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Solution: ������ � since � � ���
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6 6.896 Final Examination Name Problem 4. Short Answer (3 parts) [18 points]
(a) Prove or give a counterexample. The number of latches in a synchronous circuit re mains invariant under retiming. Solution: This is false and counterexamples are easy to ﬁnd. The correct statement
is that the number of latches around a cycle is preserved.
(b) Give a Gray code on �bit numbers.
Solution:
0000 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000 7 6.896 Final Examination Name 8 (c) Consider an �input, �output butterﬂy network. Suppose that an adversary destroys a single switch in the network, preventing some inputs from connecting to some outputs in a single pass through the network. We can still ﬁnd a large subset � of inputs and a large subset � of outputs such that every input in � can connect to every output in � . Let � � ��� ��� � � �� ��, where � and � are chosen to allow � to be as large as possible. Give a good asymptotic upp...
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This document was uploaded on 03/20/2014 for the course EECS 6.896 at MIT.
 Spring '07
 BradleyKuszmaul

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