This thesis studies data structures in the word-RAM and bitprobe models, with an emphasis on space efficiency. In the word-RAM model of computation the space cost of a data structure is measured in terms of the number of w-bit words stored in memory, and the cost of answering a query is measured in terms of the number of read, write, and arithmetic operations that must be performed. In the bitprobe model, like the word-RAM model, the space cost is measured in terms of the number of bits stored in memory, but the query cost is measured solely in terms of the number of bit accesses, or probes, that are performed. First, we examine the problem of succinctly representing a partially ordered set, or poset, in the word-RAM model with word size Theta(lg n) bits. A succinct representation of a combinatorial object is one that occupies space matching the information theoretic lower bound to within lower order terms. We show how to represent a poset on n vertices using a data structure that occupies n^2/4 + o(n^2) bits, and can answer precedence (i.e., less-than) queries in constant time. Since the transitive closure of a directed acyclic graph is a poset, this implies that we can support reachability queries on an arbitrary directed graph in the same space bound. As far as we are aware, this is the first representation of an arbitrary directed graph that supports reachability queries in constant time, and stores less than n choose 2 bits. We also consider several additional query operations. Second, we examine the problem of supporting range queries on strings of n characters (or, equivalently, arrays of n elements) in the word-RAM model with word size Theta(lg n) bits. We focus on the specific problem of answering range majority queries: i.e., given a range, report the character that is the majority among those in the range, if one exists. We show that these queries can be supported in constant time using a linear space (in words) data structure. We generalize this result in several directions, considering various frequency thresholds, geometric variants of the problem, and dynamism. These results are in stark contrast to recent work on the similar range mode problem, in which the query operation asks for the mode (i.e., most frequent) character in a given range. The current best data structures for the range mode problem take soft-Oh(n^(1/2)) time per query for linear space data structures. Third, we examine the deterministic membership (or dictionary) problem in the bitprobe model. This problem asks us to store a set of n elements drawn from a universe [1,u] such that membership queries can be always answered in t bit probes. We present several new fully explicit results for this problem, in particular for the case when n = 2, answering an open problem posed by Radhakrishnan, Shah, and Shannigrahi [ESA 2010]. We also present a general strategy for the membership problem that can be used to solve many related fundamental problems, such as rank, counting, and emptiness queries. Finally, we conclude with a list of open problems and avenues for future work.

Ranking queries are widely used in data exploration, data analysis and decision making scenarios. While most of the currently proposed ranking techniques focus on deterministic data, several emerging applications involve data that are imprecise or uncertain. Ranking uncertain data raises new challenges in query semantics and processing, making conventional methods inapplicable. Furthermore, the interplay between ranking and uncertainty models introduces new dimensions for ordering query results that do not exist in the traditional settings. This dissertation introduces new formulations and processing techniques for ranking queries on uncertain data. The formulations are based on marriage of traditional ranking semantics with possible worlds semantics under widely-adopted uncertainty models. In particular, we focus on studying the impact of tuple-level and attribute-level uncertainty on the semantics and processing techniques of ranking queries. Under the tuple-level uncertainty model, we introduce a processing framework leveraging the capabilities of relational database systems to recognize and handle data uncertainty in score-based ranking. The framework encapsulates a state space model, and efficient search algorithms that compute query answers by lazily materializing the necessary parts of the space. Under the attribute-level uncertainty model, we give a new probabilistic ranking model, based on partial orders, to encapsulate the space of possible rankings originating from uncertainty in attribute values. We present a set of efficient query evaluation algorithms, including sampling-based techniques based on the theory of Markov chains and Monte-Carlo method, to compute query answers. We build on our techniques for ranking under attribute-level uncertainty to support rank join queries on uncertain data. We show how to extend current rank join methods to handle uncertainty in scoring attributes. We provide a pipelined query operator implementation of uncertainty-aware rank join algorithm integrated with sampling techniques to compute query answers.

