problemset6 solutions

C prove or give a counterexample for any standard

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Unformatted text preview: mparator, all inputs to the ´Õ · µ Ø comparators are consistent with those of the original network. This fact is apparent in the construction. As for running time, we iterate over all Ö comparators, and at each comparator we consider all the following comparators. Thus, we have a running time of ¢´Ö ¾ µ. (c) Prove or give a counterexample: For any standard sorting network, if a comparator ´ added anywhere in the network, the network continues to sort. Solution: Simple counterexamples exist. µ, where , is Handout 14: Solution Set 6 2 Problem 6-2. A comparison network is a transposition network if each comparator connects adjacent lines. Intuitively, a transposition network represents the action over time of a linear systolic array making oblivious comparison exchanges between adjacent array elements. (a) Show that if a transposition network with Ò inputs actually sorts, then it has ª´Ò ¾ µ comparators. Solution: For simplicity, let’s consider an even Ò. If the network actually sorts, then it must sort the Ò ½ ¼ ½ Ò ¾ ½ . Each input has to be moved across exactly Ò ¾ lines. input Ò ¾ Ò ¾ · ½ In a transposition network, each comparator moves two inputs by exactly ½ line. Thus, each input must be involved in Ò ¾ comparators. Since there are Ò inputs, and each comparator can only move two inputs, ª´Ò¾ µ comparators. The same argument works for odd Ò. we need at least Ò¾ (b) Prove that a transposition network with Ò inputs is a sorting network if and only if it sorts the sequence Ò Ò ½ ½....
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