On the Average Redundancy Rate of the Lempel-Ziv Code

Wojciech Szpankowski'" Department of Computer Science Purdue University W. Lafayette, IN 47907 U.S.A. It was conjectured that the average redundancy rate, Tnl for the Lempel-Ziv code (LZ78) is 0(loglognflogn) where n is the length of the database sequence. However, it was also known that for infinitely many n the redundancy Tn is bounded from the below by 2/ log n. In this paper we settle the above conjecture in the negative by proving that for memoryless and Markov sources the average redundancy rate attains asymptotically Ern = (A +8(n))flog n +D(loglog n/log n) where A is an explicitly given constant that depends on the source characteristics, and o(x) is a fluctuating function. This result is a consequence ofrecently established second-order properties for the number of phrases in the Lempel-Ziv algorithm. We also derive the leading term for the kth moment of the number of phrases. Finally, in concluding remarks we discuss generalized Lcrnpel-Ziv codes for which the average redundancy rates are computed and compared with the original Lempel-Ziv codes_