Escalonamento de processos é um tipo de problema de otimização combinatória no qual devemos alocar máquinas à tarefas por períodos específicos de tempo. A literatura contém diversos estudos propondo técnicas para resolver modelos de escalonamento de processos como o job shop e o open shop. Esses modelos permitem que os passos no processo produtivo sejam ou completamente ordenados ou sem ordenação alguma. Com o aumento da complexidade das aplicações industriais no encontramos, mais recentemente, diversos trabalhos que propõe problemas de escalonamento de processos mais gerais para modelar mais precisamente os processos produtivos. O mixed shop, group shop e partial shop são exemplos de tais modelos. Nesse trabalho nós propomos uma busca tabu iterada para o partial shop, que é um modelo geral que inclui diversos modelos mais restritivos. Os componentes novos mais importantes da técnica são o gerador de solução inicial, a vizinhança e o limite inferior para a vizinhança. Em experimentos computacionais nós conseguimos demonstrar que a heurística genérica e única é capaz de competir, e as vezes superar, as técnicas de estado de arte desenvolvidas especificamente para partial, open, mixed e group shop. Algumas vezes uma máquina é o gargalo de um processo produtivo, e é replicada. Na literatura o caso das máquinas paralelas foi incluído em diversas extensões de problemas de escalonamento de processos. Nessa tese nós também propomos uma técnica para escalonar as máquinas paralelas, sem incluí-las explicitamente na representação do problema. Nós usamos técnicas gerais para os casos sem máquinas paralelas para produzir uma busca heurística tabu rápida, e estado da arte, para o caso do job shop com máquinas paralelas. / Shop scheduling is a combinatorial optimization type of problem in which we must allocate machines to jobs for specific periods time. A set of constraints defines which schedules are valid, and we must select one that minimizes or maximizes an objective function. In this work we use the makespan, which is the time the last job finishes. The literature contains several studies proposing techniques to solve shop problems such as the job shop and open shop. These problems allow the steps of the production processes to be either fully ordered or not ordered at all. With increasing complexity and size of industrial applications we find, more recently, several works which propose more general shop problems to model the production processes more accurately. The mixed shop, group shop and partial shop are examples of such problems In this work we propose an iterated tabu search for the partial shop, which is a general problem and includes several other more restrictive shop problems. The most important novel components of the solver are the initial solution generator, the neighbourhood, and the lower bound for the neighbourhood. In computational experiments we were able to show that the general partial shop solver is able to compete with, and sometimes surpass, the state-of-the-art solvers developed specifically for the partial, open, mixed and group shops. Sometimes a machine is a bottleneck in the production process, and is replicated. In the literature the parallel machines case has being included in several extensions of shop problems. In this thesis we also propose a technique to schedule the parallel machines heuristically, without including them explicitly in the representation of the problem. We use general techniques for the non-parallel machine cases to produce a fast tabu search heuristic results for the job shop with parallel machines.

