By Henk C. A. van Tilborg (auth.)

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So C* is not a prefix code, while C was. This is only possible if f~ is a proper prefix of some codeword fi' 1 ~ i ~ n -l. Since li ~ 1" ,this proves that fi and f,. only differ in their last coordinate. O The Huffman algorithm [Huf52] consists of two parts: The reduction process. Let S be a source which outputs symbols mi, 1 ~ i babilities PI ~ P2~ ~ n, with respective pro- ... ~ p". _1 and m" by one new symbol with pro- bability P,,_I + p". In this way one obtains a new source with one output symbol less than S.

M& /*(x) 1degree (a(x» < n ) -- il(f). O E il (h). 20 suggests checking the divisors of f . That this is sufficient will be proved later. 21 Let (Sdi;;'OE il(f)andS(x)=-r(x)//*(x). ""f[(S;}i;;'OE <=> il(h)] gcd(-r(x),/*(x»*l. Proof: Let d(x) I gcd ('t(x),/*(x» with degree (d(x» = d ~ 1. _ 't(x)/d(x) Then S (x) - /* (x )/d (x)' SO (Si};;;'O E il (f Id*). 22 S(P)(x) i) ii) il (h). The proof in the reverse direction goes exactly the same. (s;};;;,o Let be a = so+ SIX + ... + Sp_IX p - l • [S;}i;;'O 'r/II [] binary, periodic sequence, say with period p.

In this chapter we shall describe a general method for what is called data compression or source coding. }, with respective probabilities PI> P2 •.. • p". A symbol mi. 1 $; i $; n, will be encoded into a binary string fi of length li. The set {fI> f2. . • f,,} is called a code C for the source S. The idea of data compression is to use such a code that the expected value of the length of the encoded plaintext is minimal. 1) L = LPil;. i=1 over ali possible codes C for the source S. There is however an additional constraint.