Algorithms by Jeff Erickson

By Jeff Erickson

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Hobbes: You know, eleventeen, thirty-twelve, and all those. It’s a little confusing at first. Calvin: How did YOU learn all this? You’ve never even gone to school! Hobbes: Instinct. Tigers are born with it. — “Calvin and Hobbes” (January 6, 1998) It needs evaluation So let the games begin A heinous crime, a show of force A murder would be nice, of course — “Bad Horse’s Letter”, Dr. 1 Polynomials In this lecture we’ll talk about algorithms for manipulating polynomials: functions of one variable built from additions subtractions, and multiplications (but no divisions).

Bit-reversal permutation U FFT(n/2) U* P P* V FFT(n/2) V* 0000 0000 0000 0000 0001 0010 0100 1000 0010 0100 1000 0100 0011 0110 1100 1100 0100 1000 0010 0010 0101 1010 0110 1010 0110 1100 1010 0110 0111 1110 1110 1110 1000 0001 0001 0001 1001 0011 0101 1001 1010 0101 1001 0101 1011 0111 1101 1101 1100 1001 0011 0010 1101 1011 0111 1011 1110 1101 1011 0111 1111 1111 1111 1111 Butterfly network The recursive structure of the FFT algorithm. If we expand this recursive structure completely, we see that the circuit splits naturally into two parts.

N] with elements larger than A[1], whichever is longer. The longest increasing subsequence without restrictions can now be redefined as the longest increasing subsequence with elements larger than −∞. Rewriting this recursive definition into pseudocode gives us the following recursive algorithm. 4 Lecture 3: Backtracking [Fa’10] Algorithms LIS(A[1 .. n]): return LISBIGGER(−∞, A[1 .. n]) LISBIGGER(prev, A[1 .. n]): if n = 0 return 0 else max ← LISBIGGER(prev, A[2 .. n]) if A[1] > pr ev L ← 1 + LISBIGGER(A[1], A[2 ..

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