Random Walks with Multiple Step Lengths

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

In nature, search processes that use randomly oriented steps of different lengths have been observed at both the microscopic and the macroscopic scales. Physicists have analyzed in depth two such processes on grid topologies: Intermittent Search, which uses two step lengths, and Lévy Walk, which uses many. Taking a computational perspective, this paper considers the number of distinct step lengths k as a complexity measure of the considered process. Our goal is to understand what is the optimal achievable time needed to cover the whole terrain, for any given value of k. Attention is restricted to dimension one, since on higher dimensions, the simple random walk already displays a quasi linear cover time.

We say X is a k -intermittent search on the one dimensional n-node cycle if there exists a probability distribution \(\mathbf{p }=(p_i)_{i=1}^k\), and integers \(L_1,L_2,\ldots , L_k\), such that on each step X makes a jump \(\pm L_i\) with probability \(p_i\), where the direction of the jump (\(+\) or −) is chosen independently with probability 1/2. When performing a jump of length \(L_i\), the process consumes time \(L_i\), and is only considered to visit the last point reached by the jump (and not any other intermediate nodes). This assumption is consistent with biological evidence, in which entities do not search while moving ballistically.

We provide upper and lower bounds for the cover time achievable by k-intermittent searches for any integer k. In particular, we prove that in order to reduce the cover time \({\varTheta }(n^2)\) of a simple random walk to linear in n up to logarithmic factors, roughly \(\frac{\log n}{\log \log n}\) step lengths are both necessary and sufficient, and we provide an example where the lengths form an exponential sequence.

In addition, inspired by the notion of intermittent search, we introduce the Walk or Probe problem, which can be defined with respect to arbitrary graphs. Here, it is assumed that querying (probing) a node takes significantly more time than moving to a random neighbor. Hence, to efficiently probe all nodes, the goal is to balance the time spent walking randomly and the time spent probing. We provide preliminary results for connected graphs and regular graphs.

References

  1. 1.
    Adler, M., Räcke, H., Sivadasan, N., Sohler, C., Vöcking, B.: Randomized pursuit-evasion in graphs. Comb. Probab. Comput. 12(3), 225–244 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aldous, D., Fill, J.A.: Reversible Markov chains and random walks on graphs. Unfinished monograph, recompiled 2014 (2002). http://www.stat.berkeley.edu/aldous/RWG/book.html
  3. 3.
    Aleliunas, R., Karp, R.M., Lipton, R.J., Lovasz, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: SFCS. IEEE Computer Society, Washington, D.C., USA (1979)Google Scholar
  4. 4.
    Bartumeus, F., Catalan, J., Fulco, U.L., Lyra, M.L., Viswanathan, G.M.: Optimizing the encounter rate in biological interactions: Lévy versus Brownian strategies. Phys. Rev. Lett. 88, 097901 (2002)CrossRefGoogle Scholar
  5. 5.
    Bartumeus, F.: Lévy processes in animal movement: an evolutionary hypothesis. Fractals 15(02), 151–162 (2007)CrossRefGoogle Scholar
  6. 6.
    Bénichou, O., Loverdo, C., Moreau, M., Voituriez, R.: Intermittent search strategies. Rev. Mod. Phy. 83(1), 81 (2011)CrossRefMATHGoogle Scholar
  7. 7.
    Benjamini, I., Kozma, G., Wormald, N.: The mixing time of the giant component of a random graph. Random Struct. Algorithms 45(3), 383–407 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Berg, O.G., Winter, R.B., Von Hippel, P.H.: Diffusion-driven mechanisms of protein translocation on nucleic acids. 1. Models and theory. Biochemistry 20(24), 6929–6948 (1981)CrossRefGoogle Scholar
  9. 9.
    Boyer, D., et al.: Scale-free foraging by primates emerges from their interaction with a complex environment. Proc. R. Soc. Lond. B Biol. Sci. 273(1595), 1743–1750 (2006)CrossRefGoogle Scholar
  10. 10.
    Chupeau, M., Benichou, O., Voituriez, R.: Cover times of random searches. Nat. Phy. 11(10), 844 (2015)CrossRefGoogle Scholar
  11. 11.
    Cooper, C., Frieze, A.: The cover time of the Giant component of a random graph. Random Struct. Algorithms 32(4), 401–439 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Coppersmith, D., Feige, U., Shearer, J.B.: Random walks on regular and irregular graphs. SIAM J. Discret. Math. 9(2), 301–308 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Coppey, M., Bnichou, O., Voituriez, R., Moreau, M.: Kinetics of target site localization of a protein on DNA: a stochastic approach. Biophys. J. 87(3), 1640–1649 (2004)CrossRefGoogle Scholar
  14. 14.
    Czyzowicz, J., Gçsieniec, L., Georgiou, K., Kranakis, E., MacQuarrie, F.: The Beachcombers’ problem: walking and searching with mobile robots. Theor. Comput. Sci. 608(Part 3), 201–218 (2015). Structural Information and Communication ComplexityMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Edwards, A.M., et al.: Revisiting levy flight search patterns of wandering albatrosses, bumblebees and deer. Nature 449(7165), 1044 (2007)CrossRefGoogle Scholar
  16. 16.
    Einstein, A.: Investigations on the theory of the Brownian movement. Annal. Physik 34, 591–592 (1911)CrossRefGoogle Scholar
  17. 17.
    Feige, U.: Collecting coupons on trees, and the analysis of random walks. Technical report (1994)Google Scholar
  18. 18.
    Feige, U.: A tight lower bound on the cover time for random walks on graphs. Random Struct. Algorithms 6(4), 433–438 (1995)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fonio, E., Heyman, Y., Boczkowski, L., Gelblum, A., Kosowski, A., Korman, A., Feinerman, O.: A locally-blazed ant trail achieves efficient collective navigation despite limited information. eLife 5 (2016)Google Scholar
  20. 20.
    Friedman, J.: A proof of Alon’s second eigenvalue conjecture and related problems. CoRR, cs.DM/0405020 (2004)Google Scholar
  21. 21.
    Harris, T.H., et al.: Generalized Lévy walks and the role of chemokines in migration of effector CD8(+) T cells. Nature 486, 545 (2012)CrossRefGoogle Scholar
  22. 22.
    Israeli, A., Jalfon, M.: Token management schemes and random walks yield self-stabilizing mutual exclusion. In: PODC. ACM (1990)Google Scholar
  23. 23.
    Kanade, V., Mallmann-Trenn, F., Sauerwald, T.: On coalescence time in graphs-when is coalescing as fast as meeting? CoRR, abs/1611.02460 (2016)Google Scholar
  24. 24.
    Kempe, D., Kleinberg, J.M., Demers, A.J.: Spatial gossip and resource location protocols. J. ACM 51(6), 943–967 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Klafter, J., Zumofen, G.: Lévy statistics in a Hamiltonian system. Phys. Rev. E 49, 4873 (1994)CrossRefGoogle Scholar
  26. 26.
    Klafter, J., Shlesinger, M.F., Zumofen, G.: Beyond Brownian motion. Phys. Today 49(2), 33 (1996)CrossRefGoogle Scholar
  27. 27.
    Kleinberg, J.M.: The small-world phenomenon: an algorithmic perspective. In: STOC (2000)Google Scholar
  28. 28.
    Lawler, G., Limic, V.: Random Walk: A Modern Introduction. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
  29. 29.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  30. 30.
    Lomholt, M.A., et al.: Lévy strategies in intermittent search processes are advantageous. Proc. Natl. Acad. Sci. 105(32), 11055–11059 (2008)CrossRefGoogle Scholar
  31. 31.
    Loverdo, C.: Optimal search strategies and intermittent random walk: from restriction enzymes to the albatross flight, December 2009Google Scholar
  32. 32.
    Lyons, R.: Asymptotic enumeration of spanning trees. Comb. Probab. Comput. 14(4), 491–592 (2005)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Oshanin, G., et al.: Intermittent random walks for an optimal search strategy: one-dimensional case. J. Phy. Condens. Matter 19(6), 065142 (2007)CrossRefGoogle Scholar
  34. 34.
    Pearson, K.: The problem of the random walk. Nature 72, 1905 (1865)Google Scholar
  35. 35.
    Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 17:1–17:24 (2008)Google Scholar
  36. 36.
    Reynolds, G.: Navigating our world like birds and bees. The New York Times (2014)Google Scholar
  37. 37.
    Rhee, I., Shin, M., Hong, S., Lee, K., Chong, S.: On the Levy-walk nature of human mobility. In: IEEE INFOCOM 2008 (2008)Google Scholar
  38. 38.
    Travis, J.: Do wandering albatrosses care about math? Science 318(5851), 742–743 (2007)CrossRefGoogle Scholar
  39. 39.
    Viswanathan, G.M., et al.: Levy flight search patterns of wandering albatrosses. Nature 381(6581), 413 (1996)CrossRefGoogle Scholar
  40. 40.
    Viswanathan, G.M., et al.: Optimizing the success of random searches. Nature 401(6756), 911 (1999)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IRIF, CNRS and University Paris DiderotParisFrance
  2. 2.Ben Gurion University of the NegevBeershebaIsrael
  3. 3.Computer Sciences DepartmentUniversity of Wisconsin - MadisonMadisonUSA

Personalised recommendations