In this thesis, we study succinct representations of trees and graphs. A succinct representation of a combinatorial object is a space efficient representation that supports a reasonable set of operations and queries on the object in constant or near constant time on the RAM with logarithmic word size. The storage requirement of a succinct representation is intended to be optimal to within lower order terms. We first propose a uniform approach for succinct representation of various families of trees. The method is based on two recursive decompositions of trees into subtrees. The approach simplifies the existing representation of ordinal trees while allowing the full set of navigational operations and queries. The approach applied to cardinal (i.e., k-ary) trees yields a space-optimal succinct representation allowing cardinal-type operations (e.g., determining child labeled i) as well as the full set of ordinal-type operations (e.g., reporting the number of siblings to the left of a node). Previous space-optimal succinct representations had not been able to support both types of operations efficiently. We demonstrate how the approach can be applied to obtain a space-optimal succinct representation for the family of free trees where the order of children is insignificant. Furthermore, we show that our approach can be used to obtain entropy-based succinct representations. The approach adapts to match the degree-distribution entropy suggested by Jansson et al. We discuss that our approach can be made adaptive to various other entropy measures. Next, we focus on ordinal trees, and present a novel universal succinct representation. Our new representation is able to simultaneously emulate previous ordinal tree representations of the balanced parenthesis (BP), depth first unary degree sequence (DFUDS) and partitioned representations using a single instance of the data structure. They not only support the union of all the ordinal tree operations supported by these representations, but will also automatically inherit any new operations supported by these representations in the future; hence the universality title we attributed to the representation. We then move to more general graphs rather than trees, and consider the problem of encoding a graph with $n$ vertices and m edges compactly supporting adjacency, neighborhood and degree queries in constant time. The adjacency query asks whether there is an edge between two vertices, the neighborhood query reports the neighbors of a given vertex in constant time per neighbor, and the degree query reports the number of edges incident to a given vertex. The representation is to achieve the optimal space requirement as a function of n and m to within lower order terms. We prove a lower bound in the cell probe model that it is impossible to achieve the information theoretic lower bound to within lower order terms unless the graph is too sparse (namely, $m=o(n^\delta)$ for any constant \delta > 0) or too dense (namely m = \littleOmega{n^{2-\delta}}) for any constant \delta > 0). We also present a succinct encoding for graphs for all values of n,m supporting queries in constant time. The space requirement of the representation is always within a multiplicative 1+\epsilon factor of the information-theory lower bound for any constant $\epsilon > 0$. This is the best achievable space bound according to our lower bound where it applies. The space requirement of the representation achieves the information-theory lower bound tightly to within lower order terms when the graph is sparse (m=o(n^\delta) for any constant \delta > 0), or very dense (m = \littleOmega (n^2/(\sqrt{\log n})).

