Document Open Access Logo

Online Directed Spanners and Steiner Forests

Authors Elena Grigorescu , Young-San Lin , Kent Quanrud

Thumbnail PDF


  • Filesize: 0.94 MB
  • 25 pages

Document Identifiers

Author Details

Elena Grigorescu
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Young-San Lin
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Kent Quanrud
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA


We thank the anonymous reviewers for comments and suggestions that helped improve the presentation. We thank Anupam Gupta and Greg Bodwin for bringing to our attention references that we missed in previous versions of the writeup.

Cite AsGet BibTex

Elena Grigorescu, Young-San Lin, and Kent Quanrud. Online Directed Spanners and Steiner Forests. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 5:1-5:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


We present online algorithms for directed spanners and directed Steiner forests. These are well-studied network connectivity problems that fall under the unifying framework of online covering and packing linear programming formulations. This framework was developed in the seminal work of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and is based on primal-dual techniques. Specifically, our results include the following: - For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{4/5}) for graphs with general edge lengths, where n is the number of vertices of the given graph. For graphs with uniform edge lengths, we give an efficient randomized algorithm with competitive ratio Õ(n^{2/3+ε}), and an efficient deterministic algorithm with competitive ratio Õ(k^{1/2+ε}), where k is the number of terminal pairs. To the best of our knowledge, these are the first online algorithms for directed spanners. In the offline version, the current best approximation ratio for uniform edge lengths is Õ(n^{3/5 + ε}), due to Chlamt{á}č, Dinitz, Kortsarz, and Laekhanukit (SODA 2017, TALG 2020). - For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{2/3 + ε}). The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is Õ(k^{1/2 + ε})-competitive. In the offline version, the current best approximation ratio with uniform costs is Õ(n^{26/45 + ε}), due to Abboud and Bodwin (SODA 2018). To obtain efficient and competitive online algorithms, we observe that a small modification of the online covering and packing framework by Buchbinder and Naor implies a polynomial-time implementation of the primal-dual approach with separation oracles, which a priori might perform exponentially many calls to the oracle. We convert the online spanner problem into an online covering problem and complete the rounding-step analysis in a problem-specific fashion.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → Packing and covering problems
  • Theory of computation → Routing and network design problems
  • Theory of computation → Rounding techniques
  • online directed pairwise spanners
  • online directed Steiner forests
  • online covering/packing linear programming
  • primal-dual approach


