Algorithms by Jeff Erickson

By Jeff Erickson

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N] with elements larger than x. Otherwise, the LIS of A[1 .. n] with elements larger than x is either the LIS of A[2 .. n] with elements larger than x or A[1] followed by the LIS of A[2 .. 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 ..

Despite these complications, this theorem implies a unique representation of any polynomial of the form n p(x) = s (x − r j ) j=1 where the r j ’s are the roots and s is a scale factor. Once again, to represent a polynomial of degree n, we need a list of n + 1 numbers: one scale factor and n roots. Given a polynomial in this root representation, we can clearly evaluate it in O(n) time. Given two polynomials in root representation, we can easily multiply them in O(n) time by multiplying their scale factors and just concatenating the two root sequences.

5] of distinct numbers using at most 6 comparisons. ” and each leaf is an index into the array. A[1]:A[2] < A[1]:A[3] < A[2]:A[3] < A[2] > A[1] > > A[1]:A[3] < A[1] > A[2]:A[3] < A[3] A[3] > A[2] Finding the median of a 3-element array using at most 3 comparisons 10. Suppose you are given a stack of n pancakes of different sizes. You want to sort the pancakes so that smaller pancakes are on top of larger pancakes. The only operation you can perform is a flip—insert a spatula under the top k pancakes, for some integer k between 1 and n, and flip them all over.

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