abstract: The uncrossing partially ordered set $P_n$ is defined on the set of matchings on $2n$ points on a circle represented with wires. The order relation is $\tau'\leq \tau$ in $P_n$ if and only if $\tau'$ is obtained by resolving a crossing of $\tau$. %This partial order has been studied by Alman-Lian-Tran, Huang-Wen-Xie, Kenyon, and Lam. %The posets $P_n$ emerged from studies of circular planar electrical networks. Circular planar electrical networks are finite weighted undirected graphs embedded into a disk, with boundary vertices and interior vertices. By Curtis-Ingerman-Morrow and de Verdi\`ere-Gitler-Vertigan, the electrical networks can be encoded with response matrices. By Lam the space of response matrices for electrical networks has a cell structure, and this cell structure can be described by the uncrossing partial orders. %Lam proves that the posets can be identified with dual Bruhat order on affine permutations of type $(n,2n)$. Using this identification, Lam proves the poset $\hat{P}_n$, the uncrossing poset $P_n$ with a unique minimum element $\hat{0}$ adjoined, is Eulerian. This thesis consists of two sets of results: (1) flag enumeration in intervals in the uncrossing poset $P_n$ and (2) cyclic sieving phenomenon on the set $P_n$. I identify elements in $P_n$ with affine permutations of type $(0,2n)$. %This identification enables us to explicitly describe the elements in $P_n$ with the elements in $\mathcal{MP}_n$. Using this identification, I adapt a technique in Reading for finding recursions for the cd-indices of intervals in Bruhat order of Coxeter groups to the uncrossing poset $P_n$. As a result, I produce recursions for the cd-indices of intervals in the uncrossing poset $P_n$. I also obtain a recursion for the ab-indices of intervals in the poset $\hat{P}_n$, the poset $P_n$ with a unique minimum $\hat0$ adjoined. %We define an induced subposet $\mathcal{MP}_n$ of the affine permutations under Bruhat order. Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings with the $q$-Catalan polynomial. Bowling-Liang presented similar results on the set of $k$-crossing matchings for $1\leq k \leq 3$. In this dissertation, I focus on the set of all matchings on $[2n]:=\{1,2,\dots,2n\}$. I find the number of matchings fixed by $\frac{2\pi}{d}$ rotations for $d|2n$. I then find the polynomial $X_n(q)$ such that the set of matchings together with $X_n(q)$ and the cyclic group of order $2n$ exhibits the CSP. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2018

Escalonamento de processos é um tipo de problema de otimização combinatória no qual devemos alocar máquinas à tarefas por períodos específicos de tempo. A literatura contém diversos estudos propondo técnicas para resolver modelos de escalonamento de processos como o job shop e o open shop. Esses modelos permitem que os passos no processo produtivo sejam ou completamente ordenados ou sem ordenação alguma. Com o aumento da complexidade das aplicações industriais no encontramos, mais recentemente, diversos trabalhos que propõe problemas de escalonamento de processos mais gerais para modelar mais precisamente os processos produtivos. O mixed shop, group shop e partial shop são exemplos de tais modelos. Nesse trabalho nós propomos uma busca tabu iterada para o partial shop, que é um modelo geral que inclui diversos modelos mais restritivos. Os componentes novos mais importantes da técnica são o gerador de solução inicial, a vizinhança e o limite inferior para a vizinhança. Em experimentos computacionais nós conseguimos demonstrar que a heurística genérica e única é capaz de competir, e as vezes superar, as técnicas de estado de arte desenvolvidas especificamente para partial, open, mixed e group shop. Algumas vezes uma máquina é o gargalo de um processo produtivo, e é replicada. Na literatura o caso das máquinas paralelas foi incluído em diversas extensões de problemas de escalonamento de processos. Nessa tese nós também propomos uma técnica para escalonar as máquinas paralelas, sem incluí-las explicitamente na representação do problema. Nós usamos técnicas gerais para os casos sem máquinas paralelas para produzir uma busca heurística tabu rápida, e estado da arte, para o caso do job shop com máquinas paralelas. / Shop scheduling is a combinatorial optimization type of problem in which we must allocate machines to jobs for specific periods time. A set of constraints defines which schedules are valid, and we must select one that minimizes or maximizes an objective function. In this work we use the makespan, which is the time the last job finishes. The literature contains several studies proposing techniques to solve shop problems such as the job shop and open shop. These problems allow the steps of the production processes to be either fully ordered or not ordered at all. With increasing complexity and size of industrial applications we find, more recently, several works which propose more general shop problems to model the production processes more accurately. The mixed shop, group shop and partial shop are examples of such problems In this work we propose an iterated tabu search for the partial shop, which is a general problem and includes several other more restrictive shop problems. The most important novel components of the solver are the initial solution generator, the neighbourhood, and the lower bound for the neighbourhood. In computational experiments we were able to show that the general partial shop solver is able to compete with, and sometimes surpass, the state-of-the-art solvers developed specifically for the partial, open, mixed and group shops. Sometimes a machine is a bottleneck in the production process, and is replicated. In the literature the parallel machines case has being included in several extensions of shop problems. In this thesis we also propose a technique to schedule the parallel machines heuristically, without including them explicitly in the representation of the problem. We use general techniques for the non-parallel machine cases to produce a fast tabu search heuristic results for the job shop with parallel machines.

