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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A Note on the Asymptotic Behavior of the Heights in b-Tries for b Large. | A Note on the Asymptotic Behavior of the Heights in b-TỶies for b Large Charles Knessl Wojciech SzpankowskY Dept. Mathematics Statistics Computer Science Department of Computer Science University of Illinois at Chicago Purdue University Chicago Illinois 60607-7045 W. Lafayette IN 47907 U.S.A. U.S.A. knessl@uic.edu spa@cs.purdue.edu Submitted August 2 1999 Accepted August 1 2000. Abstract We study the limiting distribution of the height in a generalized trie in which external nodes are capable to store up to b items the so called b-tries . We assume that such a tree is built from n random strings items generated by an unbiased memoryless source. In this paper we discuss the case when b and n are both large. We shall identify five regions of the height distribution that should be compared to three regions obtained for fixed b. We prove that for most n the limiting distribution is concentrated at the single point ki _log2 n b 1 as n b 1. We observe that this is quite different than the height distribution for fixed b in which case the limiting distribution is of an extreme value type concentrated around 1 1 b log2 n. We derive our results by analytic methods namely generating functions and the saddle point method. We also present some numerical verification of our results. 1 Introduction We study here the most basic digital tree known as a trie the name comes from retrieval . The primary purpose of a trie is to store a set S of strings words keys say S X1 . Xn . Each word X XiX2x3 . is a finite or infinite string of symbols taken from a finite alphabet. Throughout the paper we deal only with the binary alphabet 0 1 but all our results should be extendible to a general finite alphabet. A string will be stored in a leaf an external node of the trie. The trie over S is built recursively as follows For S 0 the trie is of course empty. For S 1 trie S is a single node. If S 1 S is split into two subsets So and Si so that a string is in Sj if its first symbol is j 2 0 1 . .
