Sep 21, 2017 term paper 2

# PROVE THEOREM 8.2.1. YOU MIGHT FOLLOW THE FOLLOWING PROGRAM. (A) IF T IS A BINARY TREE GENERATED…

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Prove Theorem 8.2.1. You might follow the following program.

(a) If T is a binary tree generated by applying Huffman’s algorithm to the non-negatively weighted leaf nodes, then the two smallest node weights appear on sibling leaf nodes, by appeal to the procedure of formation. By a similar appeal, the tree obtained by deleting those two leaf nodes, thereby making their parent a leaf, is also a Huffman tree. The proof that the nodes of T can be put in Gallager order is now straightforward, by induction on the number of leaf nodes of T .

(b) Suppose T is a binary tree with non-negatively weighted nodes, with each parent weighted with the sum of the weights of its children. Suppose the nodes of T can be put in Gallager order, u1,u2,…,u2m−1. If all the weights on nodes are positive, then u1 and u2, siblings, must be leaf nodes. (Why?) If zero appears as a weight, then it is possible that one of u,uis a parent, say of u2k−1,u2k , k ≥ 2, but only if the weights on u2k−1 and u2k are both zero (verify this assertion, under the assumptions), in which case all the weights on u1,…,u2k are zero. Switch the sibling pairs u1,u2 and u2k−1,u2k , in the ordering; the new ordering is still a Gallager ordering. Switch again, if necessary, and continue switching until the first two nodes in the ordering are sibling leaf nodes. (How can you be sure that all this switching will come to an end with the desired result?) Now consider the tree T  obtained from T by deleting u1,u2. Draw the conclusion that T is a Huffman tree, by induction on the number of leaf nodes. (The important formalities are left to you.)

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