Doctor of Philosophy / Department of Computing and Information Science / John M. Hatcliff / With the advent of multi-core processors and rich language support for concurrency, the paradigm of concurrent programming has arrived; however, the cost of developing and maintaining concurrent programs is still high. Simultaneously, the increase in social ubiquity of computing is reducing the "time-to-market" factor while demanding stronger correctness requirements. These effects are amplified with ever-growing size of software systems. Consequently, there is (will be) a rise in the demand for scalable and accurate techniques to enable faster development and maintenance of correct large scale concurrent software. This dissertation presents a collection of scalable and accurate approaches to tackle the above situation. Primarily, the approaches are focused on discovering dependences (relations) between various parts of the software/program and leveraging the dependences to improve maintenance and development tasks via program slicing (comprehension) and verification. Briefly, the proposed approaches are embodied in the following specific contributions: 1. New trace-based foundation for control dependences. 2. An equivalence class based analysis to efficiently and accurately calculate escape information and intra- and inter-thread dependences. 3. A new parametric data flow style slicing algorithm with various extensions to uniformly and easily realize and reason about most existing forms of static sequential and concurrent slicing. 4. A new generic notion of property/trace sensitivity to represent and reason about richer forms of context sensitivity. 5. Program dependence based partial order reduction techniques to enable efficient and accurate state space exploration in both static and dynamic mode. In an attempt to simplify the approaches, they have been based on the basic concepts/ideas of the affected techniques (e.g. program slicing is a rooted transitive closure of dependence relation). As trace-based reasoning is well suited for concurrent systems, an attempt has been made to explore trace-based reasoning wherever possible. While providing a rigorous theoretical presentation of these techniques, this effort also validates the techniques by implementing them in a robust tool framework called Indus (available from http://indus.projects.cis.ksu.edu) and then providing experimental results that demonstrate the effectiveness of the techniques on various concurrent applications. Given the current trend towards concurrent programming and social ubiquity of computing, the approaches proposed in this dissertation provide a foundation for collectively attacking scalability, accuracy, and soundness challenges in current and emerging systems.

Les applications de transmission de messages distribués font partie du courant dominant des technologies de l'information car elles exploitent la puissance des systèmes informatiques parallèles pour produire des performances plus élevées. La conception de programmes distribués reste difficile car les développeurs doivent raisonner sur la concurrence, le non-déterminisme, la distribution de données… qui sont les principales caractéristiques des programmes distribués. En outre, il est pratiquement impossible de garantir l'exactitude de tels programmes via des approches de test classiques, car il est possible que l'on n'atteigne jamais avec succès l'exécution qui conduit à des comportements indésirables dans les programmes. Il existe donc un besoin de techniques de vérification plus puissantes. La vérification des modèles est l'une des méthodes formelles qui permet de vérifier automatiquement et efficacement certaines propriétés des modèles de systèmes informatiques en explorant tous les comportements possibles (états et transitions) du modèle de système. Cependant, les espaces d'état augmentent de façon exponentielle avec le nombre de processus simultanés, conduisant à une «explosion de l'espace d'état» .La réduction dynamique de l'ordre partiel basée sur le dépliage (UDPOR) est une technique récente mélangeant la réduction dynamique de l'ordre partiel (DPOR) avec des concepts de