Universal lossless source coding with the Burrows Wheeler transform

Abstract

Abstract-The Burrows Wheeler Transform (BWT) is a reversible sequence transformation used in a variety of practical lossless source-coding algorithms. In each, the BWT is followed by a lossless source code that attempts to exploit the natural ordering of the BWT coefficients. BWT-based compression schemes are widely touted as low-complexity algorithms giving lossless coding rates better than those of the Ziv-Lempel codes (commonly known as LZ'77 and LZ'78) and almost as good as those achieved by prediction by partial matching (PPM) algorithms. To date, the coding performance claims have been made primarily on the basis of experimental results. This work gives a theoretical evaluation of BWT-based coding. The main results of this theoretical evaluation include: 1) statistical characterizations of the BWT output on both finite strings and sequences of length , 2) a variety of very simple new techniques for BWT-based lossless source coding, and 3) proofs of the universality and bounds on the rates of convergence of both new and existing BWT-based codes for finite-memory and stationary ergodic sources. The end result is a theoretical justification and validation of the experimentally derived conclusions: BWT-based lossless source codes achieve universal lossless coding performance that converges to the optimal coding performance more quickly than the rate of convergence observed in Ziv-Lempel style codes and, for some BWT-based codes, within a constant factor of the optimal rate of convergence for finite-memory sources.Index Terms-Burrows Wheeler Transform (BWT), rate of convergence, redundancy, text compression, universal noiseless source coding.

A fast and efficient nearly-optimal adaptive Fano coding scheme

“…Adaptive coding approaches that use higher-order statistical models, and other structural models, include dictionary techniques (LZ and its enhancements) [10][11][12], prediction with partial matching (PPM) and its enhancements [13,14], block sorting compression [15,16], and grammar based compression (GBC) [17].…”

Blind Equalization Via Sample Sorting

“…Throughout this paper, we illustrate the merits of our methods by means of a simple example; specifically, we focus on a system with binary data I ∈ {1, −1} and a channel with coefficients h = [1,4,2]. Figure 2 (a) shows the successive rank pairs and Figure 2 (b) shows the sorted successor pairs.…”

“…In order to evaluate the performance of the demodulation technique proposed in Section 5 and to compare to that of other available equalization algorithms, simulations have been performed for the three-tap channel with coefficients [1,4,2].…”

“…In fact, with finite-precison observations we will demonstrate an algorithm that has linear complexity in the number of samples and does not require any arithmetic operations. Our processing technique is related to the Burrows-Wheeler transform [3,4] and to suffix arrays which have historically been used in data compression applications [4] and recently extended to general string and sequence searching applications [5]. To our knowledge this is the first application of such techniques to problems in equalization and we believe that our techniques are applicable more widely in signal processing.…”

On the possible patterns of inputs for block sorting in the Burrows–Wheeler transformation

“…There are two ways to define the result of the modified BWT. One is to use the string nb$alool itself and the other is to use the pair (nbalool, 2) of the string with $ deleted and the position of $ in the string before the deletion [4]. This modified BWT has the same merits as the original BWT as preprocessing phases of data compression.…”