arXiv subjects

Wojciech Szpankowski

Publications and source records attributed to Wojciech Szpankowski.

39 records · Page 3Linked to original sources

arXiv2012

Average redundancy of the Shannon code for Markov sources

It is known that for memoryless sources, the average and maximal redundancy of fixed-to-variable length codes, such as the Shannon and Huffman codes, exhibit two modes of behavior for long blocks. It either converges to a limit or it has an oscillatory pattern, depending on the irrationality or rationality, respectively, of certain parameters that depend on the source. In this paper, we extend these findings, concerning the Shannon code, to the case of a Markov source, which is considerably more involved. While this dichotomy, of convergent vs. oscillatory behavior, is well known in other contexts (including renewal theory, ergodic theory, local limit theorems and large deviations of discrete distributions), in information theory (e.g., in redundancy analysis) it was recognized relatively recently. To the best of our knowledge, no results of this type were reported thus far for Markov sources. We provide a precise characterization of the convergent vs. oscillatory behavior of the Shannon code redundancy for a class of irreducible, periodic and aperiodic, Markov sources. These findings are obtained by analytic methods, such as Fourier/Fejer series analysis and spectral analysis of matrices.

cs.IT↗

arXiv2011

Deinterleaving Finite Memory Processes via Penalized Maximum Likelihood

We study the problem of deinterleaving a set of finite-memory (Markov) processes over disjoint finite alphabets, which have been randomly interleaved by a finite-memory switch. The deinterleaver has access to a sample of the resulting interleaved process, but no knowledge of the number or structure of the component Markov processes, or of the switch. We study conditions for uniqueness of the interleaved representation of a process, showing that certain switch configurations, as well as memoryless component processes, can cause ambiguities in the representation. We show that a deinterleaving scheme based on minimizing a penalized maximum-likelihood cost function is strongly consistent, in the sense of reconstructing, almost surely as the observed sequence length tends to infinity, a set of component and switch Markov processes compatible with the original interleaved process. Furthermore, under certain conditions on the structure of the switch (including the special case of a memoryless switch), we show that the scheme recovers \emph{all} possible interleaved representations of the original process. Experimental results are presented demonstrating that the proposed scheme performs well in practice, even for relatively short input samples.

cs.IT↗

arXiv2005

Partial fillup and search time in LC tries

Andersson and Nilsson introduced in 1993 a level-compressed trie (in short: LC trie) in which a full subtree of a node is compressed to a single node of degree being the size of the subtree. Recent experimental results indicated a 'dramatic improvement' when full subtrees are replaced by partially filled subtrees. In this paper, we provide a theoretical justification of these experimental results showing, among others, a rather moderate improvement of the search time over the original LC tries. For such an analysis, we assume that n strings are generated independently by a binary memoryless source with p denoting the probability of emitting a 1. We first prove that the so called alpha-fillup level (i.e., the largest level in a trie with alpha fraction of nodes present at this level) is concentrated on two values with high probability. We give these values explicitly up to O(1), and observe that the value of alpha (strictly between 0 and 1) does not affect the leading term. This result directly yields the typical depth (search time) in the alpha-LC tries with p not equal to 1/2, which turns out to be C loglog n for an explicitly given constant C (depending on p but not on alpha). This should be compared with recently found typical depth in the original LC tries which is C' loglog n for a larger constant C'. The search time in alpha-LC tries is thus smaller but of the same order as in the original LC tries.

cs.DS↗