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Dynamic Graph Stream Algorithms in o(n) Space

Authors Zengfeng Huang, Pan Peng

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Zengfeng Huang
Pan Peng

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Zengfeng Huang and Pan Peng. Dynamic Graph Stream Algorithms in o(n) Space. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.ICALP.2016.18

Abstract

In this paper we study graph problems in dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require Omega(n) space, where n is the number of vertices, existing works mainly focused on designing ~O(n) space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for many applications (e.g. n is huge or the graph is sparse). In this work, we give single-pass algorithms beating this space barrier for two classes of problems. We present o(n) space algorithms for estimating the number of connected components with additive error epsilon*n and (1 + epsilon)-approximating the weight of minimum spanning tree. The latter improves previous ~O(n) space algorithm given by Ahn et al. (SODA 2012) for connected graphs with bounded edge weights. We initiate the study of approximate graph property testing in the dynamic streaming model, where we want to distinguish graphs satisfying the property from graphs that are epsilon-far from having the property. We consider the problem of testing k-edge connectivity, k-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly ~O(n^{1-epsilon}) space, which is o(n) for any constant epsilon. To complement our algorithms, we present Omega(n^{1-O(epsilon)}) space lower bounds for these problems, which show that such a dependence on epsilon is necessary.

Subject Classification

Keywords
  • dynamic graph streams
  • sketching
  • property testing
  • minimum spanning tree

