By Peter Seibt
Algorithmic info thought treats the math of many very important components in electronic details processing. it's been written as a read-and-learn e-book on concrete arithmetic, for academics, scholars and practitioners in digital engineering, desktop technological know-how and arithmetic. The presentation is dense, and the examples and routines are a number of. it really is in keeping with lectures on info know-how (Data Compaction, Cryptography, Polynomial Coding) for engineers.
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Extra info for Algorithmic Information Theory: Mathematics of Digital Information Processing (Signals and Communication Technology)
Our goal: we shall show that the Huﬀman algorithm necessarily produces an optimal binary preﬁx code. 30 1 Data Compaction But ﬁrst several characteristic properties of optimal codes: Proposition Let C be an optimal binary preﬁx code, associated with p = (p0 , p1 , . . , pN −1 ). Then we necessarily have: 1. pj > pk =⇒ lj ≤ lk . 2. The code will have an even number of words of maximal length. 3. Whenever several code words have the same length, two of them will be equal except for the last bit.
5) The situation as in exercise (4). Suppose that all probabilities are powers of 12 : pj = 2−lj , 0 ≤ j ≤ N − 1. Show that in this case the arithmetic code word of a source word s1 s2 · · · sn is equal to the Shannon code word (obtained by simple concatenation of the code words for s1 , s2 , . . , sn ). (6) A memoryless source, producing the N letters a0 , a1 , . . , aN −1 according to the (“decreasing”) probability distribution p = (p0 , p1 , . . , pN −1 ). Let s1 and s2 be two source words such that c(s1 ) = c(s2 ) (they have the same arithmetic code word).
Then C is optimal for S. Proof The lengths of the code words (L for C , l for C): L(j1 ,j2 ) + 1, if j = j1 , j2 , lj = else. Lj , One gets for the average lengths (L for C , l for C): l = j=j1 ,j2 pj lj + pj1 lj1 + pj2 lj2 = j=j1 ,j2 pj Lj + p(j1 ,j2 ) L(j1 ,j2 ) + pj1 + pj2 = L + pj1 + pj2 . ). Thus, L minimal =⇒ l minimal. 1 Entropy Coding 31 Corollary The Huﬀman algorithm produces optimal binary preﬁx codes. In particular, the average word length l of the code words is constant for all Huﬀman codes associated with a ﬁxed probability distribution p (note that you will be frequently obliged to make choices when constructing Huﬀman trees).
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