In this thesis, we study succinct representations of trees and graphs. A succinct representation of a combinatorial object is a space efficient representation that supports a reasonable set of operations and queries on the object in constant or near constant time on the RAM with logarithmic word size. The storage requirement of a succinct representation is intended to be optimal to within lower order terms. We first propose a uniform approach for succinct representation of various families of trees. The method is based on two recursive decompositions of trees into subtrees. The approach simplifies the existing representation of ordinal trees while allowing the full set of navigational operations and queries. The approach applied to cardinal (i.e., k-ary) trees yields a space-optimal succinct representation allowing cardinal-type operations (e.g., determining child labeled i) as well as the full set of ordinal-type operations (e.g., reporting the number of siblings to the left of a node). Previous space-optimal succinct representations had not been able to support both types of operations efficiently. We demonstrate how the approach can be applied to obtain a space-optimal succinct representation for the family of free trees where the order of children is insignificant. Furthermore, we show that our approach can be used to obtain entropy-based succinct representations. The approach adapts to match the degree-distribution entropy suggested by Jansson et al. We discuss that our approach can be made adaptive to various other entropy measures. Next, we focus on ordinal trees, and present a novel universal succinct representation. Our new representation is able to simultaneously emulate previous ordinal tree representations of the balanced parenthesis (BP), depth first unary degree sequence (DFUDS) and partitioned representations using a single instance of the data structure. They not only support the union of all the ordinal tree operations supported by these representations, but will also automatically inherit any new operations supported by these representations in the future; hence the universality title we attributed to the representation. We then move to more general graphs rather than trees, and consider the problem of encoding a graph with $n$ vertices and m edges compactly supporting adjacency, neighborhood and degree queries in constant time. The adjacency query asks whether there is an edge between two vertices, the neighborhood query reports the neighbors of a given vertex in constant time per neighbor, and the degree query reports the number of edges incident to a given vertex. The representation is to achieve the optimal space requirement as a function of n and m to within lower order terms. We prove a lower bound in the cell probe model that it is impossible to achieve the information theoretic lower bound to within lower order terms unless the graph is too sparse (namely, $m=o(n^\delta)$ for any constant \delta > 0) or too dense (namely m = \littleOmega{n^{2-\delta}}) for any constant \delta > 0). We also present a succinct encoding for graphs for all values of n,m supporting queries in constant time. The space requirement of the representation is always within a multiplicative 1+\epsilon factor of the information-theory lower bound for any constant $\epsilon > 0$. This is the best achievable space bound according to our lower bound where it applies. The space requirement of the representation achieves the information-theory lower bound tightly to within lower order terms when the graph is sparse (m=o(n^\delta) for any constant \delta > 0), or very dense (m = \littleOmega (n^2/(\sqrt{\log n})).

This thesis defines and designs succinct indexes for several abstract data types (ADTs). The concept is to design auxiliary data structures that ideally occupy asymptotically less space than the information-theoretic lower bound on the space required to encode the given data, and support an extended set of operations using the basic operators defined in the ADT. As opposed to succinct (integrated data/index) encodings, the main advantage of succinct indexes is that we make assumptions only on the ADT through which the main data is accessed, rather than the way in which the data is encoded. This allows more freedom in the encoding of the main data. In this thesis, we present succinct indexes for various data types, namely strings, binary relations, multi-labeled trees and multi-labeled graphs, as well as succinct text indexes. For strings, binary relations and multi-labeled trees, when the operators in the ADTs are supported in constant time, our results are comparable to previous results, while allowing more flexibility in the encoding of the given data. Using our techniques, we improve several previous results. We design succinct representations for strings and binary relations that are more compact than previous results, while supporting access/rank/select operations efficiently. Our high-order entropy compressed text index provides more efficient support for searches than previous results that occupy essentially the same amount of space. Our succinct representation for labeled trees supports more operations than previous results do. We also design the first succinct representations of labeled graphs. To design succinct indexes, we also have some preliminary results on succinct data structure design. We present a theorem that characterizes a permutation as a suffix array, based on which we design succinct text indexes. We design a succinct representation of ordinal trees that supports all the navigational operations supported by various succinct tree representations. In addition, this representation also supports two other encodings schemes of ordinal trees as abstract data types. Finally, we design succinct representations of planar triangulations and planar graphs which support the rank/select of edges in counter clockwise order in addition to other operations supported in previous work, and a succinct representation of k-page graph which supports more efficient navigation than previous results for large values of k.

