More Efficient Algorithms and Analyses for Unequal Letter Cost Prefix-Free Coding

There is a large literature devoted to the problem of finding an optimal (min-cost) prefix-free code with an unequal letter-cost encoding alphabet of size. While there is no known polynomial time algorithm for solving it optimally there are many good heuristics that all provide additive errors to optimal. The additive error in these algorithms usually depends linearly upon the largest encoding letter size. This paper was motivated by the problem of finding optimal codes when the encoding alphabet is infinite. Because the largest letter cost is infinite, the previous analyses could give infinite error bounds. We provide a new algorithm that works with infinite encoding alphabets. When restricted to the finite alphabet case, our algorithm often provides better error bounds than the best previous ones known.Comment: 29 pages;9 figures

Huffman Coding with Letter Costs: A Linear-Time Approximation Scheme

We give a polynomial-time approximation scheme for the generalization of Huffman Coding in which codeword letters have non-uniform costs (as in Morse code, where the dash is twice as long as the dot). The algorithm computes a (1+epsilon)-approximate solution in time O(n + f(epsilon) log^3 n), where n is the input size

Infinite anti-uniform sources

6 pagesInternational audienceIn this paper we consider the class of anti-uniform Huffman (AUH) codes for sources with infinite alphabet. Poisson, negative binomial, geometric and exponential distributions lead to infinite anti-uniform sources for some ranges of their parameters. Huffman coding of these sources results in AUH codes. We prove that as a result of this encoding, we obtain sources with memory. For these sources we attach the graph and derive the transition matrix between states, the state probabilities and the entropy. If c0 and c1 denote the costs for storing or transmission of symbols "0" and "1", respectively, we compute the average cost for these AUH codes

Efficient Storage of Genomic Sequences in High Performance Computing Systems

ABSTRACT: In this dissertation, we address the challenges of genomic data storage in high performance computing systems. In particular, we focus on developing a referential compression approach for Next Generation Sequence data stored in FASTQ format files. The amount of genomic data available for researchers to process has increased exponentially, bringing enormous challenges for its efficient storage and transmission. General-purpose compressors can only offer limited performance for genomic data, thus the need for specialized compression solutions. Two trends have emerged as alternatives to harness the particular properties of genomic data: non-referential and referential compression. Non-referential compressors offer higher compression rations than general purpose compressors, but still below of what a referential compressor could theoretically achieve. However, the effectiveness of referential compression depends on selecting a good reference and on having enough computing resources available. This thesis presents one of the first referential compressors for FASTQ files. We first present a comprehensive analytical and experimental evaluation of the most relevant tools for genomic raw data compression, which led us to identify the main needs and opportunities in this field. As a consequence, we propose a novel compression workflow that aims at improving the usability of referential compressors. Subsequently, we discuss the implementation and performance evaluation for the core of the proposed workflow: a referential compressor for reads in FASTQ format that combines local read-to-reference alignments with a specialized binary-encoding strategy. The compression algorithm, named UdeACompress, achieved very competitive compression ratios when compared to the best compressors in the current state of the art, while showing reasonable execution times and memory use. In particular, UdeACompress outperformed all competitors when compressing long reads, typical of the newest sequencing technologies. Finally, we study the main aspects of the data-level parallelism in the Intel AVX-512 architecture, in order to develop a parallel version of the UdeACompress algorithms to reduce the runtime. Through the use of SIMD programming, we managed to significantly accelerate the main bottleneck found in UdeACompress, the Suffix Array Construction

Efficient estimation of evolutionary distances

The advent of high throughput sequencers has lead to a dramatic increase in the size of available genomic data. Standard methods, which have worked well for many years, are not suitable for the analysis of big data sets, due to their reliance on a time-consuming alignment step. In this thesis, a new alignment-free approach for phylogeny reconstruction is introduced. The corresponding program, andi, is orders of magnitude faster than classical approaches and also superior to comparable alignment-free methods. The central data structure in andi is the enhanced suffix array. It is used to find long exact matches between sequences. In this thesis, various approaches to the construction of enhanced suffix arrays, including novel ones, are evaluated with respect to performance. Additionally, a new parallel algorithm for the computation of suffix arrays is introduced

A decomposition method for global evaluation of Shannon entropy and local estimations of algorithmic complexity

We investigate the properties of a Block Decomposition Method (BDM), which extends the power of a Coding Theorem Method (CTM) that approximates local estimations of algorithmic complexity based on Solomonoff–Levin’s theory of algorithmic probability providing a closer connection to algorithmic complexity than previous attempts based on statistical regularities such as popular lossless compression schemes. The strategy behind BDM is to find small computer programs that produce the components of a larger, decomposed object. The set of short computer programs can then be artfully arranged in sequence so as to produce the original object. We show that the method provides efficient estimations of algorithmic complexity but that it performs like Shannon entropy when it loses accuracy. We estimate errors and study the behaviour of BDM for different boundary conditions, all of which are compared and assessed in detail. The measure may be adapted for use with more multi-dimensional objects than strings, objects such as arrays and tensors. To test the measure we demonstrate the power of CTM on low algorithmic-randomness objects that are assigned maximal entropy (e.g., π) but whose numerical approximations are closer to the theoretical low algorithmic-randomness expectation. We also test the measure on larger objects including dual, isomorphic and cospectral graphs for which we know that algorithmic randomness is low. We also release implementations of the methods in most major programming languages—Wolfram Language (Mathematica), Matlab, R, Perl, Python, Pascal, C++, and Haskell—and an online algorithmic complexity calculator.Swedish Research Council (Vetenskapsrådet