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Amir Abboud and Greg Bodwin. Reachability preservers: New extremal bounds and approximation algorithms. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1865-1883. SIAM, 2018. Google Scholar
  2. Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, and Richard Spence. Graph spanners: A tutorial review, 2019. URL:
  3. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. A general approach to online network optimization problems. ACM Transactions on Algorithms (TALG), 2(4):640-660, 2006. Google Scholar
  4. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. The online set cover problem. SIAM J. Comput., 39(2):361-370, 2009. Google Scholar
  5. Noga Alon and Baruch Schieber. Optimal preprocessing for answering on-line product queries, 1987. Google Scholar
  6. Spyridon Antonakopoulos. Approximating directed buy-at-bulk network design. In International Workshop on Approximation and Online Algorithms, pages 13-24. Springer, 2010. Google Scholar
  7. Esther M Arkin, Joseph SB Mitchell, and Christine D Piatko. Bicriteria shortest path problems in the plane. In Proc. 3rd Canad. Conf. Comput. Geom, pages 153-156. Citeseer, 1991. Google Scholar
  8. Pranjal Awasthi, Madhav Jha, Marco Molinaro, and Sofya Raskhodnikova. Testing lipschitz functions on hypergrid domains. Algorithmica, 74(3):1055-1081, 2016. Google Scholar
  9. Baruch Awerbuch. Communication-time trade-offs in network synchronization. In Proceedings of the Fourth Annual ACM Symposium on Principles of Distributed Computing, PODC ’85, page 272–276, New York, NY, USA, 1985. Association for Computing Machinery. Google Scholar
  10. Baruch Awerbuch and Yossi Azar. Buy-at-bulk network design. In Proceedings 38th Annual Symposium on Foundations of Computer Science, pages 542-547. IEEE, 1997. Google Scholar
  11. Baruch Awerbuch, Yossi Azar, and Yair Bartal. On-line generalized steiner problem. Theoretical Computer Science, 324(2-3):313-324, 2004. Google Scholar
  12. Baruch Awerbuch, Yossi Azar, and Serge Plotkin. Throughput-competitive on-line routing. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science, pages 32-40. IEEE, 1993. Google Scholar
  13. Yossi Azar, Umang Bhaskar, Lisa Fleischer, and Debmalya Panigrahi. Online mixed packing and covering. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 85-100. SIAM, 2013. Google Scholar
  14. Yossi Azar, Niv Buchbinder, TH Hubert Chan, Shahar Chen, Ilan Reuven Cohen, Anupam Gupta, Zhiyi Huang, Ning Kang, Viswanath Nagarajan, Joseph Naor, et al. Online algorithms for covering and packing problems with convex objectives. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 148-157. IEEE, 2016. Google Scholar
  15. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. Randomized competitive algorithms for generalized caching. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 235-244, 2008. Google Scholar
  16. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. A primal-dual randomized algorithm for weighted paging. Journal of the ACM (JACM), 59(4):1-24, 2012. Google Scholar
  17. Surender Baswana. Streaming algorithm for graph spanners - single pass and constant processing time per edge. Inf. Process. Lett, 2008. Google Scholar
  18. Surender Baswana and Telikepalli Kavitha. Faster algorithms for all-pairs approximate shortest paths in undirected graphs. SIAM J. Comput., 39(7):2865-2896, 2010. Google Scholar
  19. MohammadHossein Bateni and MohammadTaghi Hajiaghayi. Euclidean prize-collecting steiner forest. Algorithmica, 62(3-4):906-929, 2012. Google Scholar
  20. Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Approximation algorithms for spanner problems and directed steiner forest. Information and Computation, 222:93-107, 2013. Google Scholar
  21. Piotr Berman and Chris Coulston. On-line algorithms for steiner tree problems. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 344-353, 1997. Google Scholar
  22. Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P Woodruff. Transitive-closure spanners. SIAM Journal on Computing, 41(6):1380-1425, 2012. Google Scholar
  23. Greg Bodwin and Virginia Vassilevska Williams. Better distance preservers and additive spanners. In Robert Krauthgamer, editor, SODA, pages 855-872. SIAM, 2016. Google Scholar
  24. Glencora Borradaile, Philip N Klein, and Claire Mathieu. A polynomial-time approximation scheme for euclidean steiner forest. ACM Transactions on Algorithms (TALG), 11(3):1-20, 2015. Google Scholar
  25. Niv Buchbinder and Joseph Naor. Improved bounds for online routing and packing via a primal-dual approach. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 293-304. IEEE, 2006. Google Scholar
  26. Niv Buchbinder and Joseph Naor. Online primal-dual algorithms for covering and packing. Mathematics of Operations Research, 34(2):270-286, 2009. Google Scholar
