Huffman Codes
Suppose we have 10^{5} characters in a data file. Normal Storage: 8 bits per character (ASCII)  8 x 10^{5} bits in a file. But we want to compress the file and save it compactly. Suppose only six characters appear in the file: How can we represent the data in a Compact way? (i) Fixed length Code: Each letter represented by an equal number of bits. With a fixed length code, at least 3 bits per character: For example: a 000 b 001 c 010 d 011 e 100 f 101 For a file with 10^{5} characters, we need 3 x 10^{5} bits. (ii) A variablelength code: It can do considerably better than a fixedlength code, by giving many characters short code words and infrequent character long codewords. For example: a 0 b 101 c 100 d 111 e 1101 f 1100 Number of bits = (45 x 1 + 13 x 3 + 12 x 3 + 16 x 3 + 9 x 4 + 5 x 4) x 1000 = 2.24 x 10^{5}bits Thus, 224,000 bits to represent the file, a saving of approximately 25%.This is an optimal character code for this file. Prefix Codes:The prefixes of an encoding of one character must not be equal to complete encoding of another character, e.g., 1100 and 11001 are not valid codes because 1100 is a prefix of some other code word is called prefix codes. Prefix codes are desirable because they clarify encoding and decoding. Encoding is always simple for any binary character code; we concatenate the code words describing each character of the file. Decoding is also quite comfortable with a prefix code. Since no codeword is a prefix of any other, the codeword that starts with an encoded data is unambiguous. Greedy Algorithm for constructing a Huffman Code:Huffman invented a greedy algorithm that creates an optimal prefix code called a Huffman Code. The algorithm builds the tree T analogous to the optimal code in a bottomup manner. It starts with a set of C leaves (C is the number of characters) and performs C  1 'merging' operations to create the final tree. In the Huffman algorithm 'n' denotes the quantity of a set of characters, z indicates the parent node, and x & y are the left & right child of z respectively.
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