théorie de la concurrence tels que dépliages pour atténuer efficacement l'explosion de l'espace d'états lors de la vérification des modèles de programmes simultanés. Il est optimal dans le sens où chaque trace de Mazurkiewicz, c'est-à-dire une classe d'entrelacements équivalents en commutant des actions indépendantes adjacentes, est explorée exactement une fois. Et elle s'applique aux programmes en cours d'exécution, pas seulement aux modèles de programmes.La thèse vise à adapter UDPOR pour vérifier les programmes distribués asynchrones (par exemple les programmes MPI) dans le cadre du simulateur SIMGRID d'applications distribuées. Pour ce faire, un modèle de programmation abstrait de programmes distribués asynchrones est défini et formalisé en langage TLA +, permettant de définir avec précision une relation d'indépendance, ingrédient principal de la sémantique concurrentielle. Ensuite, l'adaptation de l'UDPOR, impliquant la construction d'un dépliage, est rendue efficace par une analyse précise des dépendances dans le modèle de programmation, permettant des calculs efficaces d'opérations habituellement coûteuses. Un prototype d'implémentation d'UDPOR adapté aux programmes asynchrones distribués a été développé, donnant des résultats expérimentaux prometteurs sur un ensemble significatif de références. / Distributed message passing applications are in the mainstream of information technology since they exploit the power of parallel computer systems to produce higher performance. Designing distributed programs remains challenging because developers have to reason about concurrency, non-determinism, data distribution… that are main characteristics of distributed programs. Besides, it is virtually impossible to ensure the correctness of such programs via classical testing approaches since one may never successfully reach the execution that leads to unwanted behaviors in the programs. There is thus a need for more powerful verification techniques. Model-checking is one of the formal methods that allows to verify automatically and effectively some properties on models of computer systems by exploring all possible behaviors (states and transitions) of the system model. However, state spaces increase exponentially with the number of concurrent processes, leading to “state space explosion”.Unfolding-based Dynamic Partial Order Reduction (UDPOR) is a recent technique mixing Dynamic Partial Order Reduction (DPOR) with concepts of concurrency theory such as unfoldings to efficiently mitigate state space explosion in model-checking of concurrent programs. It is optimal in the sense that each Mazurkiewicz trace, i.e. a class of interleavings equivalent by commuting adjacent independent actions, is explored exactly once. And it is applicable to running programs, not only models of programs.The thesis aims at adapting UDPOR to verify asynchronous distributed programs (e.g. MPI programs) in the setting of the SIMGRID simulator of distributed applications. To do so, an abstract programming model of asynchronous distributed programs is defined and formalized in the TLA+ language, allowing to precisely define an independence relation, a main ingredient of the concurrency semantics. Then, the adaptation of UDPOR, involving the construction of an unfolding, is made efficient by a precise analysis of dependencies in the programming model, allowing efficient computations of usually costly operation. A prototype implementation of UDPOR adapted to distributed asynchronous programs has been developed, giving promising experimental results on a significant set of benchmarks.

The systematic assessment of rate variations across large datasets requires a systematic approach for summarizing results from individual tests. Often, this is performed by coarse-graining the phylogeny to consider rate variations at the level of sub-claded. In a phylo-geographic setting, however, one is often more interested in other partitions of the data, and in an exploratory mode a pre-specified subdivision of the data is often undesirable. We propose here to arrange rate variation data as the partially ordered set defined by the significant test results.

