Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences

The concept of symbolic sequences play important role in study of complex systems. In the work we are interested in ultrametric structure of the set of cyclic sequences naturally arising in theory of dynamical systems. Aimed at construction of analytic and numerical methods for investigation of clusters we introduce operator language on the space of symbolic sequences and propose an approach based on wavelet analysis for study of the cluster hierarchy. The analytic power of the approach is demonstrated by derivation of a formula for counting of {\it two-fold de Bruijn sequences}, the extension of the notion of de Bruijn sequences. Possible advantages of the developed description is also discussed in context of applied

Optimal Assembly for High Throughput Shotgun Sequencing

We present a framework for the design of optimal assembly algorithms for shotgun sequencing under the criterion of complete reconstruction. We derive a lower bound on the read length and the coverage depth required for reconstruction in terms of the repeat statistics of the genome. Building on earlier works, we design a de Brujin graph based assembly algorithm which can achieve very close to the lower bound for repeat statistics of a wide range of sequenced genomes, including the GAGE datasets. The results are based on a set of necessary and sufficient conditions on the DNA sequence and the reads for reconstruction. The conditions can be viewed as the shotgun sequencing analogue of Ukkonen-Pevzner's necessary and sufficient conditions for Sequencing by Hybridization.Comment: 26 pages, 18 figure

Focus: A Graph Approach for Data-Mining and Domain-Specific Assembly of Next Generation Sequencing Data

Next Generation Sequencing (NGS) has emerged as a key technology leading to revolutionary breakthroughs in numerous biomedical research areas. These technologies produce millions to billions of short DNA reads that represent a small fraction of the original target DNA sequence. These short reads contain little information individually but are produced at a high coverage of the original sequence such that many reads overlap. Overlap relationships allow for the reads to be linearly ordered and merged by computational programs called assemblers into long stretches of contiguous sequence called contigs that can be used for research applications. Although the assembly of the reads produced by NGS remains a difficult task, it is the process of extracting useful knowledge from these relatively short sequences that has become one of the most exciting and challenging problems in Bioinformatics. The assembly of short reads is an aggregative process where critical information is lost as reads are merged into contigs. In addition, the assembly process is treated as a black box, with generic assembler tools that do not adapt to input data set characteristics. Finally, as NGS data throughput continues to increase, there is an increasing need for smart parallel assembler implementations. In this dissertation, a new assembly approach called Focus is proposed. Unlike previous assemblers, Focus relies on a novel hybrid graph constructed from multiple graphs at different levels of granularity to represent the assembly problem, facilitating information capture and dynamic adjustment to input data set characteristics. This work is composed of four specific aims: 1) The implementation of a robust assembly and analysis tool built on the hybrid graph platform 2) The development and application of graph mining to extract biologically relevant features in NGS data sets 3) The integration of domain specific knowledge to improve the assembly and analysis process. 4) The construction of smart parallel computing approaches, including the application of energy-aware computing for NGS assembly and knowledge integration to improve algorithm performance. In conclusion, this dissertation presents a complete parallel assembler called Focus that is capable of extracting biologically relevant features directly from its hybrid assembly graph

New Algorithms and Lower Bounds for Sequential-Access Data Compression

This thesis concerns sequential-access data compression, i.e., by algorithms that read the input one or more times from beginning to end. In one chapter we consider adaptive prefix coding, for which we must read the input character by character, outputting each character's self-delimiting codeword before reading the next one. We show how to encode and decode each character in constant worst-case time while producing an encoding whose length is worst-case optimal. In another chapter we consider one-pass compression with memory bounded in terms of the alphabet size and context length, and prove a nearly tight tradeoff between the amount of memory we can use and the quality of the compression we can achieve. In a third chapter we consider compression in the read/write streams model, which allows us passes and memory both polylogarithmic in the size of the input. We first show how to achieve universal compression using only one pass over one stream. We then show that one stream is not sufficient for achieving good grammar-based compression. Finally, we show that two streams are necessary and sufficient for achieving entropy-only bounds.Comment: draft of PhD thesi