arXiv subjects

Tsung-Hsi Tsai

Publications and source records attributed to Tsung-Hsi Tsai.

7 recordsLinked to original sources

arXiv2024

Periodic minimum in the count of binomial coefficients not divisible by a prime

The summatory function of the number of binomial coefficients not divisible by a prime is known to exhibit regular periodic oscillations, yet identifying the less regularly behaved minimum of the underlying periodic functions has been open for almost all cases. We propose an approach to identify such minimum in some generality, solving particularly a previous conjecture of B. Wilson [Asymptotic behavior of Pascal's triangle modulo a prime, Acta Arith. 83 (1998), pp. 105-116].

math.NT↗

arXiv2022

Identities and periodic oscillations of divide-and-conquer recurrences splitting at half

We study divide-and-conquer recurrences of the form \begin{equation*} f(n) = αf(\lfloor \tfrac n2\rfloor) + βf(\lceil \tfrac n2\rceil) + g(n) \qquad(n\ge2), \end{equation*} with $g(n)$ and $f(1)$ given, where $α,β\ge0$ with $α+β>0$; such recurrences appear often in analysis of computer algorithms, numeration systems, combinatorial sequences, and related areas. We show that the solution satisfies always the simple \emph{identity} \begin{equation*} f(n) = n^{\log_2(α+β)} P(\log_2n) - Q(n) \end{equation*} under an optimum (iff) condition on $g(n)$. This form is not only an identity but also an asymptotic expansion because $Q(n)$ is of a smaller order. Explicit forms for the \emph{continuity} of the periodic function $P$ are provided, together with a few other smoothness properties. We show how our results can be easily applied to many dozens of concrete examples collected from the literature, and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary, but leads to the strongest types of results for all examples to which our theory applies.

cs.DS↗

arXiv2015

Random unfriendly seating arrangement in a dining table

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.

math.PR↗

arXiv2014

Probabilistic analysis of the (1+1)-evolutionary algorithm

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.

math.PR↗

arXiv2011

Threshold phenomena in k-dominant skylines of random samples

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.

cs.DS↗

arXiv2010

Multivariate records based on dominance

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.

math.ST↗

arXiv2009

Simple, efficient maxima-finding algorithms for multidimensional samples

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.

cs.DS↗