Rewriting Codes for Joint Information Storage in Flash Memories

Memories whose storage cells transit irreversibly between states have been common since the start of the data storage technology. In recent years, flash memories have become a very important family of such memories. A flash memory cell has q states—state 0.1.....q-1 - and can only transit from a lower state to a higher state before the expensive erasure operation takes place. We study rewriting codes that enable the data stored in a group of cells to be rewritten by only shifting the cells to higher states. Since the considered state transitions are irreversible, the number of rewrites is bounded. Our objective is to maximize the number of times the data can be rewritten. We focus on the joint storage of data in flash memories, and study two rewriting codes for two different scenarios. The first code, called floating code, is for the joint storage of multiple variables, where every rewrite changes one variable. The second code, called buffer code, is for remembering the most recent data in a data stream. Many of the codes presented here are either optimal or asymptotically optimal. We also present bounds to the performance of general codes. The results show that rewriting codes can integrate a flash memory’s rewriting capabilities for different variables to a high degree

Trajectory Codes for Flash Memory

Flash memory is well-known for its inherent asymmetry: the flash-cell charge levels are easy to increase but are hard to decrease. In a general rewriting model, the stored data changes its value with certain patterns. The patterns of data updates are determined by the data structure and the application, and are independent of the constraints imposed by the storage medium. Thus, an appropriate coding scheme is needed so that the data changes can be updated and stored efficiently under the storage-medium's constraints. In this paper, we define the general rewriting problem using a graph model. It extends many known rewriting models such as floating codes, WOM codes, buffer codes, etc. We present a new rewriting scheme for flash memories, called the trajectory code, for rewriting the stored data as many times as possible without block erasures. We prove that the trajectory code is asymptotically optimal in a wide range of scenarios. We also present randomized rewriting codes optimized for expected performance (given arbitrary rewriting sequences). Our rewriting codes are shown to be asymptotically optimal.Comment: Submitted to IEEE Trans. on Inform. Theor

The Parametric Ordinal-Recursive Complexity of Post Embedding Problems

Post Embedding Problems are a family of decision problems based on the interaction of a rational relation with the subword embedding ordering, and are used in the literature to prove non multiply-recursive complexity lower bounds. We refine the construction of Chambart and Schnoebelen (LICS 2008) and prove parametric lower bounds depending on the size of the alphabet.Comment: 16 + vii page

Some recent asynchronous system design methodologies

Journal ArticleWe present an in-depth study of some techniques for asynchronous system design, analysis, and verification. After defining basic terminology, we take one simple example - a four-phase t o two-phase converter - and present its design using (a) classical flow-tables; (b) Signal Transition Graphs of [8]; and (c) Trace Theory of [15]. We then present necessary and sufficient conditions for Delay Insensitivity, proposed by [38], and illustrate it on our example. Finally, we present the work of [13] on the verification of asynchronous circuits, and illustrate it on the circuits derived in the paper. The following points are emphasized: (i) presentation of techniques at more depth than in a general survey; (ii) illustration of all t h e aspects discussed on a common example; (hi) comparative study of the works presented. Many interesting works had to be left out, solely because of our lack of space and time

Improving Data Structures and Algorithms

This thesis addresses important algorithms and data structures used in sequence analysis for applications such as read mapping. First, we give an overview on state-of-the-art FM indices and present the latest improvements. In particular, we will introduce a recently published FM index based on a new data structure: EPR dictionaries. This rank data structures allows search steps in constant time for unidirectional and bidirectional FM indices. To our knowledge this is the first and only constant-time implementation of a bidirectional FM index at the time of writing. We show that its running time is not only optimal in theory, but currently also outperforms all available FM index implementations in practice. Second, we cover approximate string matching in bidirectional indices. To improve the running time and make higher error rates suitable for index-based searches, we introduce an integer linear program for finding optimal search strategies. We show that it is significantly faster than other search strategies in indices and cover additional improvements such as hybrid approaches of index-based searches with in-text verification, i.e., at some point the partially matched string is located and verified directly in the text. Finally, we present a yet unpublished algorithm for fast computation of the mappability of genomic sequences. Mappability is a measure for the uniqueness of a genome by counting how often each $k$-mer of the sequence occurs with a certain error threshold in the genome itself. We suggest two applications of mappability with prototype implementations: First, a read mapper incorporating the mappability information to improve the running time when mapping reads that match highly repetitive regions, and second, we use the mappability information to identify phylogenetic markers in a set of similar strains of the same species by the example of E. coli. Unique regions allow identifying and distinguishing even highly similar strains using unassembled sequencing data. The findings in this thesis can speed up many applications in bioinformatics as we demonstrate for read mapping and computation of mappability, and give suggestions for further research in this field