 languages. A second example of a ligature, also in all computer keyboards, is the "at sign" character @, a contraction for at, as used in e-mail addresses to indicate the domain name. Interestingly, its origins can be traced back to the Middle Ages to designate weight or liquid capacity (in Spanish and Portuguese, amphora is arroba, which yields the current French name of arobase). The character has also been commonly used in English as a contraction of at to designate prices. See http://en.wikipedia.org/wiki/At_sign, Optimal coding and compression 1000.0 100.0 y = 113.12e-0.0116x Digram Figure 9.7 Plot of the 676 English-letter digram frequencies (logarithmic scale) out of a 10 000-letter count, showing that the distribution is very nearly exponential (from data in Fig. 9.5). English digrams contain less information, or uncertainty, than the mere concatenation of letters by pairs. For the probability distribution, this implies that p(xy) > p(x)p(y), most generally for any symbol pair X, y, meaning that English letters are most often cor- related to a certain extent, and correlation means less information. For instance, we find for the first four leading digrams (xy = TH, HE, AN, IN) in Fig. 9.6 (dividing counts by 10 000), the following probabilities: p(TH) =0.0313>p(T)p(H) =0.0055, = = p(HE) = 0.0249 > p(H)p(E) = 0.0077, p(AN) = 0.0171 > p(A)p(N) = 0.0054, p(IN) = 0.0168 > p(I)p(N) = 0.0055, which indeed illustrates a strong correlation attached to the four digrams TH, HE, AN, and IN. We shall consider next the coding of English digrams through binary codewords. This will provide a good test for comparing the efficiencies of Shannon-Fano and Huffman codes. Owing to the size of the source, the calculation of the Huffman coding-tree must be performed with a computer. 17 From my own computer program and test runs I found that the Shannon-Fano and Huffman codeword lengths for the English-digram source vary from l = 5 to l = 17 (but obviously with different assignments), leading to the 17 Different algorithms and s