## Journal of Applied Probability

### A binomial splitting process in connection with corner parking problems

#### 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.

#### Article information

Source
J. Appl. Probab., Volume 51, Number 4 (2014), 971-989.

Dates
First available in Project Euclid: 20 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1421763322

Mathematical Reviews number (MathSciNet)
MR3301283

Zentralblatt MATH identifier
1321.60012

#### Citation

Fuchs, Michael; Hwang, Hsien-Kuei; Itoh, Yoshiaki; Mahmoud, Hosam H. A binomial splitting process in connection with corner parking problems. J. Appl. Probab. 51 (2014), no. 4, 971--989. https://projecteuclid.org/euclid.jap/1421763322

#### References

• Archibald, M., Knopfmacher, A. and Prodinger, H. (2006). The number of distinct values in a geometrically distributed sample. Europ. J. Combinatorics 27, 1059–1081.
• Evans, J. W. (1993). Random and cooperative sequential adsorption. Rev. Modern Phys. 65, 1281–1330.
• Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.
• Flajolet, P., Gourdon, X. and Dumas, P. (1995). Mellin transforms and asymptotics: harmonic sums. Theoret. Comput. Sci. 144, 3–58.
• 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, 1–36.
• Hayman, W. K. (1956). A generalisation of Stirling's formula. J. Reine Angew. Math. 196, 67–95.
• Hwang, H.-K. and Janson, S. (2008). Local limit theorems for finite and infinite urn models. Ann. Prob. 36, 992–1022.
• Hwang, H.-K., Fuchs, M. and Zacharovas, V. (2010). Asymptotic variance of random symmetric digital search trees. Discrete Math. Theoret. Comput. Sci. 12, 103–165.
• Itoh, Y. and Solomon, H. (1986). Random sequential coding by Hamming distance. J. Appl. Prob. 23, 688–695.
• 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, 65–69 (in Japanese).
• Jacquet, P. and Szpankowski, W. (1998). Analytical de-Poissonization and its applications. Theoret. Comput. Sci. 201, 1–62.
• Janson, S. and Szpankowski, W. (1997). Analysis of an asymmetric leader election algorithm. Electron. J. Combinatorics 4, Research Paper 17.
• Janson, S., Lavault, C. and Louchard, G. (2008). Convergence of some leader election algorithms. Discrete Math. Theoret. Comput. Sci. 10, 171–196.
• Kalpathy, R. and Mahmoud, H. (2014). Perpetuities in fair leader election algorithms. Adv. Appl. Prob. 46 203–216.
• Kirschenhofer, P. and Prodinger, H. (1987). Asymptotische Untersuchungen über charakteristische Parameter von Suchbäumen. Zahlentheoretische Analysis II, Lecture Notes in Math., 1262, 93–107, Springer, Berlin.
• 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, 379–401.
• Mahmoud, H. M. (1992). Evolution of Random Search Trees. John Wiley, New York.
• Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York.
• Pittel, B. (1986). Paths in a random digital tree: limiting distributions. Adv. Appl. Prob. 18, 139–155. (Correction: 24 (1992), 759.)
• Prodinger, H. (2004). Periodic oscillations in the analysis of algorithms and their cancellations. J. Iranian Statist. Soc. 3, 251–270.
• Rais, B., Jacquet, P. and Szpankowski, W. (1993). Limiting distribution for the depth in PATRICIA tries. SIAM J. Discrete Math. 6, 197–213.
• Dutour Sikirić, M. and Itoh, Y. (2011). Random Sequential Packing of Cubes. World Scientific, Hackensack, NJ.