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Public Const APPNAME As String = Text Tools Custom data type for undoing Type OrigData OldText As Variant Address As String End Type

In the original paper of Ziv and Lempel [603], the authors described the basic LZ77 algorithm and proved that it compressed any string as well as any nite-state compressor acting on that string. However, they did not prove that this algorithm achieved asymptotic optimality (i.e., that the compression ratio converged to the entropy for an ergodic source). This result was proved by Wyner and Ziv [591]. The proof relies on a simple lemma due to Kac: the average length of time that you need to wait to see a particular symbol is the reciprocal of the probability of a symbol. Thus, we are likely to see the high-probability strings within the window and encode these strings ef ciently. The strings that we do not nd within the window have low probability, so that asymptotically, they do not in uence the compression achieved. Instead of proving the optimality of the practical version of LZ77, we will present a simpler proof for a different version of the algorithm, which, though not practical, captures some of the basic ideas. This algorithm assumes that both the sender and receiver have access to the in nite past of the string, and represents a string of length n by pointing to the last time it occurred in the past. We assume that we have a stationary and ergodic process de ned for time from to , and that both the encoder and decoder have access to . . . , X 2 , X 1 , the in nite past of the sequence. Then to encode X0 , X1 , . . . , Xn 1 (a block of length n), we nd the last time we have seen these n symbols in the past. Let Rn (X0 , X1 , . . . , Xn 1 ) = max{j < 0 : (X j , X j +1 . . . X j +n 1 ) = (X0 , . . . , Xn 1 )}. (13.60)

IPv4 Options A Review Security Loose source routing Strict source routing Record route Stream ID Internet timestamp

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