Advertisement

The Language (and Series) of Hammersley-Type Processes

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10881)

Abstract

We study languages and formal power series associated to (variants of) the Hammersley process. We show that the ordinary Hammersley process yields a regular language and the Hammersley tree process yields deterministic context-free (but non-regular) languages. For the Hammersley interval process we show that there are two relevant variants of formal languages. One of them leads to the same language as the ordinary Hammersley tree process. The other one yields non-context-free languages.

The results are motivated by the problem of studying the analog of the famous Ulam-Hammersley problem for heapable sequences. Towards this goal we also give an algorithm for computing formal power series associated to the Hammersley process. We employ this algorithm to settle the nature of the scaling constant, conjectured in previous work to be the golden ratio. Our results provide experimental support to this conjecture.

References

  1. 1.
    Aldous, D., Diaconis, P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theor. Relat. Fields 103(2), 199–213 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aldous, D., Diaconis, P.: Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bull. A.M.S. 36(4), 413–432 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balogh, J., Bonchiş, C., Diniş, D., Istrate, G., Todincã, I.: Heapability of partial orders. arXiv preprint arXiv:1706.01230 (2017)
  4. 4.
    Basdevant, A.-L., Gerin, L., Gouéré, J.-B., Singh, A.: From Hammersley’s lines to Hammersley’s trees. Prob. Theory Related Fields 171(1–2), 1–51 (2018).  https://doi.org/10.1007/s00440-017-0772-2MathSciNetCrossRefGoogle Scholar
  5. 5.
    Basdevant, A.-L., Singh, A.: Almost-sure asymptotic for the number of heaps inside a random sequence. arXiv preprint arXiv:1702.06444 (2017)
  6. 6.
    Berstel, J., Reutenauer, C.: Noncommutative Rational Series with Applications, vol. 137. Cambridge University Press, Cambridge (2011)MATHGoogle Scholar
  7. 7.
    Byers, J., Heeringa, B., Mitzenmacher, M., Zervas, G.: Heapable sequences and subseqeuences. In: Proceedings of ANALCO 2011, pp. 33–44. SIAM Press (2011)Google Scholar
  8. 8.
    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Istrate, G., Bonchiş, C.: Partition into heapable sequences, heap tableaux and a multiset extension of Hammersley’s process. In: Cicalese, F., Porat, E., Vaccaro, U. (eds.) CPM 2015. LNCS, vol. 9133, pp. 261–271. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-19929-0_22CrossRefGoogle Scholar
  10. 10.
    Istrate, G., Bonchiş, C.: Heapability, interactive particle systems, partial orders: results and open problems. In: Câmpeanu, C., Manea, F., Shallit, J. (eds.) DCFS 2016. LNCS, vol. 9777, pp. 18–28. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-41114-9_2CrossRefMATHGoogle Scholar
  11. 11.
    Justicz, J., Scheinerman, E.R., Winkler, P.M.: Random intervals. Am. Math. Mon. 97(10), 881–889 (1990)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liggett, T.: Interacting Particle Systems. Springer, Heidelberg (2005).  https://doi.org/10.1007/b138374CrossRefMATHGoogle Scholar
  13. 13.
    Moore, C., Lakdawala, P.: Queues, stacks and transcendentality at the transition to chaos. Physica D 135(1–2), 24–40 (2000)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Petre, I., Salomaa, A.: Algebraic systems and pushdown automata. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, pp. 257–289. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-01492-5_7CrossRefGoogle Scholar
  15. 15.
    Porfilio, J.: A combinatorial characterization of heapability. Master’s thesis, Williams College, May 2015. https://unbound.williams.edu/theses/islandora/object/studenttheses%3A907. Accessed Dec 2017
  16. 16.
    Reutenauer, C.: Séries formelles et algebres syntactiques. J. Algebra 66(2), 448–483 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Romik, D.: The Surprising Mathematics of Longest Increasing Subsequences. Cambridge University Press, Cambridge (2015)MATHGoogle Scholar
  18. 18.
    Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Springer, New York (1978).  https://doi.org/10.1007/978-1-4612-6264-0CrossRefMATHGoogle Scholar
  19. 19.
    Szpankowski, W.: Average Case of Algorithms on Sequences. Wiley, New York (2001)CrossRefGoogle Scholar
  20. 20.
    Welsh, D.: Complexity: Knots, Colourings and Counting. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar
  21. 21.
    Xie, H.: Grammatical Complexity and One-Dimensional Dynamical Systems. Directions in Chaos, vol. 6. World Scientific, Singapore (1996)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Cosmin Bonchiş
    • 1
    • 2
  • Gabriel Istrate
    • 1
    • 2
  • Vlad Rochian
    • 1
  1. 1.Department of Computer ScienceWest University of TimişoaraTimişoaraRomania
  2. 2.e-Austria Research InstituteTimişoaraRomania

Personalised recommendations