problemset6 solutions

for a base case consider the network before any

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Unformatted text preview: Ü ½ ܾ ÜÒ to a transposition network and the state of the network Solution: Consider an input . after the -th comparator. We denote the data held by line after the -th comparator by Lemma 1 Consider the descending input Consider also any other input ܽ , then . ܾ Ò Ò ½ ½ to some Ò-input transposition network. ÜÒ . For any pairs of lines and with , if Proof. We use induction over the Ö comparators in order. , For a base case, consider the network before any comparators. For any and with ¼ ¼ because the input is descending. Thus, we make no claim about any of the . ¼ Assume that the claim holds after the ½-th comparator. Consider the -th comparator. Because we have a transposition network, we compare two adjacent lines and · ½. We need to show that the claim holds (for all pairs) of lines after the -th comparator. That is, consider any two lines and with ½ Ö. Suppose that . Then we need to show that . We have several cases: Case 1 Suppose that and · ½. Note that ´ · ½µ because of how a comparator works, so ´ · ½µ . , · ½, , and · ½. Then lines and are not involved in Case 2 Suppose that the comparator. Thus, ½ ½ . By our inductive assumption, we have . ½ ½ , which yields ½ ½ , and . Then we have Ñ Ò´ Case 3 Suppose that ½ ½ ´ · ½µ ½ µ Ñ Ü´ ´ · ½µ µ. By the inductive assumption, and ½ ½ ½ ½ ½ ´ · ½µ . Thus, Ñ Ò´ ´ · ½µ µ . ½ ½ ½ ½ Case 4 Suppose that , but ·½. Then we have Ñ Ü´ ½ ½ ´ · ½µ ½ µ. If ½ ´ · ½µ ½, then we have ½ ...
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