  27. Niv Buchbinder and Joseph Seffi Naor. The design of competitive online algorithms via a primal-dual approach. Foundations and Trendsregistered in Theoretical Computer Science, 3(2-3):93-263, 2009. Google Scholar
  28. Deeparnab Chakrabarty, Alina Ene, Ravishankar Krishnaswamy, and Debmalya Panigrahi. Online buy-at-bulk network design. SIAM J. Comput., 47(4):1505-1528, 2018. Google Scholar
  29. Moses Charikar, Chandra Chekuri, To-yat Cheung, Zuo Dai, Ashish Goel, Sudipto Guha, and Ming Li. Approximation algorithms for directed steiner problems. Journal of Algorithms, 33(1):73-91, 1999. Google Scholar
  30. Shuchi Chawla, Tim Roughgarden, and Mukund Sundararajan. Optimal cost-sharing mechanisms for steiner forest problems. In International Workshop on Internet and Network Economics, pages 112-123. Springer, 2006. Google Scholar
  31. Shiri Chechik. Approximate distance oracles with improved bounds. In Rocco A. Servedio and Ronitt Rubinfeld, editors, STOC, pages 1-10. ACM, 2015. Google Scholar
  32. Chandra Chekuri, Guy Even, Anupam Gupta, and Danny Segev. Set connectivity problems in undirected graphs and the directed steiner network problem. ACM Transactions on Algorithms (TALG), 7(2):1-17, 2011. Google Scholar
  33. Chandra Chekuri, Mohammad Taghi Hajiaghayi, Guy Kortsarz, and Mohammad R Salavatipour. Approximation algorithms for nonuniform buy-at-bulk network design. SIAM Journal on Computing, 39(5):1772-1798, 2010. Google Scholar
  34. Eden Chlamtáč, Michael Dinitz, Guy Kortsarz, and Bundit Laekhanukit. Approximating spanners and directed steiner forest: Upper and lower bounds. ACM Transactions on Algorithms (TALG), 16(3):1-31, 2020. Google Scholar
  35. Lenore Cowen and Christopher G. Wagner. Compact roundtrip routing in directed networks. J. Algorithms, 50(1):79-95, 2004. Google Scholar
  36. Bilel Derbel, Cyril Gavoille, and David Peleg. Deterministic distributed construction of linear stretch spanners in polylogarithmic time. In Andrzej Pelc, editor, DISC, volume 4731 of Lecture Notes in Computer Science, pages 179-192. Springer, 2007. Google Scholar
  37. Bilel Derbel, Cyril Gavoille, David Peleg, and Laurent Viennot. On the locality of distributed sparse spanner construction. In Rida A. Bazzi and Boaz Patt-Shamir, editors, PODC, pages 273-282. ACM, 2008. Google Scholar
  38. Michael Dinitz and Robert Krauthgamer. Directed spanners via flow-based linear programs. In STOC, pages 323-332, 2011. Google Scholar
  39. Michael Dinitz, Yasamin Nazari, and Zeyu Zhang. Lasserre integrality gaps for graph spanners and related problems. arXiv preprint arXiv:1905.07468, 2019. Google Scholar
  40. Michael Dinitz and Zeyu Zhang. Approximating low-stretch spanners. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms, pages 821-840. SIAM, 2016. Google Scholar
  41. Dorit Dor, Shay Halperin, and Uri Zwick. All-pairs almost shortest paths. SIAM J. Comput., 29(5):1740-1759, 2000. Google Scholar
  42. Michael Elkin. Computing almost shortest paths. ACM Trans. Algorithms, 1(2):283-323, 2005. Google Scholar
  43. Michael Elkin. Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners. ACM Trans. Algorithms, 7(2):20:1-20:17, 2011. Google Scholar
  44. Michael Elkin and David Peleg. The client-server 2-spanner problem with applications to network design. In Francesc Comellas, Josep Fàbrega, and Pierre Fraigniaud, editors, SIROCCO 8, Proceedings of the 8th International Colloquium on Structural Information and Communication Complexity, Vall de Núria, Girona-Barcelona, Catalonia, Spain, 27-29 June, 2001, volume 8 of Proceedings in Informatics, pages 117-132. Carleton Scientific, 2001. Google Scholar
  45. Michael Elkin and David Peleg. The hardness of approximating spanner problems. Theory Comput. Syst., 41(4):691-729, 2007. Google Scholar
  46. Moran Feldman, Guy Kortsarz, and Zeev Nutov. Improved approximation algorithms for directed steiner forest. Journal of Computer and System Sciences, 78(1):279-292, 2012. Google Scholar
  47. Manuel Fernandez, David P. Woodruff, and Taisuke Yasuda. Graph spanners in the message-passing model. In Thomas Vidick, editor, ITCS, volume 151 of LIPIcs, pages 77:1-77:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  48. Arnold Filtser, Michael Kapralov, and Navid Nouri. Graph spanners by sketching in dynamic streams and the simultaneous communication model. arXiv preprint arXiv:2007.14204, 2020. Google Scholar
  49. Lisa Fleischer, Jochen Könemann, Stefano Leonardi, and Guido Schäfer. Simple cost sharing schemes for multicommodity rent-or-buy and stochastic steiner tree. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 663-670, 2006. Google Scholar
  50. Naveen Garg, Goran Konjevod, and Ramamoorthi Ravi. A polylogarithmic approximation algorithm for the group steiner tree problem. Journal of Algorithms, 37(1):66-84, 2000. Google Scholar
  51. Michel X. Goemans and David P. Williamson. A general approximation technique for constrained forest problems. SIAM J. Comput., 24(2):296-317, 1995. Google Scholar