This thesis defines and designs succinct indexes for several abstract data types (ADTs). The concept is to design auxiliary data structures that ideally occupy asymptotically less space than the information-theoretic lower bound on the space required to encode the given data, and support an extended set of operations using the basic operators defined in the ADT. As opposed to succinct (integrated data/index) encodings, the main advantage of succinct indexes is that we make assumptions only on the ADT through which the main data is accessed, rather than the way in which the data is encoded. This allows more freedom in the encoding of the main data. In this thesis, we present succinct indexes for various data types, namely strings, binary relations, multi-labeled trees and multi-labeled graphs, as well as succinct text indexes. For strings, binary relations and multi-labeled trees, when the operators in the ADTs are supported in constant time, our results are comparable to previous results, while allowing more flexibility in the encoding of the given data. Using our techniques, we improve several previous results. We design succinct representations for strings and binary relations that are more compact than previous results, while supporting access/rank/select operations efficiently. Our high-order entropy compressed text index provides more efficient support for searches than previous results that occupy essentially the same amount of space. Our succinct representation for labeled trees supports more operations than previous results do. We also design the first succinct representations of labeled graphs. To design succinct indexes, we also have some preliminary results on succinct data structure design. We present a theorem that characterizes a permutation as a suffix array, based on which we design succinct text indexes. We design a succinct representation of ordinal trees that supports all the navigational operations supported by various succinct tree representations. In addition, this representation also supports two other encodings schemes of ordinal trees as abstract data types. Finally, we design succinct representations of planar triangulations and planar graphs which support the rank/select of edges in counter clockwise order in addition to other operations supported in previous work, and a succinct representation of k-page graph which supports more efficient navigation than previous results for large values of k.

We focus on space-efficient, namely succinct, representations of static ordinal unlabeled trees. These structures have space complexity which is optimal up to a lower-order term, yet they support a reasonable set of operations in constant time. This topic has been studied in the last 27 years by numerous authors who came with several distinct solutions to this problem. It is not only of an academic interest, the succinct tree data structures has been used in several data-intensive applications, such as XML processing and representation of suffix trees. In this thesis, we describe the current state of knowledge in this area, compare the many different approaches, and propose several either new or alternative algorithms for operations in the representations alongside. Powered by TCPDF (www.tcpdf.org)

L'évolution rapide des techniques de génération et de stockage de données a permis à de nombreux organismes la création de bases de données volumineuses, pour stocker l'information nécessaire à leurs activités. Ces bases de données qui deviennent de plus en plus importantes sont réellement peu exploitées, alors qu'elles cachent des connaissances potentiellement utiles pour l'organisation. L'extraction de ces informations enfouies dans ces masses de données est traitée par la fouille de données ("Data Mining"). Ce projet de mémoire traite plus particulièrement le problème d'extraction des informations sous forme de règles d'associations. Le problème de la pertinence et de l'utilité des règles extraites est un problème majeur de l'extraction des règles d'associations. Ce problème est lié au nombre important de règles extraites et à la présence d'une forte proportion de règles redondantes. Nombreuses techniques de réduction de la famille de règles ont été publiées. Dans ce contexte, les résultats obtenus par l'analyse formelle des concepts (AFC) ont permis de définir un sous-ensemble de l'ensemble des règles d'associations valides appelés bases informatives. La génération de ces bases informatives se fait par une extraction efficace des itemsets fermés fréquents et leurs générateurs minimaux associés. Les générateurs minimaux composent les prémisses minimales de ces règles alors que leurs fermetures composent les conclusions maximales de ces règles. Cependant un survol de la littérature montre que les générateurs minimaux composant l'antécédent et la conséquence de ces bases, contiennent encore de la redondance. Une représentation réduite de ces générateurs minimaux est utile pour révéler la relation d'équivalence parmi les générateurs minimaux. Une étude a été menée dernièrement dans ce sens dans laquelle l'algorithme DSFS_MINER a été proposé et validé, permettant l'extraction d'une représentation succincte sans perte d'informations des générateurs minimaux. Notre contribution dans ce projet réside d'une part, dans l'étude et l'expérimentation d'approches de représentations succinctes des générateurs minimaux, et d'autre part, dans la proposition d'un algorithme d'expansion permettant la dérivation de tous les générateurs minimaux afin de constituer la famille entière des générateurs minimaux du contexte d'extraction. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Data Mining, Règles d'associations, Analyse formelle des concepts, Générateurs minimaux, Itemset fermés, Générateur minimal, Représentation succincte des générateurs minimaux.

