On the Average Redundancy Rate of the Lempel-Ziv Code with K-Error Protocol
DCC '00 Proceedings of the Conference on Data Compression
Precise Average Redundancy Of An Idealized Arithmetic Coding
DCC '02 Proceedings of the Data Compression Conference
Redundancy estimates for the Lempel-Ziv algorithm of data compression
Discrete Applied Mathematics
Analytic combinatorics: a calculus of discrete structures
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
IEEE Transactions on Information Theory
| Hi-index | 754.90 |
In this paper, we settle a long-standing open problem concerning the average redundancy rn of the Lempel-Ziv'78 (LZ78) code. We prove that for a memoryless source the average redundancy rate attains asymptotically Ern=(A+δ(n))/log n+ O(log log n/log2 n), where A is an explicitly given constant that depends on source characteristics, and δ(x) is a fluctuating function with a small amplitude. We also derive the leading term for the kth moment of the number of phrases. We conclude by conjecturing a precise formula on the expected redundancy for a Markovian source. The main result of this paper is a consequence of the second-order properties of the Lempel-Ziv algorithm obtained by Jacquet and Szpankowski (1995). These findings have been established by analytical techniques of the precise analysis of algorithms. We give a brief survey of these results since they are interesting in their own right, and shed some light on the probabilistic behavior of pattern matching based data compression