  52. Elena Grigorescu, Young-San Lin, and Kent Quanrud. Online directed spanners and steiner forests. CoRR, 2021. URL:
  53. Anupam Gupta, Amit Kumar, Martin Pál, and Tim Roughgarden. Approximation via cost-sharing: a simple approximation algorithm for the multicommodity rent-or-buy problem. In 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings., pages 606-615. IEEE, 2003. Google Scholar
  54. Anupam Gupta and Viswanath Nagarajan. Approximating sparse covering integer programs online. Mathematics of Operations Research, 39(4):998-1011, 2014. Google Scholar
  55. Anupam Gupta, R Ravi, Kunal Talwar, and Seeun William Umboh. Last but not least: Online spanners for buy-at-bulk. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 589-599. SIAM, 2017. Google Scholar
  56. Anupam Gupta, Kunal Talwar, and Udi Wieder. Changing bases: Multistage optimization for matroids and matchings. In International Colloquium on Automata, Languages, and Programming, pages 563-575. Springer, 2014. Google Scholar
  57. Refael Hassin. Approximation schemes for the restricted shortest path problem. Mathematics of Operations research, 17(1):36-42, 1992. Google Scholar
  58. Christopher S Helvig, Gabriel Robins, and Alexander Zelikovsky. An improved approximation scheme for the group steiner problem. Networks: An International Journal, 37(1):8-20, 2001. Google Scholar
  59. Makoto Imase and Bernard M Waxman. Dynamic steiner tree problem. SIAM Journal on Discrete Mathematics, 4(3):369-384, 1991. Google Scholar
  60. Michael Kapralov and David P. Woodruff. Spanners and sparsifiers in dynamic streams. In Magnús M. Halldórsson and Shlomi Dolev, editors, PODC, pages 272-281. ACM, 2014. Google Scholar
  61. Anna R. Karlin, Mark S. Manasse, Lyle A. McGeoch, and Susan S. Owicki. Competitive randomized algorithms for non-uniform problems. Algorithmica, 11(6):542-571, 1994. Google Scholar
  62. Vikram Khurana, Jian Peng, Chee Yeun Chung, Pavan K Auluck, Saranna Fanning, Daniel F Tardiff, Theresa Bartels, Martina Koeva, Stephen W Eichhorn, Hadar Benyamini, et al. Genome-scale networks link neurodegenerative disease genes to α-synuclein through specific molecular pathways. Cell systems, 4(2):157-170, 2017. Google Scholar
  63. Jochen Könemann, Stefano Leonardi, Guido Schäfer, and Stefan van Zwam. From primal-dual to cost shares and back: a stronger lp relaxation for the steiner forest problem. In International Colloquium on Automata, Languages, and Programming, pages 930-942. Springer, 2005. Google Scholar
  64. Jochen Könemann, Stefano Leonardi, Guido Schäfer, and Stefan HM van Zwam. A group-strategyproof cost sharing mechanism for the steiner forest game. SIAM Journal on Computing, 37(5):1319-1341, 2008. Google Scholar
  65. Guy Kortsarz. On the hardness of approximating spanners. Algorithmica, 30:432-450, 2001. Google Scholar
  66. Dean H Lorenz and Danny Raz. A simple efficient approximation scheme for the restricted shortest path problem. Operations Research Letters, 28(5):213-219, 2001. Google Scholar
  67. Jakub Pachocki, Liam Roditty, Aaron Sidford, Roei Tov, and Virginia Vassilevska Williams. Approximating cycles in directed graphs: Fast algorithms for girth and roundtrip spanners. In Artur Czumaj, editor, SODA, pages 1374-1392. SIAM, 2018. Google Scholar
  68. Mihai Patrascu and Liam Roditty. Distance oracles beyond the Thorup-Zwick bound. SIAM J. Comput., 43(1):300-311, 2014. Google Scholar
  69. David Peleg and Alejandro A. Schäffer. Graph spanners. Journal of Graph Theory, 13(1):99-116, 1989. URL:
  70. David Peleg and Jeffrey D. Ullman. An optimal synchronizer for the hypercube. SIAM J. Comput., 18(4):740-747, 1989. Google Scholar
  71. Leila Pirhaji, Pamela Milani, Mathias Leidl, Timothy Curran, Julian Avila-Pacheco, Clary B Clish, Forest M White, Alan Saghatelian, and Ernest Fraenkel. Revealing disease-associated pathways by network integration of untargeted metabolomics. Nature methods, 13(9):770-776, 2016. Google Scholar
  72. Liam Roditty, Mikkel Thorup, and Uri Zwick. Roundtrip spanners and roundtrip routing in directed graphs. ACM Trans. Algorithms, 4(3):29:1-29:17, 2008. Google Scholar
  73. Tim Roughgarden and Mukund Sundararajan. Optimal efficiency guarantees for network design mechanisms. In International Conference on Integer Programming and Combinatorial Optimization, pages 469-483. Springer, 2007. Google Scholar
  74. Xiangkun Shen and Viswanath Nagarajan. Online covering with l_q-norm objectives and applications to network design. Mathematical Programming, 184, 2020. Google Scholar
  75. Andrew Chi-Chih Yao. Space-time tradeoff for answering range queries (extended abstract). In STOC '82, 1982. Google Scholar
  76. Neal Young. The k-server dual and loose competitiveness for paging. Algorithmica, 11(6):525-541, 1994. Google Scholar
  77. Alexander Zelikovsky. A series of approximation algorithms for the acyclic directed steiner tree problem. Algorithmica, 18(1):99-110, 1997. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail