Skip to main content Accessibility help
×
Home

A Binomial Splitting Process in Connection with Corner Parking Problems

  • Michael Fuchs (a1), Hsien-Kuei Hwang (a2), Yoshiaki Itoh (a3) and Hosam H. Mahmoud (a4)

Abstract

This paper studies a special type of binomial splitting process. Such a process can be used to model a high dimensional corner parking problem as well as determining the depth of random PATRICIA (practical algorithm to retrieve information coded in alphanumeric) tries, which are a special class of digital tree data structures. The latter also has natural interpretations in terms of distinct values in independent and identically distributed geometric random variables and the occupancy problem in urn models. The corresponding distribution is marked by a logarithmic mean and a bounded variance, which is oscillating, if the binomial parameter p is not equal to ½, and asymptotic to one in the unbiased case. Also, the limiting distribution does not exist as a result of the periodic fluctuations.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A Binomial Splitting Process in Connection with Corner Parking Problems
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      A Binomial Splitting Process in Connection with Corner Parking Problems
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      A Binomial Splitting Process in Connection with Corner Parking Problems
      Available formats
      ×

Copyright

Corresponding author

Postal address: Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan. Email address: mfuchs@math.nctu.edu.tw
∗∗ Postal address: Institute of Statistical Science and Institute of Information Science, Academia Sinica, Taipei, 115, Taiwan.
∗∗∗ Postal address: Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo, 190-8562, Japan.
∗∗∗∗ Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.

References

Hide All
[1] Archibald, M., Knopfmacher, A. and Prodinger, H. (2006). The number of distinct values in a geometrically distributed sample. Europ. J. Combinatorics 27, 10591081.
[2] Evans, J. W. (1993). Random and cooperative sequential adsorption. Rev. Modern Phys. 65, 12811330.
[3] Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.
[4] Flajolet, P., Gourdon, X. and Dumas, P. (1995). Mellin transforms and asymptotics: harmonic sums. Theoret. Comput. Sci. 144, 358.
[5] Fuchs, M., Hwang, H.-K. and Zacharovas, V. (2014). An analytic approach to the asymptotic variance of trie statistics and related structures. Theoret. Comput. Sci. 527, 136.
[6] Hayman, W. K. (1956). A generalisation of Stirling's formula. J. Reine Angew. Math. 196, 6795.
[7] Hwang, H.-K. and Janson, S. (2008). Local limit theorems for finite and infinite urn models. Ann. Prob. 36, 9921022.
[8] Hwang, H.-K., Fuchs, M. and Zacharovas, V. (2010). Asymptotic variance of random symmetric digital search trees. Discrete Math. Theoret. Comput. Sci. 12, 103165.
[9] Itoh, Y. and Solomon, H. (1986). Random sequential coding by Hamming distance. J. Appl. Prob. 23, 688695.
[10] Itoh, Y. and Ueda, S. (1983). On packing density by a discrete random sequential packing of cubes in a space of m dimension. Proc. Inst. Statist. Math. 31, 6569 (in Japanese).
[11] Jacquet, P. and Szpankowski, W. (1998). Analytical de-Poissonization and its applications. Theoret. Comput. Sci. 201, 162.
[12] Janson, S. and Szpankowski, W. (1997). Analysis of an asymmetric leader election algorithm. Electron. J. Combinatorics 4, Research Paper 17.
[13] Janson, S., Lavault, C. and Louchard, G. (2008). Convergence of some leader election algorithms. Discrete Math. Theoret. Comput. Sci. 10, 171196.
[14] Kalpathy, R. and Mahmoud, H. (2014). Perpetuities in fair leader election algorithms. Adv. Appl. Prob. 46 203216.
[15] Kirschenhofer, P. and Prodinger, H. (1987). Asymptotische Untersuchungen über charakteristische Parameter von Suchbäumen. Zahlentheoretische Analysis II, Lecture Notes in Math., 1262, 93107, Springer, Berlin.
[16] Kirschenhofer, P., Prodinger, H. and Szpankowski, W. (1996). Analysis of a splitting process arising in probabilistic counting and other related algorithms. Random Structures Algorithms 9, 379401.
[17] Mahmoud, H. M. (1992). Evolution of Random Search Trees. John Wiley, New York.
[18] Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York.
[19] Pittel, B. (1986). Paths in a random digital tree: limiting distributions. Adv. Appl. Prob. 18, 139155. (Correction: 24 (1992), 759.)
[20] Prodinger, H. (2004). Periodic oscillations in the analysis of algorithms and their cancellations. J. Iranian Statist. Soc. 3, 251270.
[21] Rais, B., Jacquet, P. and Szpankowski, W. (1993). Limiting distribution for the depth in PATRICIA tries. SIAM J. Discrete Math. 6, 197213.
[22] Dutour Sikirić, M. and Itoh, Y. (2011). Random Sequential Packing of Cubes. World Scientific, Hackensack, NJ.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed