A CTW Scheme for Non-Tree Sources

zs denotes the concatenation of a data zz and a string s, l(s) is the length of a string s, 0 < ps < 1 is a constant given to a string s if 0 < l(s) 5 D 1, ,Bs = 0 if 2(s) = D (D > 1: some integer), Q,(x;) is the product of the predicted probabilities over the time instances such that the string defined below is s. The CTW assumes that the source is some tree source: the probability of x1+1 E A depends on the previous shortest string s = ~t-d+r . . . xi with some depth d (d 2 0: some integer) which is not be a postfix of another string. Although it is known that the computation of the CTW in coding/decoding is O(Dn), the redundancy gets worse in the case where the source is outside the tree sources. The proposed scheme deals with a more wider source class: the strings are expressed as elements in (A U {*})D instead of those in U~<~<DA~, where ‘*’ denotes -the don’t care symbol. The extended tree sources include the original tree sources. The recursive equation is modified as