On redundancy rate of FDLZ algorithm and its variants

We use the machinery developed by Wyner [1], for the sources satisfying Markov condition, to obtain an upper bound on the contribution of pointer bits to the compression ratio for fixed database Lempel-Ziv (FDLZ) algorithm to be H + O(1/ log2n) which is an improvement from the previous bound of H + H(1 + o(1))log2log2n/ log2n . We use the definition of compression ratio as in Yang and Kieffer [2]. Here H is the entropy rate of the source and n is the size of the database. Then using the same definition of compression ratio we obtain an upper bound on the contribution of phrase length bits for the variant of FDLZ suggested in [3] to be O(1/ log2n), which gives an upper bound of O(1/ log2n) on the redundancy rate itself for this version of FDLZ.