Burrows-wheeler transform in secondary memory

Master’s Thesis in Computer EngineeringA suffix array is an index, a data structure that allows searching for sequences of characters. Such structures are of key importance for a large set of problems related to sequences of characters. An especially important use of suffix arrays is to compute the Burrows-Wheeler Transform, which can be used for compressing text. This procedure is the base of the UNIX utility bzip2. The Burrows-Wheeler transform is a key step in the construction of more sophisticated indexes. For large sequences of characters, such as DNA sequences of about 10 GB, it is not possible to calculate the Burrows-Wheeler transform in an average computer without using secondary memory. In this dissertation we will study the state-of-the-art algorithms to construct the Burrows-Wheeler transform in secondary memory. Based on this research we propose an algorithm and compare it against the previous ones to determine its relative performance. Our algorithm is based on the classical external Heapsort. The novelty lies in a heap that is especially designed for suffix arrays, which we call String Heap. This algorithm aims to be space-conscious, while trying to handle the disk access dominance over main memory access. We divide our solution in two parts, splitting and merging suffix arrays, the latter is the main application of the String Heap. The merging part produces the BWT, as a side effect of merging a set of partial suffix arrays of a text. We also compare its performance against the other algorithms. We also study a second version of the algorithm that accesses secondary memory in blocks

On Sorting, Heaps, and Minimum Spanning Trees

Let A be a set of size m. Obtaining the first k ≤ m elements of A in ascending order can be done in optimal O(m + k log k) time. We present Incremental Quicksort (IQS), an algorithm (online on k) which incrementally gives the next smallest element of the set, so that the first k elements are obtained in optimal expected time for any k. Based on IQS, we present the Quickheap (QH), a simple and efficient priority queue for main and secondary memory. Quickheaps are comparable with classical binary heaps in simplicity, yet are more cache-friendly. This makes them an excellent alternative for a secondary memory implementation. We show that the expected amortized CPU cost per operation over a Quickheap of m elements is O(logm), and this translates into O((1/B) log(m/M)) I/O cost with main memory size M and block size B, in a cache-oblivious fashion. As a direct application, we use our techniques to implement classical Minimum Spanning Tree (MST) algorithms. We use IQS to implement Kruskal’s MST algorithm and QHs to implement Prim’s. Experimental results show that IQS, QHs, external QHs, and our Kruskal’s and Prim’s MST variants are competitive, and in many cases better in practice than current state-of-the-art alternative (and much more sophisticated) implemen- tations.Supported in part by the Millennium Nucleus Center for Web Research, Grant P04-067-F, Mideplan, Chile; Yahoo! Research grant “Compact Data Structures”; and Fondecyt grant 1-080019, Chile

Efficient Data Structures for Partial Orders, Range Modes, and Graph Cuts

This thesis considers the study of data structures from the perspective of the theoretician, with a focus on simplicity and practicality. We consider both the time complexity as well as space usage of proposed solutions. Topics discussed fall in three main categories: partial order representation, range modes, and graph cuts. We consider two problems in partial order representation. The first is a data structure to represent a lattice. A lattice is a partial order where the set of elements larger than any two elements x and y are all larger than an element z, known as the join of x and y; a similar condition holds for elements smaller than any two elements. Our data structure is the first correct solution that can simultaneously compute joins and the inverse meet operation in sublinear time while also using subquadratic space. The second is a data structure to support queries on a dynamic set of one-dimensional ordered data; that is, essentially any operation computable on a binary search tree. We develop a data structure that is able to interpolate between binary search trees and efficient priority queues, offering more-efficient insertion times than the former when query distribution is non-uniform. We also consider static and dynamic exact and approximate range mode. Given one-dimensional data, the range mode problem is to compute the mode of a subinterval of the data. In the dynamic range mode problem, insertions and deletions are permitted. For the approximate problem, the element returned is to have frequency no less than a factor (1+epsilon) of the true mode, for some epsilon > 0. Our results include a linear-space dynamic exact range mode data structure that simultaneously improves on best previous operation complexity and an exact dynamic range mode data structure that breaks the Theta(n^(2/3)) time per operation barrier. For approximate range mode, we develop a static succinct data structure offering a logarithmic-factor space improvement and give the first dynamic approximate range mode data structure. We also consider approximate range selection. The final category discussed is graph and dynamic graph algorithms. We develop an optimal offline data structure for dynamic 2- and 3- edge and vertex connectivity. Here, the data structure is given the entire sequence of operations in advance, and the dynamic operations are edge insertion and removal. Finally, we give a simplification of Karger's near-linear time minimum cut algorithm, utilizing heavy-light decomposition and iteration in place of dynamic programming in the subroutine to find a minimum cut of a graph G that cuts at most two edges of a spanning tree T of G