9-Path Length and Huffman Codes

# 9-Path Length and Huffman Codes - Path Length and Huffman...

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Path Length and Huffman Codes The path length of a binary tree t with nonnegative weights attached to its nodes is W(t) = ∑ w(i) d(i) where node i has weight w(i) and is at depth d(i) in t. All nodes i in t Example: t: 2 / \ 1 0 / \ \ 2 3 5 W(t) = 2x1 + 1x2 + 0x2 + 2x3 + 3x3 + 5x3 = 34 Three ways to compute the path length: 1. Directly from the definition as above. 2. Assign a value to each node equal to the sum of the weights in its subtree and then add up the values 13 / \ 6 5 13 + 6 + 5 + 2 + 3 + 5 = 34 / \ \ 2 3 5 3. Let SW be the sum of all the weights. Then W(t) = Sw + W(tL) + W(tR) = 13 + 11 + 10 = 34 Huffman Codes: Given messages with frequencies or weights w 1 , w 2 , . . . w n find a set of codewords for the messages, each composed of 0’s and 1’a, so the average number of 0’s and 1’s is minimized. That is, the n ∑ w i l(i) is minimized. i=1 where l(i) is the length of codeword i. Also - useful for compression. Discuss example to show the problem and then relate to binary trees. Show connection between the above minimization and minimizing path length. Show that optimal tree must be full. Discuss algorithm involving a search of all trees and then Huffman’s

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• Fall '16
• James Korsh
• Graph Theory, Tn, Binary heap, path length

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