Sublinear Estimation of Weighted Matchings in Dynamic Data Streams

  • Marc BuryEmail author
  • Chris Schwiegelshohn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)


This paper presents an algorithm for estimating the weight of a maximum weighted matching by augmenting any estimation routine for the size of an unweighted matching. The algorithm is implementable in any streaming model including dynamic graph streams. We also give the first constant estimation for the maximum matching size in a dynamic graph stream for planar graphs (or any graph with bounded arboricity) using \(\tilde{O}(n^{4/5})\) space which also extends to weighted matching. Using previous results by Kapralov, Khanna, and Sudan (2014) we obtain a polylog(n) approximation for general graphs using polylog(n) space in random order streams, respectively. In addition, we give a space lower bound of Ω(n1 − ε) for any randomized algorithm estimating the size of a maximum matching up to a 1 + O(ε) factor for adversarial streams.


Planar Graph Maximum Match Pseudorandom Generator Weighted Match Stream Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahn, K., Guha, S.: Graph sparsification in the semi-streaming model. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part II. LNCS, vol. 5556, pp. 328–338. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Ahn, K., Guha, S., McGregor, A.: Analyzing graph structure via linear measurements. In: SODA, pp. 459–467 (2012)Google Scholar
  3. 3.
    Ahn, K., Guha, S., McGregor, A.: Graph sketches: sparsification, spanners, and subgraphs. In: PODS, pp. 5–14 (2012)Google Scholar
  4. 4.
    Ahn, K., Guha, S., McGregor, A.: Spectral sparsification in dynamic graph streams. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds.) RANDOM 2013 and APPROX 2013. LNCS, vol. 8096, pp. 1–10. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Assadi, S., Khanna, S., Li, Y., Yaroslavtsev, G.: Tight bounds for linear sketches of approximate matchings. CoRR, abs/1505.01467 (2015)Google Scholar
  6. 6.
    Bar-Yossef, Z., Jayram, T.S., Kerenidis, I.: Exponential separation of quantum and classical one-way communication complexity. SIAM J. Comput. 38(1), 366–384 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chitnis, R., Cormode, G., Esfandiari, H., Hajiaghayi, M., McGregor, A., Monemizadeh, M., Vorotnikova, S.: Kernelization via sampling with applications to dynamic graph streams. CoRR, abs/1505.01731 (2015)Google Scholar
  8. 8.
    Clarkson, K., Woodruff, D.: Numerical linear algebra in the streaming model. In: STOC, pp. 205–214 (2009)Google Scholar
  9. 9.
    Crouch, M., McGregor, A., Stubbs, D.: Dynamic graphs in the sliding-window model. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 337–348. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  10. 10.
    Crouch, M., Stubbs, D.: Improved streaming algorithms for weighted matching, via unweighted matching. In: APPROX/RANDOM 2014, pp. 96–104 (2014)Google Scholar
  11. 11.
    Epstein, L., Levin, A., Mestre, J., Segev, D.: Improved approximation guarantees for weighted matching in the semi-streaming model. SIAM J. Discrete Math. 25(3), 1251–1265 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Epstein, L., Levin, A., Segev, D., Weimann, O.: Improved bounds for online preemptive matching. In: STACS, pp. 389–399 (2013)Google Scholar
  13. 13.
    Esfandiari, H., Hajiaghayi, M., Liaghat, V., Monemizadeh, M., Onak, K.: Streaming algorithms for estimating the matching size in planar graphs and beyond. In: SODA, pp. 1217–1233 (2015)Google Scholar
  14. 14.
    Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: On graph problems in a semi-streaming model. Theor. Comput. Sci. 348(2-3), 207–216 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gavinsky, D., Kempe, J., Kerenidis, I., Raz, R., de Wolf, R.: Exponential separation for one-way quantum communication complexity, with applications to cryptography. SIAM J. Comput. 38(5), 1695–1708 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Goel, A., Kapralov, M., Khanna, S.: On the communication and streaming complexity of maximum bipartite matching. In: SODA, pp. 468–485 (2012)Google Scholar
  17. 17.
    Henzinger, M., Raghavan, P., Rajagopalan, S.: Computing on data streams (1998)Google Scholar
  18. 18.
    Indyk, P.: Stable distributions, pseudorandom generators, embeddings and data stream computation. In: FOCS, pp. 189–197 (2000)Google Scholar
  19. 19.
    Kapralov, M.: Better bounds for matchings in the streaming model. In: SODA, pp. 1679–1697 (2013)Google Scholar
  20. 20.
    Kapralov, M., Khanna, S., Sudan, M.: Approximating matching size from random streams. In: SODA, pp. 734–751 (2014)Google Scholar
  21. 21.
    Konrad, C.: Maximum matching in turnstile streams. CoRR, abs/1505.01460 (2015)Google Scholar
  22. 22.
    Konrad, C., Magniez, F., Mathieu, C.: Maximum matching in semi-streaming with few passes. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX 2012 and RANDOM 2012. LNCS, vol. 7408, pp. 231–242. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Li, Y., Nguyen, H., Woodruff, D.: On sketching matrix norms and the top singular vector. In: SODA, pp. 1562–1581 (2014)Google Scholar
  24. 24.
    Li, Y., Nguyen, H., Woodruff, D.: Turnstile streaming algorithms might as well be linear sketches. In: STOC, pp. 174–183 (2014)Google Scholar
  25. 25.
    Lovász, L.: On determinants, matchings, and random algorithms. In: FCT, pp. 565–574 (1979)Google Scholar
  26. 26.
    McGregor, A.: Finding graph matchings in data streams. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 170–181. Springer, Heidelberg (2005)Google Scholar
  27. 27.
    McGregor, A.: Graph stream algorithms: a survey. SIGMOD Record 43(1), 9–20 (2014)CrossRefGoogle Scholar
  28. 28.
    Nisan, N.: Pseudorandom generators for space-bounded computation. Combinatorica 12(4), 449–461 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tutte, W.: The factorization of linear graphs. J. London Math. Soc. 22, 107–111 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Uehara, R., Chen, Z.: Parallel approximation algorithms for maximum weighted matching in general graphs. Inf. Process. Lett. 76(1-2), 13–17 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Verbin, E., Yu, W.: The streaming complexity of cycle counting, sorting by reversals, and other problems. In: SODA, pp. 11–25. SIAM (2011)Google Scholar
  32. 32.
    Zelke, M.: Weighted matching in the semi-streaming model. Algorithmica 62(1-2), 1–20 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Efficient Algorithms and Complexity TheoryTU DortmundDortmundGermany

Personalised recommendations