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References

  1. Kook Jin Ahn, Graham Cormode, Sudipto Guha, Andrew McGregor, and Anthony Wirth. Correlation clustering in data streams. In Proceedings of the 32nd International Conference on Machine Learning, ICML, pages 6-11, 2015. Google Scholar
  2. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 459-467. SIAM, 2012. Google Scholar
  3. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Proceedings of the 31st symposium on Principles of Database Systems, pages 5-14. ACM, 2012. Google Scholar
  4. Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 20-29. ACM, 1996. Google Scholar
  5. Sepelir Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. Maximum matchings in dynamic graph streams and the simultaneous communication model. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 1345-1364. SIAM, 2016. URL: http://dl.acm.org/citation.cfm?id=2884435.2884528.
  6. Sayan Bhattacharya, Monika Henzinger, Danupon Nanongkai, and Charalampos E Tsourakakis. Space-and time-efficient algorithm for maintaining dense subgraphs on one-pass dynamic streams. In ACM Symposium on Theory of Computing, 2015. Google Scholar
  7. Marc Bury and Chris Schwiegelshohn. Sublinear estimation of weighted matchings in dynamic data streams. ESA, 2015. Google Scholar
  8. Bernard Chazelle, Ronitt Rubinfeld, and Luca Trevisan. Approximating the minimum spanning tree weight in sublinear time. SIAM Journal on computing, 34(6):1370-1379, 2005. Google Scholar
  9. Rajesh Chitnis, Graham Cormode, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Andrew McGregor, Morteza Monemizadeh, and Sofya Vorotnikova. Kernelization via sampling with applications to dynamic graph streams. SODA, 2016. Google Scholar
  10. Rajesh Chitnis, Graham Cormode, MohammadTaghi Hajiaghayi, and Morteza Monemizadeh. Parameterized streaming: maximal matching and vertex cover. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1234-1251. SIAM, 2015. Google Scholar
  11. Artur Czumaj, Funda Ergün, Lance Fortnow, Avner Magen, Ilan Newman, Ronitt Rubinfeld, and Christian Sohler. Approximating the weight of the euclidean minimum spanning tree in sublinear time. SIAM Journal on Computing, 35(1):91-109, 2005. Google Scholar
  12. Artur Czumaj, Morteza Monemizadeh, Krzysztof Onak, and Christian Sohler. Planar graphs: Random walks and bipartiteness testing. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 423-432. IEEE, 2011. Google Scholar
  13. Artur Czumaj and Christian Sohler. Estimating the weight of metric minimum spanning trees in sublinear time. SIAM Journal on Computing, 39(3):904-922, 2009. Google Scholar
  14. Hossein Esfandiari, Mohammad T Hajiaghayi, Vahid Liaghat, Morteza Monemizadeh, and Krzysztof Onak. Streaming algorithms for estimating the matching size in planar graphs and beyond. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1217-1233. SIAM, 2015. Google Scholar
  15. Hossein Esfandiari, MohammadTaghi Hajiaghayi, and David P Woodruff. Applications of uniform sampling: Densest subgraph and beyond. arXiv preprint arXiv:1506.04505, 2015. Google Scholar
  16. Stefan Fafianie and Stefan Kratsch. Streaming kernelization. In Mathematical Foundations of Computer Science 2014, pages 275-286. Springer, 2014. Google Scholar
  17. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theoretical Computer Science, 348(2):207-216, 2005. Google Scholar
  18. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. Graph distances in the data-stream model. SIAM Journal on Computing, 38(5):1709-1727, 2008. Google Scholar
  19. Gereon Frahling, Piotr Indyk, and Christian Sohler. Sampling in dynamic data streams and applications. In Proceedings of the twenty-first annual symposium on Computational geometry, pages 142-149. ACM, 2005. Google Scholar
  20. Oded Goldreich. Introduction to testing graph properties. In Property testing, pages 105-141. Springer, 2011. Google Scholar
  21. Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM (JACM), 45(4):653-750, 1998. Google Scholar
  22. Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, 32:302-343, 2002. Google Scholar
  23. Sudipto Guha, Andrew McGregor, and David Tench. Vertex and hyperedge connectivity in dynamic graph streams. In Proceedings of the 34th ACM Symposium on Principles of Database Systems, pages 241-247. ACM, 2015. Google Scholar
  24. Monika Rauch Henzinger, Prabhakar Raghavan, and Sridhar Rajagopalan. Computing on data streams. In External Memory Algorithms, Proceedings of a DIMACS Workshop, New Brunswick, New Jersey, USA, May 20-22, 1998, pages 107-118, 1998. Google Scholar
  25. Hossein Jowhari. Estimating the number of connected components in graph streams. Personal communication. Google Scholar
  26. Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Approximating matching size from random streams. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 734-751. SIAM, 2014. Google Scholar
  27. Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Streaming lower bounds for approximating max-cut. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1263-1282. SIAM, 2015. Google Scholar
  28. Michael Kapralov, Yin Tat Lee, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 561-570. IEEE, 2014. Google Scholar
  29. Michael Kapralov and David Woodruff. Spanners and sparsifiers in dynamic streams. In Proceedings of the 2014 ACM symposium on Principles of distributed computing, pages 272-281. ACM, 2014. Google Scholar
  30. Dmitry Kogan and Robert Krauthgamer. Sketching cuts in graphs and hypergraphs. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, pages 367-376. ACM, 2015. Google Scholar
  31. Christian Konrad. Maximum matching in turnstile streams. ESA, 2015. Google Scholar
  32. Andrew McGregor. Graph stream algorithms: A survey. ACM SIGMOD Record, 43(1):9-20, 2014. Google Scholar
  33. Andrew McGregor, David Tench, Sofya Vorotnikova, and Hoa T Vu. Densest subgraph in dynamic graph streams. MFCS, 2015. Google Scholar
  34. S Muthukrishnan. Data streams: Algorithms and applications. Theoretical Computer Science, 1(2):117-236, 2005. Google Scholar
  35. Yaron Orenstein and Dana Ron. Testing eulerianity and connectivity in directed sparse graphs. Theoretical Computer Science, 412(45):6390-6408, 2011. Google Scholar
  36. Michal Parnas and Dana Ron. Testing the diameter of graphs. Random Structures &Algorithms, 20(2):165-183, 2002. Google Scholar
  37. Eric Price. Efficient sketches for the set query problem. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 41-56. SIAM, 2011. Google Scholar
  38. Dana Ron. Algorithmic and analysis techniques in property testing. Foundations and Trendsregistered in Theoretical Computer Science, 5(2):73-205, 2010. Google Scholar
  39. Ronitt Rubinfeld and Asaf Shapira. Sublinear time algorithms. SIAM Journal on Discrete Mathematics, 25(4):1562-1588, 2011. Google Scholar
  40. Xiaoming Sun and David P. Woodruff. Tight bounds for graph problems in insertion streams. In The 18th. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX'2015), 2015. Google Scholar
  41. Elad Verbin and Wei Yu. The streaming complexity of cycle counting, sorting by reversals, and other problems. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 11-25. SIAM, 2011. Google Scholar
  42. Yuichi Yoshida and Hiro Ito. Property testing on k-vertex-connectivity of graphs. In Automata, Languages and Programming, pages 539-550. Springer, 2008. Google Scholar
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