Document Open Access Logo

Pairwise Independent Random Walks Can Be Slightly Unbounded

Author Shyam Narayanan



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2019.63.pdf
  • Filesize: 488 kB
  • 19 pages

Document Identifiers

Author Details

Shyam Narayanan
  • Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Acknowledgements

I would like to thank Prof. Jelani Nelson for advising this work, as well as for problem suggestions, forwarding me many papers from the literature, and providing helpful feedback on my writeup.

Cite AsGet BibTex

Shyam Narayanan. Pairwise Independent Random Walks Can Be Slightly Unbounded. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 63:1-63:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.63

Abstract

A family of problems that have been studied in the context of various streaming algorithms are generalizations of the fact that the expected maximum distance of a 4-wise independent random walk on a line over n steps is O(sqrt{n}). For small values of k, there exist k-wise independent random walks that can be stored in much less space than storing n random bits, so these properties are often useful for lowering space bounds. In this paper, we show that for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in +/- 1 and expected maximum distance Omega(sqrt{n} lg n) from the origin. We also show that this bound is tight for the first and second moment, i.e. the expected maximum square distance of a 2-wise independent random walk is always O(n lg^2 n). Also, for any even k >= 4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O(n^{k/2}). The previous two results generalize to random walks tracking insertion-only streams, and provide higher moment bounds than currently known. We also prove a generalization of Kolmogorov’s maximal inequality by showing an asymptotically equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of Błasiok.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • k-wise Independence
  • Random Walks
  • Moments
  • Chaining

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Noga Alon, Yossi Matias, and Mario Szegedy. The Space Complexity of Approximating the Frequency Moments. J. Comput. Syst. Sci., 58(1):137-147, 1999. URL: https://doi.org/10.1006/jcss.1997.1545.
  2. Christophe Andrieu, Nando de Freitas, Arnaud Doucet, and Michael I. Jordan. An Introduction to MCMC for Machine Learning. Machine Learning, 50(1):5-43, January 2003. URL: https://doi.org/10.1023/A:1020281327116.
  3. Itai Benjamini, Gady Kozma, and Dan Romik. Random walks with k-wise independent increments. Electron. Commun. Probab., 11:100-107, 2006. URL: https://doi.org/10.1214/ECP.v11-1201.
  4. Howard Berg. Random Walks in Biology. Princeton University Press, 1993. Google Scholar
  5. Jarosław Błasiok. Optimal streaming and tracking distinct elements with high probability. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 2432-2448, 2018. URL: https://doi.org/10.1137/1.9781611975031.156.
  6. Jarosław Błasiok, Jian Ding, and Jelani Nelson. Continuous Monitoring of 𝓁_p Norms in Data Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2017, August 16-18, 2017, Berkeley, CA, USA, pages 32:1-32:13, 2017. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.32.
  7. Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. Google Scholar
  8. Vladimir Braverman, Stephen R. Chestnut, Nikita Ivkin, Jelani Nelson, Zhengyu Wang, and David P. Woodruff. BPTree: An 𝓁₂ heavy hitters algorithm using constant memory. In Proceedings of the 36th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2017, Chicago, IL, USA, May 14-19, 2017, pages 361-376, 2017. URL: https://doi.org/10.1145/3034786.3034798.
  9. Vladimir Braverman, Stephen R. Chestnut, Nikita Ivkin, and David P. Woodruff. Beating CountSketch for heavy hitters in insertion streams. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 740-753, 2016. URL: https://doi.org/10.1145/2897518.2897558.
  10. Larry Carter and Mark N. Wegman. Universal Classes of Hash Functions. J. Comput. Syst. Sci., 18(2):143-154, 1979. URL: https://doi.org/10.1016/0022-0000(79)90044-8.
  11. R. M. Dudley. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. Journal of Functional Analysis, 1(3):290-330, 1967. URL: https://doi.org/10.1016/0022-1236(67)90017-1.
  12. M. P. Etienne and P. Vallois. Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score. Methodology And Computing In Applied Probability, 6(3):255-275, September 2004. URL: https://doi.org/10.1023/B:MCAP.0000026559.87023.ec.
  13. Eugene F. Fama. Random Walks in Stock Market Prices. Financial Analysts Journal, 21(5):55-59, 1965. URL: http://www.jstor.org/stable/4469865.
  14. Pierre-Gilles Gennes. Scaling Concepts in Polymer Physics. Cornell University Press, 1 edition, November 1979. Google Scholar
  15. Uffe Haagerup. The best constants in the Khintchine inequality. Studia Mathematica, 70(3):231-283, 1981. URL: http://eudml.org/doc/218383.
  16. A. Kolmogoroff. Über die summen durch den zufall bestimmter unabhängiger größen. Mathematische Annalen, 99:309-319, 1928. URL: http://eudml.org/doc/159258.
  17. László Lovász. Random Walks on Graphs: A Survey, 1993. Google Scholar
  18. Jelani Nelson. Chaining introduction with some computer science applications. Bulletin of the EATCS, 120, 2016. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/450.
  19. Anna Pagh, Rasmus Pagh, and Milan Ružić. Linear Probing with Constant Independence. SIAM J. Comput., 39(3):1107-1120, 2009. URL: https://doi.org/10.1137/070702278.
  20. Mihai Pǎtraşcu and Mikkel Thorup. On the k-Independence Required by Linear Probing and Minwise Independence. ACM Trans. Algorithms, 12(1):8:1-8:27, 2016. URL: https://doi.org/10.1145/2716317.
  21. Yao-Feng Ren and Han-Ying Liang. On the best constant in Marcinkiewicz–Zygmund inequality. Statistics & Probability Letters, 53(3):227-233, 2001. URL: https://doi.org/10.1016/S0167-7152(01)00015-3.
  22. Mathias Knudsen (via Jelani Nelson). Personal communication. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail