Processing Succinct Matrices and Vectors

We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for MTDD_+-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of CSR 201

Parameterized Compilation Lower Bounds for Restricted CNF-formulas

We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that - there are CNF formulas of size $n$ and modular incidence treewidth $k$ whose smallest DNNF-encoding has size $n^{\Omega(k)}$, and - there are CNF formulas of size $n$ and incidence neighborhood diversity $k$ whose smallest DNNF-encoding has size $n^{\Omega(\sqrt{k})}$. These results complement recent upper bounds for compiling CNF into DNNF and strengthen---quantitatively and qualitatively---known conditional low\-er bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth

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

In-Memory Storage for Labeled Tree-Structured Data

In this thesis, we design in-memory data structures for labeled and weights trees, so that various types of path queries or operations can be supported with efficient query time. We assume the word RAM model with word size w, which permits random accesses to w-bit memory cells. Our data structures are space-efficient and many of them are even succinct. These succinct data structures occupy space close to the information theoretic lower bounds of the input trees within lower order terms. First, we study the problems of supporting various path queries over weighted trees. A path counting query asks for the number of nodes on a query path whose weights lie within a query range, while a path reporting query requires to report these nodes. A path median query asks for the median weight on a path between two given nodes, and a path selection query returns the k-th smallest weight. We design succinct data structures to support path counting queries in O(lg σ/ lg lg n + 1) time, path reporting queries in O((occ + 1)(lg σ/ lg lg n + 1)) time, and path median and path selection queries in O(lg σ/ lg lg σ) time, where n is the size of the input tree, the weights of nodes are drawn from [1..σ] and occ is the size of the output. Our results not only greatly improve the best known data structures [31, 75, 65], but also match the lower bounds for path counting, median and selection queries [86, 87, 71] when σ = Ω(n/polylog(n)). Second, we study the problem of representing labeled ordinal trees succinctly. Our new representations support a much broader collection of operations than previous work. In our approach, labels of nodes are stored in a preorder label sequence, which can be compressed using any succinct representation of strings that supports access, rank and select operations. Thus, we present a framework for succinct representations of labeled ordinal trees that is able to handle large alphabets. This answers an open problem presented by Geary et al. [54], which asks for representations of labeled ordinal trees that remain space-efficient for large alphabets. We further extend our work and present the first succinct representations for dynamic labeled ordinal trees that support several label-based operations including finding the level ancestor with a given label. Third, we study the problems of supporting path minimum and semigroup path sum queries. In the path minimum problem, we preprocess a tree on n weighted nodes, such that given an arbitrary path, the node with the smallest weight along this path can be located. We design novel succinct indices for this problem under the indexing model, for which weights of nodes are read-only and can be accessed with ranks of nodes in the preorder traversal sequence of the input tree. One of our index structures supports queries in O(α(m,n)) time, and occupies O(m) bits of space in addition to the space required for the input tree, where m is an integer greater than or equal to n and α(m, n) is the inverse-Ackermann function. Following the same approach, we also develop succinct data structures for semigroup path sum queries, for which a query asks for the sum of weights along a given query path. Then, using the succinct indices for path minimum queries, we achieve three different time-space tradeoffs for path reporting queries. Finally, we study the problems of supporting various path queries in dynamic settings. We propose the first non-trivial linear-space solution that supports path reporting in O((lgn/lglgn)^2 +occlgn/lglgn)) query time, where n is the size of the input tree and occ is the output size, and the insertion and deletion of a node of an arbitrary degree in O(lg^{2+ε} n) amortized time, for any constant ε ∈ (0, 1). Obvious solutions based on directly dynamizing solutions to the static version of this problem all require Ω((lg n/ lg lg n)^2) time for each node reported. We also design data structures that support path counting and path reporting queries in O((lg n/ lg lg n)^2) time, and insertions and deletions in O((lg n/ lg lg n)^2) amortized time. This matches the best known results for dynamic two-dimensional range counting [62] and range selection [63], which can be viewed as special cases of path counting and path selection