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Unformatted text preview: tters Character Frequency a e 15 i 12 s 3 t 4 (space) 13 (newline) 1 Total 68 10 58 Compression step 1 (b) Build a min-heap Character Sorted by frequency Frequency a e 69 10: i (space) 13: sp 13 1 Total 12: i 4 (newline) 15: e 3 t 3: s 12 s 4: t 15 i 1: nl 10 58 Compression step 2 Build the tree Start with a “forest” of trees: Repeat Take the two trees that have the lowest frequency The next two removals from the heap Make them children of a new node Keep track of the total frequency of that node 70 And stick that tree back into the heap 71 72 The final Huffman coding tree Character Code a 001 e 01 i 10 s 00000 t 0001 (space) (newline) 73 11 00001 Resulting encoding table Character Code Frequency Total bits a 001 10 30 e 01 15 30 i 10 12 24 s 00000 3 15 t 0001 4 16 11 13 26 00001 1 5 58 146 (space) (newline) Total Total encoding is 146 bits 74 Is Huffman encoding optimal? First define what optimal means: The Huffman code for a given alphabet achieves the minimum average number of bits per letter of any prefix code To prove this, we must first two things: Optimal sub-structure Greedy choice property We’ll show the second one first 75 Step 1 • Lemma 16.2: Let C be an alphabet in which each character c C has frequency c.freq. Let x and y be two characters in C having the lowest frequencies. Then there exists an optimal prefix code for C in which the codewords for x and y have the same len...
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This note was uploaded on 02/25/2014 for the course CS 4102 taught by Professor Horton during the Spring '10 term at UVA.

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