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To find the position of P[k] in this permutation, write k in binary, and then read the bits backward. For example, in an 8-element bit-reversal permutation, P[3] = P[0112 ] ends up in position 6 = 1102 . The right half of the FFT circuit is a butterfly network. Butterfly networks are often used to route between processors in massively-parallel computers, because they allow any two processors to communicate in only O(log n) steps. Exercises 1. For any two sets X and Y of integers, the Minkowski sum X + Y is the set of all pairwise sums {x + y | x ∈ X , y ∈ Y }.

You are at a political convention with n delegates, each one a member of exactly one political party. It is impossible to tell which political party any delegate belongs to; in particular, you will be summarily ejected from the convention if you ask. However, you can determine whether any two delegates belong to the same party or not by introducing them to each other—members of the same party always greet each other with smiles and friendly handshakes; members of different parties always greet each other with angry stares and insults.

N], x): y ← P[n] for i ← n − 1 downto 0 y ← x · y + P[i] return y The addition algorithm also runs in O(n) time, and this is clearly the best we can do. The multiplication algorithm, however, runs in O(n2 ) time. In the previous lecture, we saw a divide and conquer algorithm (due to Karatsuba) for multiplying two n-bit integers in only O(nlg 3 ) steps; precisely the same algorithm can be applied here. Even cleverer divide-and-conquer strategies lead to multiplication algorithms whose running times are arbitrarily close to linear—O(n1+ ) for your favorite value e > 0—but with great cleverness comes great confusion.

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Algorithms by Jeff Erickson

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