Sources of Superlinearity in Davenport-Schinzel Sequences

A generalized Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed forbidden subsequence. One of the fundamental problems in this area is bounding (asymptotically) the maximum length of such sequences. Following Klazar, let Ex(\sigma,n) be the maximum length of a sequence over an alphabet of size n avoiding subsequences isomorphic to \sigma. It has been proved that for every \sigma, Ex(\sigma,n) is either linear or very close to linear; in particular it is O(n 2^{\alpha(n)^{O(1)}}), where \alpha is the inverse-Ackermann function and O(1) depends on \sigma. However, very little is known about the properties of \sigma that induce superlinearity of \Ex(\sigma,n). In this paper we exhibit an infinite family of independent superlinear forbidden subsequences. To be specific, we show that there are 17 prototypical superlinear forbidden subsequences, some of which can be made arbitrarily long through a simple padding operation. Perhaps the most novel part of our constructions is a new succinct code for representing superlinear forbidden subsequences

Efficient union-find for planar graphs and other sparse graph classes

AbstractWe solve the Union-Find Problem (UF) efficiently for the case the input is restricted to several graph classes, namely partial k-trees for any fixed k, d-dimensional grids for any fixed dimension d and for planar graphs. The result on grids allows us to perform region growing techniques that are used for image segmentation in linear time. For planar graphs we develop a technique of decomposing such a graph into small subgraphs, patching, that might be useful for other algorithmic problems on planar graphs, too.By efficiency we do not only mean linear time in a theoretical setting but also a practical reorganization of memory such that a dynamic data structures for UF is allocated consecutively

Complexity of PQR tree construction

Orientador: João MeidanisDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: As árvores PQR são estruturas de dados usadas para tratar o problema dos uns consecutivos e problemas relacionados. Aplicações incluem reconhecimento de grafos de intervalos, de grafos planares, e problemas envolvendo moléculas de DNA. A presente dissertação busca consolidar o conhecimento sobre árvores PQR e, principalmente, sua construção incremental, visando fornecer uma base teórica para o uso desta estrutura em aplicações. Este trabalho apresenta uma descrição detalhada do projeto do algoritmo para construção online de árvores PQR, partindo de uma implementação inocente das operações sugeridas e refinando sucessivamente o algoritmo até alcançar a complexidade de tempo quase-linear. Neste projeto, lidamos com um obstáculo que surge com a utilização de estruturas de union-find que não havia sido tratado anteriormente. A demonstração da complexidade de tempo do algoritmo apresentada aqui também é nova e mais clara. Além disso, o projeto é acompanhado de uma implementação em Java dos algoritmos descritosAbstract: PQR trees are data structures used to solve the consecutive ones problem and other related problems. Applications include interval or planar graph recognition, and problems involving DNA molecules. This dissertation aims at consolidating existing and new knowledge about PQR trees and, primarily, their online construction, thus providing a theoretical basis for the use of this structure in applications. This work presents a detailed description of the online PQR tree construction algorithm's design, starting with a naive implementation of the suggested operations and refining them successively, culminating with an almost-linear time complexity. In this project, we dealt with an obstacle that arises with the use of union-find structures and that has never been addressed before. The proof presented here for the time complexity is also novel and clearer. Furthermore, the project is accompanied by a Java implementation of all the algorithms describedMestradoCiência da ComputaçãoMestre em Ciência da Computaçã