Lossless Data Compression via Substring Enumeration for k-th Order Markov Sources with a Finite Alphabet

Abstract

Dubé and Beaudoin have proposed a technique of lossless data compression called compression via substring enumeration (CSE) for a binary source alphabet [1]. Dubé and Yokoo [2] proved that CSE has a linear complexity both in time and in space worst-case performance for the length of string to be encoded. Dubé and Yokoo have specified appropriate predictors of the uniform and combinatorial prediction models for CSE, and proved that CSE has the asymptotic optimality for stationary binary ergodic sources [2,3]. Our previous study [4] evaluated the worst-case maximum redundancy of the modified CSE for an arbitrary binary string from the class of k-th order Markov sources. We propose a generalization of CSE for k-th order Markov sources with a finite alphabet X based on Ota and Morita [5] in this study. Consequently, we analyze the worst-case maximum redundancy of CSE for k-th order Markov sources with a finite alphabet. We also clarify that the compression ratio of CSE asymptotically converges to the optimal one with rate ( |X | k (|X |−1)+1 ) log n n for k-th Markov sources, if the length n of a source string tends to infinity, where |X | denotes the cardinality of X .

Two-dimensional source coding by means of subblock enumeration

“…does not satisfy (10) or such that |b i+1 | r ≥ 2 does not satisfy (16), then T (B(p), p, i + 1) = T (B(p), p, i).…”

“…is the element of X having the largest index in B(x) and note that (10) was first shown in [10]. Note that in…”

“…As for b i (= a : w : c) in (C-ii), satisfying (10) is the same that w is a c-core. Moreover, since a, w, c ∈ D( x) and (3) holds, number of candidates of b i for encoding in (C-ii) is polynomial order with n. The details are described in the bottom of this section.…”

“…The left-hand side term in (10) is given by the difference between the 3rd term and the 1st term in (12). Therefore, if (10) does not hold, then the 1st and the 3rd terms are equal.…”

“…Moreover, an antidictionary coding proposed in [9] provided the first CSE for q-ary (q > 2) alphabet sources as a byproduct. Iwata and Arimura proposed the modified algorithm and evaluated the maximum redundancy rate of the CSE for the kth order Markov sources [10].…”