A multivariate view of random bucket digital search trees

  • Authors:

  • Friedrich Hubalek;Hsien-Kuei Hwang;William Lew;Hosam Mahmoud;Helmut Prodinger

  • Affiliations:

  • Institut für Finanzund Versicherungsmathematik, Technische Universität Wien, 1040 Vienna, Austria;Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan;Center for Technology & Engineering, The General Accounting Office, Washington, DC;Department of Statistics, The George Washington University, Washington, DC;School of Mathematics, University of the Witwatersrand, PO Wits, 2050 Johannesburg, South Africa

  • Venue:
  • Journal of Algorithms - Analysis of algorithms
  • Year:
  • 2002

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Abstract

We take a multivariate view of digital search trees by studying the number of nodes of different types that may coexist in a bucket digital search tree as it grows under an arbitrary memory management system. We obtain the mean of each type of node, as well as the entire covariance matrix between types, whereupon weak laws of large numbers follow from the orders of magnitude (the norming constants include oscillating functions). The result can be easily interpreted for practical systems like paging, heaps and UNIX's buddy system. The covariance results call for developing a Mellin convolution method, where convoluted numerical sequences are handled by convolutions of their Mellin transforms. Furthermore, we use a method of moments to show that the distribution is asymptotically normal. The method of proof is of some generality and is applicable to other parameters like path length and size in random tries and Patricia tries.