Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths

Authors Brieuc Guinard , Amos Korman

Thumbnail PDF


  • Filesize: 494 kB
  • 14 pages

Document Identifiers

Author Details

Brieuc Guinard
  • IRIF, CNRS, University of Paris, France
Amos Korman
  • IRIF, CNRS, University of Paris, France

Cite AsGet BibTex

Brieuc Guinard and Amos Korman. Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Search patterns of randomly oriented steps of different lengths have been observed on all scales of the biological world, ranging from microscopic to the ecological, including in protein motors, bacteria, T-cells, honeybees, marine predators, and more, see e.g., [Humphries et al., 2010; Jansen et al., 2012; Reynolds et al., 2017; Schuster and Levandowsky, 1996; Humphries et al., 2010; Viswanathan et al., 1996; Viswanathan et al., 1999]. Through different models, it has been demonstrated that adopting a variety in the magnitude of the step lengths can greatly improve the search efficiency. However, the precise connection between the search efficiency and the number of step lengths in the repertoire of the searcher has not been identified. Motivated by biological examples in one-dimensional terrains, a recent paper studied the best cover time on an n-node cycle that can be achieved by a random walk process that uses k step lengths [Boczkowski et al., 2018]. By tuning the lengths and corresponding probabilities the authors therein showed that the best cover time is roughly n^{1+Θ(1/k)}. While this bound is useful for large values of k, it is hardly informative for small k values, which are of interest in biology [Auger-Méthé et al., 2015; Bénichou et al., 2011; Lomholt et al., 2008; {Reynolds}, 2014]. In this paper, we provide a tight bound for the cover time of such a walk, for every integer k> 1. Specifically, up to lower order polylogarithmic factors, the cover time is n^{1+1/(2k-1)}. For k=2,3, 4 and 5 the bound is thus n^{4/3}, n^{6/5}, n^{8/7}, and n^{10/9}, respectively. Informally, our result implies that, as long as the number of step lengths k is not too large, incorporating an additional step length to the repertoire of the process enables to improve the cover time by a polynomial factor, but the extent of the improvement gradually decreases with k.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Applied computing → Computational biology
  • Computational Biology
  • Randomness in Computing
  • Search Algorithms
  • Random Walks
  • Lévy Flights
  • Intermittent Search
  • CCRW


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


  1. D. Aldous and J. A. Fill. Reversible markov chains and random walks on graphs, 2002. Google Scholar
  2. M. Auger-Méthé, A. Derocher, M. Plank, E. Codling, and M.. Lewis. Differentiating the lévy walk from a composite correlated random walk. Methods in Ecology and Evolution, 6(10):1179-1189, 2015. Google Scholar
  3. Simon Benhamou and Julien Collet. Ultimate failure of the lévy foraging hypothesis: Two-scale searching strategies outperform scale-free ones even when prey are scarce and cryptic. Journal of theoretical biology, 387, October 2015. Google Scholar
  4. Olivier Bénichou, C Loverdo, M Moreau, and R Voituriez. Intermittent search strategies. Reviews of Modern Physics, 83(1), 2011. Google Scholar
  5. Otto G. Berg, Robert B. Winter, and Peter H. Von Hippel. Diffusion-driven mechanisms of protein translocation on nucleic acids. 1. models and theory. Biochemistry, 20(24), 1981. Google Scholar
  6. L. Boczkowski, O. Feinerman, A. Korman, and E. Natale. Limits for rumor spreading in stochastic populations. In 9th Innovations in Theoretical Computer Science Conference, pages 49:1-49:21, 2018. Google Scholar
  7. L. Boczkowski, B. Guinard, A. Korman, Z. Lotker, and M. Renault. Random walks with multiple step lengths. LATIN 2018: Theoretical Informatics, pages 174-186, January 2018. Google Scholar
  8. L. Boczkowski, E. Natale, O. Feinerman, and A. Korman. Limits on reliable information flows through stochastic populations. PLOS Computational Biology, 14(6):1-15, June 2018. Google Scholar
  9. D. Boyer, G. Ramos-Fernandez, O. Miramontes, J. Mateos, G. Cocho, H. Larralde, H. Ramos, and F. Rojas. Scale-free foraging by primates emerges from their interaction with a complex environment. Proceedings of the Royal Society of London B: Biological Sciences, 273(1595), 2006. Google Scholar
  10. Bernard Chazelle. Natural algorithms. In Proceedings of the twentieth annual ACM-SIAM symposium on Discrete algorithms, pages 422-431, 2009. Google Scholar
  11. Bernard Chazelle. The convergence of bird flocking. J. ACM, 61(4):21:1-21:35, 2014. Google Scholar
  12. M. Chupeau, O. Bénichou, and R. Voituriez. Cover times of random searches. Nature Physics, 11(10):844, 2015. Google Scholar
  13. A. Comtet and S. N. Majumdar. Precise asymptotics for a random walker’s maximum. Journal of Statistical Mechanics: Theory and Experiment, 6:06013, June 2005. Google Scholar
  14. M. Coppey, O. Bénichou, R. Voituriez, and M. Moreau. Kinetics of Target Site Localization of a Protein on DNA: A Stochastic Approach. Biophysical Journal, 87(3), 2004. Google Scholar
  15. Monique de Jager, Franz J Weissing, Peter MJ Herman, Bart A Nolet, and Johan van de Koppel. Lévy walks evolve through interaction between movement and environmental complexity. Science, 332(6037):1551-1553, 2011. Google Scholar
  16. Ofer Feinerman and Amos Korman. Theoretical distributed computing meets biology: A review. In International Conference on Distributed Computing and Internet Technology, pages 1-18. Springer, 2013. Google Scholar
  17. Ofer Feinerman and Amos Korman. The ANTS problem. Distributed Computing, 30(3):149-168, 2017. Google Scholar
  18. T. Harris, E. Banigan, D. Christian, et al. Generalized lévy walks and the role of chemokines in migration of effector cd8(+) t cells. Nature, 486(7404), 2012. Google Scholar
  19. T. Hills, P. Todd, D. Lazer, A. Redish, I. Couzin, and Cognitive Search Research Group. Exploration versus exploitation in space, mind, and society. Trends in cognitive sciences, 19(1):46-54, 2015. Google Scholar
  20. B. D. Hughes, M. F. Shlesinger, and E. W. Montroll. Random Walks with Self-Similar Clusters. Proceedings of the National Academy of Science, 78:3287-3291, June 1981. Google Scholar
  21. N. E. Humphries, N. Queiroz, J. R. M. Dyer, N. G. Pade, M. K. Musyl, K. M. Schaefer, D. W. Fuller, J. M. Brunnschweiler, T. K. Doyle, J. D. R. Houghton, G. C. Hays, C. S. Jones, L. R. Noble, V. J. Wearmouth, E. J. Southall, and D. W. Sims. Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature, 465:1066-1069, June 2010. Google Scholar
  22. Vincent A. A. Jansen, Alla Mashanova, and Sergei Petrovskii. Comment on quotedblleftlévy walks evolve through interaction between movement and environmental complexityquotedblright. Science, 335(6071):918-918, 2012. Google Scholar
  23. A. Koelzsch, A. Alzate, F. Bartumeus, M. de Jager, E. Weerman, G. Hengeveld, M. Naguib, B. Nolet, and J. van de Koppel. Experimental evidence for inherent lévy search behaviour in foraging animals. Proceedings of the Royal Society B, 282, May 2015. Google Scholar
  24. David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, 2008. Google Scholar
  25. Michael Lomholt, Koren Tal, Ralf Metzler, and Joseph Klafter. Lévy strategies in intermittent search processes are advantageous. Proceedings of the National Academy of Sciences, 105(32), 2008. Google Scholar
  26. Cameron Musco, Hsin-Hao Su, and Nancy A. Lynch. Ant-inspired density estimation via random walks: Extended abstract. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25-28, 2016, pages 469-478, 2016. Google Scholar
  27. G. Oshanin, H. Wio, K. Lindenberg, and S. Burlatsky. Intermittent random walks for an optimal search strategy: one-dimensional case. Journal of Physics: Condensed Matter, 19(6), 2007. Google Scholar
  28. G. H. Pyke. Understanding movements of organisms: it’s time to abandon the lévy foraging hypothesis. Methods in Ecology and Evolution, 6(1):1-16, 2015. Google Scholar
  29. D. A. Raichlen, B. M. Wood, A. D. Gordon, A. Z. P. Mabulla, F. W. Marlowe, and H. Pontzer. Evidence of Lévy walk foraging patterns in human hunter-gatherers. Proceedings of the National Academy of Science, 111:728-733, January 2014. Google Scholar
  30. A. Reynolds. Mussels realize Weierstrassian Lévy walks as composite correlated random walks. Scientific Reports, 4:4409, March 2014. Google Scholar
  31. A. Reynolds, G. Santini, G. Chelazzi, and S. Focardi. The weierstrassian movement patterns of snails. Royal Society Open Science, 4(6):160941, 2017. Google Scholar
  32. I. Rhee, M. Shin, S. Hong, K. Lee, and S. Chong. On the lévy-walk nature of human mobility. In IEEE INFOCOM 2008, 2011. Google Scholar
  33. F. L. Schuster and M. Levandowsky. Chemosensory responses of acanthamoeba castellanii: Visual analysis of random movement and responses to chemical signals. Journal of Eukaryotic Microbiology, 43(2):150-158, 1996. Google Scholar
  34. D. W. Sims, E. J. Southall, N. E. Humphries, G. C. Hays, C. J. A. Bradshaw, J. W. Pitchford, A. James, M. Z. Ahmed, A. S. Brierley, M. A. Hindell, D. Morritt, M. K. Musyl, D. Righton, E. L. C. Shepard, V. J. Wearmouth, R. P. Wilson, M. J. Witt, and J. D. Metcalfe. Scaling laws of marine predator search behaviour. Nature, 451:1098-1102, February 2008. Google Scholar
  35. G. Viswanathan, V. Afanasyevt, S. Buldyrev, E. Murphyt, P. Princet, and H. Stanley. Lévy flight search patterns of wandering albatrosses. Nature, 381(6581), 1996. Google Scholar
  36. G. Viswanathan, S. Buldyrev, S. Havlin, M. da Luz, E. Raposo, and E. Stanley. Optimizing the success of random searches. Nature, 401(6756), 1999. Google Scholar