Trees are one of the most fundamental structures in computer science. Standard pointer-based representations consume a significant amount of space while only supporting a small set of navigational operations. Succinct data structures have been developed to overcome these difficulties. A succinct data structure for an object from a given class of objects occupies space close to the information-theoretic lower-bound for representing an object from the class, while supporting the required operations on the object efficiently. In this thesis we consider representing trees succinctly. Various succinct representations have been designed for representing different classes of trees, namely, ordinal trees, cardinal trees and labelled trees. Barring a few, most of these representations are static in that they do not support inserting and deleting nodes. We consider succinct representations for cardinal trees that also support updates (insertions and deletions), i.e., dynamic cardinal trees. A cardinal tree of degree k, also referred to as a k-ary cardinal tree or simply a k-ary tree is a tree where each node has place for up to k children with labels from 1 to k. The information-theoretic lower bound for representing a k-ary cardinal tree on n nodes is roughly (2n+n log k) bits. Representations that take (2n+n log k+ o(n log k ) ) bits have been designed that support basic navigations operations like finding the parent, i-th child, child-labeled j, size of a subtree etc. in constant time. But these could not support updates efficiently. The only known succinct dynamic representation was given by Diego, who gave a structure that still uses (2n+n log k+o(n log k ) ) bits and supports basic navigational operations in O((log k+log log n) ) time, and updates in O((log k + log log n)(1+log k /log (log k + log log n))) amortized time. We improve the times for the operations without increasing the space complexity, for the case when k is reasonably small compared to n. In particular, when k=(O(√(log n ))) our representation supports all the navigational operations in constant time while supporting updates in O(√(log log n )) amortized time.

How do we design for a local community while respecting heritage and touching their hearts? We know through our minds and we understand through our hearts. Consciousness touches minds and experience touches hearts. Since phenomenology is the study of both consciousness and experience, this phenomenological architectural thesis is designed to touch both minds and hearts. Instead of replacement, we need embracement in order to root one's social identity. Only then we will elevate cultural heritage in any context, for example African. This thesis includes a case study of light followed by a 'pataphysical design proposal for Tapestry: a new library at Mzuzu University in Malawi. The library proposal is introduced through poems and visual information in the following sets: metaphysical, physical, 'pataphysical. Through the inquiry in haiku writing style, this poetry collection evaluates corners, windows, light, intensity, form. Not only we propose an exciting and unique library design, but we also discover that dignity is the key to unlocking the spirit of light in any project, regardless of its form. / Master of Architecture

This study addresses eighteenth-century English-speaking evangelicals' understandings of church history, through the lens of published attempts to represent preceding Christian centuries panoramically or comprehensively. Sources entail several short reflections on history emerging in the early years of the transatlantic Revival (1730s-1740s) and subsequent, more substantial efforts by evangelical leaders John Gillies, Jonathan Edwards, John Wesley, Joseph and Isaac Milner, and Thomas Haweis. Little scholarly analysis exists on these sources, aside from the renaissance of interest in recent decades in Edwards. This is surprising, considering the acknowledged prominence of history-writing in the eighteenth century and the influence attributed, then and now, to the works of authors such as Gibbon, Hume, and Robertson. The aim is, first, to elucidate each of the above evangelicals' interpretations of the Christian past, both in overview and according to what they said on a roster of particular historical events, people and movements, and then to consider shared and divergent aspects. These aspects range from points of detail to paradigmatic theological convictions. Secondarily, evangelical church histories are analyzed in relation to earlier Protestant as well as eighteenth-century 'enlightened' historiography, in part through attention to evangelical authors' explicit engagement with these currents. This contextualization assists in determining the unique qualities of evangelical interpretations. Is there, then, evidence of a characteristically 'evangelical' perspective on church history? An examination of this neglected area illumines patterns and particulars of evangelicals' historical thought, and these in turn communicate the self-perceptions and the defining features of evangelicalism itself. Findings support the primary contention that evangelical leaders made use of a dynamic pattern of revival and declension as a means of accounting for the full history of Christianity. Beyond displaying the central place of 'revival' for evangelicals, these church histories demonstrate evangelicalism‘s complex relationship—involving both receptivity and critique—with Protestant and Enlightenment currents of historical inquiry.