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Dive into the research topics where Hsien-Kuei Hwang is active.

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Featured researches published by Hsien-Kuei Hwang.


Journal of Algorithms | 2002

An asymptotic theory for Cauchy---Euler differential equations with applications to the analysis of algorithms

Hua-Huai Chern; Hsien-Kuei Hwang; Tsung-Hsi Tsai

Cauchy-Euler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper a very general framework for Cauchy-Euler equations and propose an asymptotic theory that covers almost all applications where Cauchy-Euler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.


Annals of Probability | 2008

Local limit theorems for finite and infinite urn models.

Hsien-Kuei Hwang; Svante Janson

Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation.


Combinatorics, Probability & Computing | 2002

Quickselect and the Dickman Function

Hsien-Kuei Hwang; Tsung-Hsi Tsai

We show that the limiting distribution of the number of comparisons used by Hoares quickselect algorithm when given a random permutation of n elements for finding the mth-smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived.


Advances in Applied Probability | 2002

Profiles of random trees: correlation and width of random recursive trees and binary search trees

Michael Drmota; Hsien-Kuei Hwang

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.


ACM Transactions on Algorithms | 2007

Phase changes in random point quadtrees

Hua-Huai Chern; Michael Fuchs; Hsien-Kuei Hwang

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 Devroyes 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.


Journal of Combinatorial Theory | 2001

Limit Theorems for the Number of Summands in Integer Partitions

Hsien-Kuei Hwang

Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cramer-type large deviations and are proved by Mellin transform and the two-dimensional saddle-point method. Applications of these results include partitions into positive integers, into powers of integers, into integers js], s>1, into aj+b, etc.


Random Structures and Algorithms | 2005

Maxima in hypercubes

Zhidong Bai; Luc Devroye; Hsien-Kuei Hwang; Tsung-Hsi Tsai

We derive a Berry-Esseen bound, essentially of the order of the square of the standard deviation, for the number of maxima in random samples from (0, 1)d. The bound is, although not optimal, the first of its kind for the number of maxima in dimensions higher than two. The proof uses Poisson processes and Steins method. We also propose a new method for computing the variance and derive an asymptotic expansion. The methods of proof we propose are of some generality and applicable to other regions such as d-dimensional simplex.


SIAM Journal on Computing | 2006

Partial Match Queries in Random k -d Trees

Hua-Huai Chern; Hsien-Kuei Hwang

We solve the open problem of characterizing the leading constant in the asymptotic approximation to the expected cost used for random partial match queries in random k-d trees. Our approach is new and of some generality; in particular, it is applicable to many problems involving differential equations (or difference equations) with polynomial coefficients.


Random Structures and Algorithms | 1998

Normal approximations of the number of records in geometrically distributed random variables

Zhi-Dong Bai; Hsien-Kuei Hwang; Wen-Qi Liang

We establish the asymptotic normality of the number of upper records in a sequence of iid geometric random variables. Large deviations and local limit theorems as well as approximation theorems for the number of lower records are also derived.


Random Structures and Algorithms | 2000

Distribution of the number of consecutive records

Hua-Huai Chern; Hsien-Kuei Hwang; Yeong-Nan Yeh

We study the distribution of the number n;r of r consecutive records in a sequence of n independent and identically distributed random variables from a common continuous distribution, or equivalently, in a random permutation ofn elements. We show that the asymptotic distribution of n;r is Poisson for r = 1; 2 and non-Poisson for r 3. Precise asymptotic results are derived for four probability distances of the associated approximations: Fortet-Mourier, total variation, Kolmogorov, and point metric. In particular, the distributions of n;r have the specic property that the last three distances are asymptotically of the same behaviors for r 2. We also provide interesting combinatorial bijections for n;2 and compute explicitly the limiting law for n;3 in terms of Kummer’s conuent hypergeometric functions.

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Michael Fuchs

National Chiao Tung University

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Hua-Huai Chern

National Center for Science Education

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Wei-Mei Chen

National Taiwan University of Science and Technology

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Wen-Qi Liang

National University of Singapore

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Zhidong Bai

Northeast Normal University

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Chern-Ching Chao

National University of Singapore

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