Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping

In this paper, we provide for the first time a systematic comparison of distribution matching (DM) and sphere shaping (SpSh) algorithms for short blocklength probabilistic amplitude shaping. For asymptotically large blocklengths, constant composition distribution matching (CCDM) is known to generate the target capacity-achieving distribution. As the blocklength decreases, however, the resulting rate loss diminishes the efficiency of CCDM. We claim that for such short blocklengths and over the additive white Gaussian channel (AWGN), the objective of shaping should be reformulated as obtaining the most energy-efficient signal space for a given rate (rather than matching distributions). In light of this interpretation, multiset-partition DM (MPDM), enumerative sphere shaping (ESS) and shell mapping (SM), are reviewed as energy-efficient shaping techniques. Numerical results show that MPDM and SpSh have smaller rate losses than CCDM. SpSh--whose sole objective is to maximize the energy efficiency--is shown to have the minimum rate loss amongst all. We provide simulation results of the end-to-end decoding performance showing that up to 1 dB improvement in power efficiency over uniform signaling can be obtained with MPDM and SpSh at blocklengths around 200. Finally, we present a discussion on the complexity of these algorithms from the perspective of latency, storage and computations.Comment: 18 pages, 10 figure

Optimal Prefix Free Codes with Partial Sorting

We describe an algorithm computing an optimal prefix free code for n unsorted positive weights in less time than required to sort them on many large classes of instances, identified by a new measure of difficulty for this problem, the alternation alpha. This asymptotical complexity is within a constant factor of the optimal in the algebraic decision tree computational model, in the worst case over all instances of fixed size n and alternation alpha. Such results refine the state of the art complexity in the worst case over instances of size n in the same computational model, a landmark in compression and coding since 1952, by the mere combination of van Leeuwen\u27s algorithm to compute optimal prefix free codes from sorted weights (known since 1976), with Deferred Data Structures to partially sort multisets (known since 1988)

Compressed Text Indexes:From Theory to Practice!

A compressed full-text self-index represents a text in a compressed form and still answers queries efficiently. This technology represents a breakthrough over the text indexing techniques of the previous decade, whose indexes required several times the size of the text. Although it is relatively new, this technology has matured up to a point where theoretical research is giving way to practical developments. Nonetheless this requires significant programming skills, a deep engineering effort, and a strong algorithmic background to dig into the research results. To date only isolated implementations and focused comparisons of compressed indexes have been reported, and they missed a common API, which prevented their re-use or deployment within other applications. The goal of this paper is to fill this gap. First, we present the existing implementations of compressed indexes from a practitioner's point of view. Second, we introduce the Pizza&Chili site, which offers tuned implementations and a standardized API for the most successful compressed full-text self-indexes, together with effective testbeds and scripts for their automatic validation and test. Third, we show the results of our extensive experiments on these codes with the aim of demonstrating the practical relevance of this novel and exciting technology