<p>Cilj ove teze je da se ispitaju&nbsp; lanci u parcijalnim uredjenjima (P(X), &sub;),&nbsp;pri čemu je P(X) skup domena izomorfnih podstruktura relacijske strukture&nbsp;X. Po&scaron;to se svaki lanac u parcijalnom uredjenju može produžiti do maksimalnog lanca, dovoljno je ispitati maksimalne lance u P(X). Dokazano je da, ako je X ultrahomogena relacijska struktura koja ima netrivijalne izomorfne&nbsp;podstrukture, onda je svaki maksimalan lanac u (P(X) &cup; {&empty;}&nbsp; , &sub;) kompletno&nbsp;linearno uredjenje koje se utapa u R i ima neizolovan minimum. Ako &nbsp;je X&nbsp;relacijska struktura, dat je dovoljan uslov da za svako kompletno linearno uredjenje L koje se utapa&nbsp; u R i ima neizolovan minimum, postoji maksimalan lanac u (P(X) &cup; {&empty;}&nbsp; , &sub;) izomorfan L.&nbsp; Dokazano je i da ako je&nbsp;X neka od sledećih relacijskih struktura: Rado graf, Hensonov graf, random poset, ultrahomogeni&nbsp; poset Bn&nbsp; ili&nbsp; ultrahomogeni&nbsp; poset Cn; onda je&nbsp;L izomorfno maksimalnom lancu u (P(X) &cup; {&empty;}&nbsp; , &sub;) ako i samo ako je &nbsp;L&nbsp;kompletno,&nbsp; utapa se u R i ima neizolovan minimum. Ako je X prebrojiv&nbsp;antilanac ili disjunktna unija &micro; kompletnih&nbsp; grafova sa &nu; tačaka za &micro;&nu; = &omega;, onda je L izomorfno maksimalnom lancu u (P(X) &cup; {&empty;}&nbsp; , &sub;) ako i samo ako&nbsp;je bulovsko,&nbsp; utapa se u R i ima neizolovan minimum.</p> / <p>The purpose of this thesis is to investigate chains in partial orders (P(X), &sub;), where P(X) is the set of domains of isomorphic substructures of a relational structure X. Since each chain in a partial&nbsp; order can be extended to a maximal one, it is enough to describe maximal chains in P(X). It is proved that, if X is an ultrahomogeneous relational structure with non-trivial isomorphic substructures, then each maximal&nbsp; chain in (P(X)&cup; {&empty;}&nbsp; , &sub;) is a complete, R-embeddable linear order with minimum&nbsp; non-isolated. If X is a relational structure, a condition is given for X, which is sufficient&nbsp; for (P(X) &cup; {&empty;}&nbsp; , &sub;) to embed each complete,&nbsp; R-embeddable&nbsp; linear order with minimum non-isolated as a maximal&nbsp; chain.&nbsp; It is also proved that if X is one of the follow- ing relational structures: Rado graph, Henson graph, random poset, ultrahomogeneous poset Bn or ultrahomogeneous poset Cn; then L is isomorphic to a maximal&nbsp; chain in (P(X) &cup; {&empty;}&nbsp; , &sub;) if and only if L is complete, R-embeddable with minimum non-isolated. If X is a countable&nbsp; antichain&nbsp; or disjoint union of &micro; complete graphs with &nu; points where &micro;&nu; = &omega;, then L is isomorphic to a maximal&nbsp; chain&nbsp; in (P(X) &cup; {&empty;}&nbsp; , &sub;) if and only if L is Boolean, R-embeddable with minimum non-isolated.</p>

We propose a fully automated and dynamic method for generating likely invariants from multithreaded programs and then leveraging these invariants to infer atomic regions and diagnose concurrency errors in the software code. Although existing methods for dynamic invariant generation perform reasonably well on sequential programs, for multithreaded programs, their effectiveness often reduces dramatically in terms of both the number of invariants that they can generate and the likelihood of them being true invariants. We solve this problem by developing a new dynamic invariant generator, which consists of a new LLVM based code instrumentation tool, an INSPECT based thread interleaving explorer, and a customized inference engine inside Daikon. We have evaluated the resulting system on public domain multithreaded C/C++ benchmarks. Our experiments show that the new method is effective in generating high-quality invariants. Furthermore, the state and transition invariants generated by our new method have been proved useful both in error diagnosis and in identifying likely atomic regions in the concurrent software code. / Master of Science