinitz, M., Krauthgamer, R.: Directed spanners via flow-based linear programs. In: Vadhan, S. (ed.) STOC. ACM, New York (to appear, 2011)
Dodis, Y., Khanna, S.: Designing networks with bounded pairwise distance. In: STOC, pp. 750–759 (1999)
Elkin, M.: Computing almost shortest paths. In: PODC, pp. 53–62 (2001)
Elkin, M., Peleg, D.: Strong inapproximability of the basic k-spanner problem. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 636–647. Springer, Heidelberg (2000)
Elkin, M., Peleg, D.: The client-server 2-spanner problem with applications to network design. In: SIROCCO, pp. 117–132 (2001)
Elkin, M., Peleg, D.: Approximating k-spanner problems for k ≥ 2. Theor. Comput. Sci. 337(1-3), 249–277 (2005)
Elkin, M., Peleg, D.: The hardness of approximating spanner problems. Theory Comput. Syst. 41(4), 691–729 (2007)
Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: Graph distances in the data-stream model. SIAM J. Comput. 38(5), 1709–1727 (2008)
Feldman, M., Kortsarz, G., Nutov, Z.: Improved approximating algorithms for Directed Steiner Forest. In: Mathieu, C. (ed.) SODA, pp. 922–931. SIAM, Philadelphia (2009)
Jha, M., Raskhodnikova, S.: Testing and reconstruction of Lipschitz functions with applications to data privacy. Electronic Colloquium on Computational Complexity (ECCC) TR11-057 (2011)
Kortsarz, G., Peleg, D.: Generating sparse 2-spanners. J. Algorithms 17(2), 222–236 (1994)
Kortsarz, G.: On the hardness of approximating spanners. Algorithmica 30(3), 432–450 (2001)
Peleg, D., Schäffer, A.A.: Graph spanners. J. Graph Theory 13(1), 99–116 (1989)
Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM J. Comput. 18(4), 740–747 (1989)
Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. JACM 36(3), 510–530 (1989)
Raskhodnikova, S.: Transitive-closure spanners: a survey. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 167–196. Springer, Heidelberg (2010)
Roditty, L., Thorup, M., Zwick, U.: Roundtrip spanners and roundtrip routing in directed graphs. ACM Transactions on Algorithms 4(3) (2008)
Thorup, M., Zwick, U.: Compact routing schemes. In: SPAA, pp. 1–10. ACM, New York (2001)
Thorup, M., Zwick, U.: Approximate distance oracles. JACM 52(1), 1–24 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berman, P., Bhattacharyya, A., Makarychev, K., Raskhodnikova, S., Yaroslavtsev, G. (2011). Improved Approximation for the Directed Spanner Problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-22006-7_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22005-0
Online ISBN: 978-3-642-22006-7
eBook Packages: Computer ScienceComputer Science (R0)