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

Grammar compressed sequences with rank/select support

An early partial version of this paper appeared in Proc. SPIRE 2014: G. Navarro, A. Ordóñez Grammar compressed sequences with rank/select support, Proc. 21st International Symposium on String Processing and Information Retrieval, LNCS, SPIRE, vol. 8799 (2014), pp. 31–44The final publication is available at Springer via http://dx.doi.org/10.1016/j.jda.2016.10.001[Abstract] Sequence representations supporting not only direct access to their symbols, but also rank/select operations, are a fundamental building block in many compressed data structures. Several recent applications need to represent highly repetitive sequences, and classical statistical compression proves ineffective. We introduce, instead, grammar-based representations for repetitive sequences, which use up to 6% of the space needed by statistically compressed representations, and support direct access and rank/select operations within tens of microseconds. We demonstrate the impact of our structures in text indexing applications.Chile. Fondo Nacional de Desarrollo Científico y Tecnológico; 140796Ministerio de Economía, Industria y Competitividad; 00645663/ITC-20133062Ministerio de Economía, Industria y Competitividad; TIN2009-14560-C03-02Ministerio de Economía, Industria y Competitividad; TIN2010-21246-C02-01Ministerio de Economía, Industria y Competitividad; TIN2013-46238-C4-3-RMinisterio de Economía, Industria y Competitividad; TIN2013-47090-C3-3-PMinisterio de Economía, Industria y Competitividad; AP2010-6038Xunta de Galicia; GRC2013/05

Space-Efficient Data Structures in the Word-RAM and Bitprobe Models

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