arXiv subjects
Publications and source records attributed to Hsien-Kuei Hwang.
A detailed study is made of the number of occupied seats in an unfriendly seating scheme with two rows of seats. An unusual identity is derived for the probability generating function, which is itself an asymptotic expansion. The identity implies particularly a local limit theorem with optimal convergence rate. Our approach relies on the resolution of Riccati equations.
We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein's method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.
We give a detailed analysis of the cost used by the (1+1)-evolutionary algorithm. The problem has been approached in the evolutionary algorithm literature under various views, formulation and degree of rigor. Our asymptotic approximations for the mean and the variance represent the strongest of their kind. The approach we develop is also applicable to characterize the limit laws and is based on asymptotic resolution of the underlying recurrence. While most approximations have their simple formal nature, we elaborate on the delicate error analysis required for rigorous justifications.
A special type of binomial splitting process is studied. Such a process can be used to model a high-dimensional corner parking problem, as well as the depth of random PATRICIA tries (a special class of digital tree data structures). The later also has natural interpretations in terms of distinct values in iid geometric random variables and the occupancy problem in urn models. The corresponding distribution is marked by logarithmic mean and bounded variance, which is oscillating, if the binomial parameter $p$ is not equal to 1/2, and asymptotic to 1 in the unbiased case. Also, the limiting distribution does not exist owing to periodic fluctuations.
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.
We consider sequences of random variables whose probability generating functions are polynomials all of whose roots lie on the unit circle. The distribution of such random variables has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth normalized (by the standard deviation) central moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance goes unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms.
We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual G_{n,p} -model where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n^{c\log n}, which we explore from several different perspectives. Also we collect many instances where such an order appears, from algorithmics to analysis, from probability to algebra. The limiting distribution of the cost required by the algorithm under a purely idealized random model is proved to be normal. The approach we develop is of some generality and is amenable for other graph algorithms.
Skylines emerged as a useful notion in database queries for selecting representative groups in multivariate data samples for further decision making, multi-objective optimization or data processing, and the $k$-dominant skylines were naturally introduced to resolve the abundance of skylines when the dimensionality grows or when the coordinates are negatively correlated. We prove in this paper that the expected number of $k$-dominant skylines is asymptotically zero for large samples when $1\le k\le d-1$ under two reasonable (continuous) probability assumptions of the input points, $d$ being the (finite) dimensionality, in contrast to the asymptotic unboundedness when $k=d$. In addition to such an asymptotic zero-infinity property, we also establish a sharp threshold phenomenon for the expected ($d-1$)-dominant skylines when the dimensionality is allowed to grow with $n$. Several related issues such as the dominant cycle structures and numerical aspects, are also briefly studied.
We consider three types of multivariate records in this paper and derive the mean and the variance of their numbers for independent and uniform random samples from two prototype regions: hypercubes $[0,1]^d$ and $d$-dimensional simplex. Central limit theorems with convergence rates are established when the variance tends to infinity. Effective numerical procedures are also provided for computing the variance constants to high degree of precision.
Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising $n(\log n)^2$-variance for certain notions of total path-length is also clarified.
An unusual and surprising expansion of the form \[ p_n = ρ^{-n-1}(6n +\tfrac{18}5+ \tfrac{336}{3125} n^{-5}+\tfrac{1008}{3125} n^{-6} +\text{smaller order terms}), \] as $n\to\infty$, is derived for the probability $p_n$ that two randomly chosen binary search trees are identical (in shape and in labels of all corresponding nodes). A quantity arising in the analysis of phylogenetic trees is also proved to have a similar asymptotic expansion. Our method of proof is new in the literature of discrete probability and analysis of algorithms, and based on the psi-series expansions for nonlinear differential equations. Such an approach is very general and applicable to many other problems involving nonlinear differential equations; many examples are discussed and several attractive phenomena are discovered.
Random increasing k-trees represent an interesting, useful class of strongly dependent graphs for which analytic-combinatorial tools can be successfully applied. We study in this paper a notion called connectivity-profile and derive asymptotic estimates for it; some interesting consequences will also be given.
New algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are very simple and easily coded and modified for practical needs. The expected complexity of some measures related to the performance of the algorithms is analyzed. We also compare the efficiency of the algorithms with a few major ones used in practice, and apply our algorithms to find the maximal layers and the longest common subsequences of multiple sequences.
By a suitable shifting-the-mean parametrization at the Dirichlet series level and Delange's Tauberian theorems, we show that the number of factors in random ordered factorizations of integers is asymptotically normally distributed.
A new approach to Poisson approximation is proposed. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. A selected survey on diverse Poisson approximation results is also given.
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.
We propose a new, direct, correlation-free approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width and almost sure convergence. It is widely applicable to random trees of logarithmic height, including recursive trees, binary search trees, quad trees, plane-oriented ordered trees and other varieties of increasing trees.