Almost sure optimality of sliding window Lempel-Ziv algorithm and variants

On match lengths and the asymptotic behavior of Sliding Window Lempel-Ziv algorithm for zero entropy sequences

“…CONVERGENCE RATE OF THE COMPRESSION RATIO It has been proved in [6] that the compression ratio given by H +O( log log nw log nw ) is achieved by SWLZ algorithm for an aperiodic and irreducible Markov source with positive entropy. Following the discussion in Section III, in this section we prove that a compression ratio given by O( log nw nw a ), where 0 < a < 1, is achieved by Fixed Shift Variant of SWLZ (FSLZ) algorithm [1] and SWLZ algorithm for stationary and ergodic processes generated by an irrational rotation. Further, we give a general expression of the compression ratio for zero entropy cases under the setting described by Eq.…”

On Match Lengths, Zero Entropy, and Large Deviations—With Application to Sliding Window Lempel–Ziv Algorithm

“…More precisely, we first consider the convergence rate of the compression ratio of the SWLZ algorithm for zero entropy processes generated by irrational rotation and then give a general result over a class of stationary and ergodic zero entropy sources. Jacob and Bansal [5] proved the asymptotic optimality of the SWLZ algorithm in almost sure sense by modifying the technique used by Wyner and Ziv [47]. Here, following the method used by Jacob and Bansal, we have obtained faster convergence rates for SWLZ algorithm for zero entropy sources as compared to that for positive entropy.…”

“…Next, for the class of zero entropy processes, we establish through corollaries 2 and 3, the behavior of match length asymptotics for irrational rotation and a general zero entropy case respectively. Further, we imitated the proofs given in [47] and [5] of FSLZ and SWLZ algorithms by choosing a L o given by Eq. (10) in contrast to their choice and showed that these algorithms achieve faster convergence rate of the compression ratio for zero entropy sequences as compared to those with positive entropy.…”

“…(11). Note the contrast in the choice of match length that is used in [47] and [5] for positive entropy stationary and ergodic processes.The algorithm is described as follows: 1) Initialization: A window of size n w is fixed. Let j = 1 and n j = n w .…”

“…However, unlike them the class of partitions considered here are obtained by shifting the partition used by them. By this the optimality of SWLZ algorithm is obtained in almost sure sense in [5]. Consider the intervals defined as I r = [r + n w , r + n w + L o − 1], r = 1, 2, ..., N ′ .…”