Average-Case Complexity of Shellsort

We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a $p$-pass Shellsort for any incremental sequence is \Omega (pn^{1 + 1/p) for all $p \leq \log n$. Using similar arguments, we analyze the average-case complexity of several other sorting algorithms.Comment: 11 pages. Submitted to ICALP'9

The Average-Case Area of Heilbronn-Type Triangles

From among ${n \choose 3}$ triangles with vertices chosen from $n$ points in the unit square, let $T$ be the one with the smallest area, and let $A$ be the area of $T$. Heilbronn's triangle problem asks for the maximum value assumed by $A$ over all choices of $n$ points. We consider the average-case: If the $n$ points are chosen independently and at random (with a uniform distribution), then there exist positive constants $c$ and $C$ such that $c/n^3 < \mu_n < C/n^3$ for all large enough values of $n$, where $\mu_n$ is the expectation of $A$. Moreover, $c/n^3 < A < C/n^3$, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in ``general position.''Comment: 13 pages, LaTeX, 1 figure,Popular treatment in D. Mackenzie, On a roll, {\em New Scientist}, November 6, 1999, 44--4

Spin-the-bottle Sort and Annealing Sort: Oblivious Sorting via Round-robin Random Comparisons

We study sorting algorithms based on randomized round-robin comparisons. Specifically, we study Spin-the-bottle sort, where comparisons are unrestricted, and Annealing sort, where comparisons are restricted to a distance bounded by a \emph{temperature} parameter. Both algorithms are simple, randomized, data-oblivious sorting algorithms, which are useful in privacy-preserving computations, but, as we show, Annealing sort is much more efficient. We show that there is an input permutation that causes Spin-the-bottle sort to require $\Omega(n^2\log n)$ expected time in order to succeed, and that in $O(n^2\log n)$ time this algorithm succeeds with high probability for any input. We also show there is an implementation of Annealing sort that runs in $O(n\log n)$ time and succeeds with very high probability.Comment: Full version of a paper appearing in ANALCO 2011, in conjunction with SODA 201

Teorias da Aleatoriedade

Este trabalho apresenta uma revisão bibliográfica sobre a definição de “seqüência aleatória”. Nós enfatizamos a definição de Martin-Löf e a definição baseada em incompressividade (complexidade de Kolmogorov). Complexidade de Kolmogorov é uma teoria sofisticada e profunda da informação e da aleatoriedade baseada na máquina de Turing. Estas duas definições resolvem todos os problemas das outras abordagens e satisfazem o nosso conceito intuitivo de aleatoriedade, sendo matematicamente corretas. Adicionalmente, apresentamos a abordagem de Schnorr que inclui um requisito de efetividade (computabilidade) em sua definição. São apresentadas as relações entre estas diversas definições de forma crítica. Palavras-chave: aleatoriedade, complexidade de Kolmogorov, máquina de Turing, computabilidade, probabilidade.This work is a survey about the definition of “random sequence”. We emphasize the definition of Martin-Löf and the definition based on incompressibility (Kolmogorov complexity). Kolmogorov complexity is a profound and sofisticated theory of information and randomness based on Turing machines. These two definitions solve all the problems of the other approaches, satisfying our intuitive concept of randomness, and both are mathematically correct. Furthermore, we show the Schnorr’s approach, that includes a requisite of effectiveness (computability) in his definition. We show the relations between all definitions in a critical way. Keywords: randomness, Kolmogorov complexity, Turing machine, computability, probability

The Frobenius Problem in a Free Monoid

Given positive integers c1,c2,...,ck with gcd(c1,c2,...,ck) = 1, the Frobenius problem (FP) is to compute the largest integer g(c1,c2,...,ck) that cannot be written as a non-negative integer linear combination of c1,c2,...,ck. The Frobenius problem in a free monoid (FPFM) is a non-commutative generalization of the Frobenius problem. Given words x1,x2,...,xk such that there are only finitely many words that cannot be written as concatenations of words in {x1,x2,...,xk}, the FPFM is to find the longest such words. Unlike the FP, where the upper bound g(c1,c2,...,ck)≤max 1≤i≤k ci2 is quadratic, the upper bound on the length of the longest words in the FPFM can be exponential in certain measures and some of the exponential upper bounds are tight. For the 2FPFM, where the given words over Σ are of only two distinct lengths m and n with 1<m<n, the length of the longest omitted words is ≤g(m, m|Σ|n-m + n - m). In Chapter 1, I give the definition of the FP in integers and summarize some of the interesting properties of the FP. In Chapter 2, I give the definition of the FPFM and discuss some general properties of the FPFM. Then I mainly focus on the 2FPFM. I discuss the 2FPFM from different points of view and present two equivalent problems, one of which is about combinatorics on words and the other is about the word graph. In Chapter 3, I discuss some variations on the FPFM and related problems, including input in other forms, bases with constant size, the case of infinite words, the case of concatenation with overlap, and the generalization of the local postage-stamp problem in a free monoid. In Chapter 4, I present the construction of some essential examples to complement the theory of the 2FPFM discussed in Chapter 2. The theory and examples of the 2FPFM are the main contribution of the thesis. In Chapter 5, I discuss the algorithms for and computational complexity of the FPFM and related problems. In the last chapter, I summarize the main results and list some open problems. Part of my work in the thesis has appeared in the papers