Succinct Indexes

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

Succinct Representations of Dynamic Strings

The rank and select operations over a string of length n from an alphabet of size $\sigma$ have been used widely in the design of succinct data structures. In many applications, the string itself need be maintained dynamically, allowing characters of the string to be inserted and deleted. Under the word RAM model with word size $w=\Omega(\lg n)$, we design a succinct representation of dynamic strings using $nH_0 + o(n)\lg\sigma + O(w)$ bits to support rank, select, insert and delete in $O(\frac{\lg n}{\lg\lg n}(\frac{\lg \sigma}{\lg\lg n}+1))$ time. When the alphabet size is small, i.e. when \sigma = O(\polylog (n)), including the case in which the string is a bit vector, these operations are supported in $O(\frac{\lg n}{\lg\lg n})$ time. Our data structures are more efficient than previous results on the same problem, and we have applied them to improve results on the design and construction of space-efficient text indexes

Compact Binary Relation Representations with Rich Functionality

Binary relations are an important abstraction arising in many data representation problems. The data structures proposed so far to represent them support just a few basic operations required to fit one particular application. We identify many of those operations arising in applications and generalize them into a wide set of desirable queries for a binary relation representation. We also identify reductions among those operations. We then introduce several novel binary relation representations, some simple and some quite sophisticated, that not only are space-efficient but also efficiently support a large subset of the desired queries.Comment: 32 page

LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations

LRM-Trees are an elegant way to partition a sequence of values into sorted consecutive blocks, and to express the relative position of the first element of each block within a previous block. They were used to encode ordinal trees and to index integer arrays in order to support range minimum queries on them. We describe how they yield many other convenient results in a variety of areas, from data structures to algorithms: some compressed succinct indices for range minimum queries; a new adaptive sorting algorithm; and a compressed succinct data structure for permutations supporting direct and indirect application in time all the shortest as the permutation is compressible.Comment: 13 pages, 1 figur

Prospects and limitations of full-text index structures in genome analysis

The combination of incessant advances in sequencing technology producing large amounts of data and innovative bioinformatics approaches, designed to cope with this data flood, has led to new interesting results in the life sciences. Given the magnitude of sequence data to be processed, many bioinformatics tools rely on efficient solutions to a variety of complex string problems. These solutions include fast heuristic algorithms and advanced data structures, generally referred to as index structures. Although the importance of index structures is generally known to the bioinformatics community, the design and potency of these data structures, as well as their properties and limitations, are less understood. Moreover, the last decade has seen a boom in the number of variant index structures featuring complex and diverse memory-time trade-offs. This article brings a comprehensive state-of-the-art overview of the most popular index structures and their recently developed variants. Their features, interrelationships, the trade-offs they impose, but also their practical limitations, are explained and compared

Compressed Representations of Permutations, and Applications

We explore various techniques to compress a permutation $\pi$ over n integers, taking advantage of ordered subsequences in $\pi$, while supporting its application $\pi$(i) and the application of its inverse $\pi^{-1}(i)$ in small time. Our compression schemes yield several interesting byproducts, in many cases matching, improving or extending the best existing results on applications such as the encoding of a permutation in order to support iterated applications $\pi^k(i)$ of it, of integer functions, and of inverted lists and suffix arrays

Dynamic Data Structures for Document Collections and Graphs

In the dynamic indexing problem, we must maintain a changing collection of text documents so that we can efficiently support insertions, deletions, and pattern matching queries. We are especially interested in developing efficient data structures that store and query the documents in compressed form. All previous compressed solutions to this problem rely on answering rank and select queries on a dynamic sequence of symbols. Because of the lower bound in [Fredman and Saks, 1989], answering rank queries presents a bottleneck in compressed dynamic indexing. In this paper we show how this lower bound can be circumvented using our new framework. We demonstrate that the gap between static and dynamic variants of the indexing problem can be almost closed. Our method is based on a novel framework for adding dynamism to static compressed data structures. Our framework also applies more generally to dynamizing other problems. We show, for example, how our framework can be applied to develop compressed representations of dynamic graphs and binary relations