By Seibt P.
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Extra info for Algorithmic Information Theory. Mathematics of Digital Information Processing
AN −1 according to the ﬁxed probability distribution p = (p0 , p1 , . . , pN −1 ). Now consider all associated binary preﬁx codes. Such a binary preﬁx code C is optimal : ⇐⇒ The average word length l = p0 l0 + p1 l1 + · · · + pN −1 lN −1 of its words is minimal. (Note that l = l(l0 , l1 , . . , lN −1 ) is a function of the lengths of the N code words associated with a0 , a1 , . . , aN −1 ; the probabilities p0 , p1 , . . ) Our goal: we shall show that the Huﬀman algorithm necessarily produces an optimal binary preﬁx code.
E. with a pointer doubled). (b) Does there exist sequences of LZW code words of the type . . ( )( )( ) . . e. with a pointer tripled)? 2 The LZW Decoder A First Approach The principal goal of the decoder is the reconstruction of the dictionary of the encoder. It has to correctly interpret the stream of code words (pointers) that it receives. The down-to-earth decoding (the identiﬁcation of the code words) is a part of this task. The current string s, candidate for admission to the dictionary, will still remain in the centre of the algorithm.
The situation A memoryless source, producing the N letters a0 , a1 , . . , aN −1 , according to the probability distribution p = (p0 , p1 , . . , pN −1 ). We shall always suppose p0 ≥ p1 ≥ · · · ≥ pN −1 . The arithmetic encoder will associate with a stream of source symbols aj1 aj2 · · · ajn · · · (which could be theoretically unlimited), a bitstream α1 α2 α3 · · · αl · · · (which would then also be unlimited). But let us stop after n encoding steps: The code word α1 α2 α3 · · · αl of l bits associated with the n ﬁrst source symbols aj1 aj2 · · · ajn will be the code word c(aj1 aj2 · · · ajn ) of a Shannon block encoding formally adapted to recursiveness according to the device: “every step yields a tree-antecedent to the next step”.
Algorithmic Information Theory. Mathematics of Digital Information Processing by Seibt P.