Tsung-Hsi Tsai

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

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.

Quickselect and the Dickman Function

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.

Maxima in hypercubes

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.

Probabilistic Analysis of the (1 + 1)-Evolutionary Algorithm

We give a detailed analysis of the optimization time of the -Evolutionary Algorithm under two simple fitness functions (OneMax and LeadingOnes). The problem has been approached in the evolutionary algorithm literature in various ways and with different degrees of rigor. Our asymptotic approximations for the mean and the variance represent the strongest of their kind. The approach we develop is based on an asymptotic resolution of the underlying recurrences and can also be extended to characterize the corresponding limiting distributions. While most of our approximations can be derived by simple heuristic calculations based on the idea of matched asymptotics, the rigorous justifications are challenging and require a delicate error analysis.

An asymptotic theory for recurrence relations based on minimization and maximization

We derive asymptotic approximations for the sequence f(n) defined recursively by f(n)=min1?j

Maxima-finding algorithms for multidimensional samples: A two-phase approach

Simple, two-phase algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are 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.

Empirical law of the iterated logarithm for Markov chains with a countable state space

We find conditions which are sufficient and nearly necessary for the compact and bounded law of the iterated logarithm for Markov chains with a countable state space.

Efficient computation of the iteration of functions

Given a function f from {0,1,...,N-1} to {0,1,...,N-1}, we prove that f^m(x), the mth iterate of f at x, can be computed in time O(logN) for each natural number m and each x by using O(N) information that is generated in a preprocessing procedure. Two types of optimal orbit decompositions of functional graphs are proposed for the preprocess. Both preprocesses require only linear time and linear space. Our decompositions minimize the number of recursions in the computation of f^m(x) and solve some open problems in Tsaban (Discrete Applied Mathematics 155 (2007) 386-393).

The CLT for Markov chains with a countable state space embedded in the space lp

We find necessary and sufficient conditions for the CLT for Markov chains with a countable state space embedded in the space lp for p[greater-or-equal, slanted]1. This result is an extension of the uniform CLT over the family of indicator functions in Levental (Stochastic Processes Appl. 34 (1990) 245-253), where the result is equivalent to our case p=1. A similar extension for the uniform CLT over a family of possibly unbounded functions in Tsai (Taiwan. J. Math. 1(4) (1997) 481-498) is also obtained.