problemset9 solutions

Proof we can prove this lemma with a simple

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Unformatted text preview: bvious. Since the leaves are in a straight line, the distance between the leftmost and rightmost leaves must be at least � �. By the triangle inequality, one of the root-to-leaf paths (just considering the leftmost and rightmost leaves) is at least � � �� � � � �. � � � � �� � Theorem 3For any VLSI layout of a complete binary tree with -leaves in a straight line, the total wire area is �� �� ��� �. Proof. We can prove this lemma with a simple recurrence. Consider any VLSI layout of a complete binary tree with -leaves in a straight line. We let � � be the minimum total wire area, not counting the longest root-to-leaf path in the tree, of any such layout. Then we will show that the recurrence is � � � � � �� � � � � � � � �� � � �, which can be solved with the master method to get � � � �� ��� �. � �� �� �� �� � �� �� � �� �� Consider any complete subtree � with � leaves in the VLSI layout. The tree � is composed of two � ��-leaf subtrees, �� and ��, with a root node connecting to the roots of � � and ��. Without loss of generality, the longest root-to-leaf path in � includes t...
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