Unbounded Length Contexts for Ppm Ppm*c Model after Processing the String

\Compression of individual sequences via variable rate coding", IEEE Transactions on Information Theory, 24(5), 530{536. [] a[] abra[] abrac ac ad bra[] brac c d ra[] rac a r d [] r c a a a a b b []abracadabra a[]abracadabr abra[]abracad abracadabra[] acadabra[]abr adabra[]abrac bra[]abracada bracadabra[]a cadabra[]abra dabra[]abraca ra[]abracadab racadabra[]ab abracadabra[] []abracadabra a[]abracadabr ra[]abracadab bra[]abracada abra[]abracad dabra[]abraca adabra[]abrac cadabra[]abra acadabra[]abr racadabra[]ab bracadabra[]a M M' I L sort M'' L rearrange FIGURE 6. BW compression of the string abracadabraa]. Figure 6, then sort the strings alphabetically to produce M 0. Two parameters are extracted from the sorted matrix. The rst, I, is an integer that records which row number corresponds to the original string. The second, L, is the character string that constitutes the last column. In this example, I = 4 and L = ard]rcaaaabb. Strange as it may seem, the input string is completely speciied by I and L: the reverse transformation for reconstructing the original is explained in (Burrows & Wheeler, 1994). Moreover, L can be transmitted very economically because it has the property that the same letters often fall together into long runs. M 00 is the same as M 0 but with L highlighted and symbols not needed to form unique contexts suppressed. It is clear then that the unique strings in M 00 correspond one-to-one with the leaves of the trie in Figure 5. The characters in L are those that lie immediately before each of the unique contexts { thus BW can be seen as very similar to the process of predicting from right to left { always predicting the next character to the left and using the same contexts as PPM*. In summary, whereas BW can be viewed as exploiting contexts of unbounded length by sorting them after the whole input string has been processed, PPM* works adaptively by predicting the next character from previous , unbounded-length, contexts. 6. CONCLUSIONS A new lossless compression mechanism, PPM*, has been described. Its major contribution is that it shows that the full information available by considering all substrings of the input string can be eeectively used to generate high quality predictions. The information in the PPM* model subsumes that used in many current high quality models including the LZ family Signiicant work remains in eeectively extracting this information, however. In another paper in this issue, Bunton (1997) shows a number of signiicant improvements to the …