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Extra resources for Algorithms

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The running time T (n) clearly satisfies the recurrence T (n) ≤ 2T (n − 1) + O(1), which we can solve using either recursion trees or annihilators (or just guessing) to obtain the upper bound T (n) = O(2n ). In the worst case, the recursion tree for this algorithm is a complete binary tree with depth n. Here is a similar recursive algorithm that actually constructs a subset of X that sums to T , if one exists. This algorithm also runs in O(2n ) time. CONSTRUCTSUBSET(X [1 .. n], T ): if T = 0 return ∅ if T < 0 or n = 0 return NONE Y ← CONSTRUCTSUBSET(X [2 ..

B) Describe an algorithm that determines whether a given set of n integers contains three elements whose sum is zero, in O(n2 ) time. (c) Now suppose the input set X contains only integers between −10000n and 10000n. Describe an algorithm that determines whether X contains three elements whose sum is zero, in O(n log n) time. ] c Copyright 2010 Jeff Erickson. 0/). Free distribution is strongly encouraged; commercial distribution is expressly forbidden. edu/~jeffe/teaching/algorithms/ for the most recent revision.

There’s essentially no correlation between the roots of p, the roots of q, and the roots of p + q. We could 1 I’m going to assume in this lecture that each arithmetic operation takes O(1) time. This may not be true in practice; in fact, one of the most powerful applications of FFTs is fast integer multiplication. One of the fastest integer multiplication algorithms, ∗ due to Schönhage and Strassen, multiplies two n-bit binary numbers using O(n log n log log n log log log n · · · 2O(log n) ) bit operations.

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