Browsing by Author "Hwang, Hsien-Kuei"

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A BINOMIAL SPLITTING PROCESS IN CONNECTION WITH CORNER PARKING PROBLEMS
(2014-12-01)
Fuchs, Michael
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Hwang, Hsien-Kuei
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Itoh, Yoshiaki
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Mahmoud, Hosam H.
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應用數學系
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 1/2, and asymptotic to one in the unbiased case. Also, the limiting distribution does not exist as a result of the periodic fluctuations.
An analytic approach to the asymptotic variance of trie statistics and related structures
(2014-03-27)
Fuchs, Michael
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Hwang, Hsien-Kuei
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Zacharovas, Vytas
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應用數學系
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Department of Applied Mathematics
We develop analytic tools for the asymptotics of general trie statistics, which are particularly advantageous for clarifying the asymptotic variance. Many concrete examples are discussed for which new Fourier expansions are given. The tools are also useful for other splitting processes with an underlying binomial distribution. We specially highlight Philippe Flajolet's contribution in the analysis of these random structures. (C) 2014 Elsevier B.V. All rights reserved.
Dependence and phase changes in random m-ary search trees
(2017-05-01)
Chern, Hua-Huai
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Fuchs, Michael
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Hwang, Hsien-Kuei
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Neininger, Ralph
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應用數學系
We study the joint asymptotic behavior of the space requirement and the total path length (either summing over all root-key distances or over all root-node distances) in random m-ary search trees. The covariance turns out to exhibit a change of asymptotic behavior: it is essentially linear when 3m13, but becomes of higher order when m14. Surprisingly, the corresponding asymptotic correlation coefficient tends to zero when 3m26, but is periodically oscillating for larger m, and we also prove asymptotic independence when 3m26. Such a less anticipated phenomenon is not exceptional and our results can be extended in two directions: one for more general shape parameters, and the other for other classes of random log-trees such as fringe-balanced binary search trees and quadtrees. The methods of proof combine asymptotic transfer for the underlying recurrence relations with the contraction method. (c) 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 353-379, 2017
DEPENDENCE BETWEEN PATH-LENGTH AND SIZE IN RANDOM DIGITAL TREES
(2017-12-01)
Fuchs, Michael
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Hwang, Hsien-Kuei
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應用數學系
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Department of Applied Mathematics
We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.
FROM COIN TOSSING TO ROCK-PAPER-SCISSORS AND BEYOND: A LOG-EXP GAP THEOREM FOR SELECTING A LEADER
(2017-03-01)
Fuchs, Michael
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Hwang, Hsien-Kuei
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Itoh, Yoshiaki
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應用數學系
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Department of Applied Mathematics
A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.
Phase Changes in Random Point Quadtrees
(2007-05-01)
Chern, Hua-Huai
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Fuchs, Michael
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Hwang, Hsien-Kuei
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應用數學系
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Department of Applied Mathematics
We show that a wide class of linear cost measures (such as the number of leaves) in random d-dimensional point quadtrees undergo a change in limit laws: If the dimension d = 1, ..., 8, then the limit law is normal; if d >= 9 then there is no convergence to a fixed limit law. Stronger approximation results such as convergence rates and local limit theorems are also derived for the number of leaves, additional phase changes being unveiled. Our approach is new and very general, and also applicable to other classes of search trees. A brief discussion of Devroye's grid trees (covering m-ary search trees and quadtrees as special cases) is given. We also propose an efficient numerical procedure for computing the constants involved to high precision.
Profiles of random trees: Limit theorems for random recursive trees and binary search trees
(2006-11-01)
Fuchs, Michael
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Hwang, Hsien-Kuei
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Neininger, Ralph
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應用數學系
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Department of Applied Mathematics
We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio alpha of the level and the logarithm of tree size lies in [0, e). Convergence of all moments is shown to hold only for a E [0, 1] (with only convergence of finite moments when alpha is an element of (1, e)). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for alpha = 0 and a "quicksort type" limit law for alpha = 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on the contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.