Divide-and-conquer
1. Describe a divide-and-conquer algorithm to find the minimum of a
set of nnumbers. Your algorithm should clearly describe how the
problem is divided into subproblems. Either give some psuedo-code
or describe your algorithm clearly in words. Assume that the set is
stored in an array A= [A[1], . . . , A[n]].
(Only a divide-and-conquer algorithm will be accepted here. Of course
you can find the minimum in time O(n), but the point is to understand
the divide-and-conquer paradigm.)
What is the running time of your algorithm?
2. Use the Master Method to find an asymptotic expression for T(n).
(Assume that T(n) = cfor n≤d, for some constants c > 0 and
d≥1.)
(a) T(n) = 2T(n/2) + log n
(b) T(n) = 7T(n/3) + n2
(c) T(n) = 9T(n/3) + n3log n
(d) T(n) = 16T(n/2) + (nlog n)4
(e) T(n) = 4T(n/2) + n2
(f) T(n) = 2T(n/2) + 10n
(g) T(n) = 8T(n/2) + 1000n2
(h) T(n) = 3T(n/3) + √n
(i) T(n) = 2T(n/2) + nlog n
(j) T(n) = 3T(n/2) + n2
(k) T(n) = T(n/2) + 2n
3. Suppose that a queue has Npeople in it, where each person has a
different height from one another. At the front of the queue, you can
see Pdifferent people, and from the rear of the queue you can see
Rdifferent people (this is because of the differences in heights, tall
people block people shorter than themselves).
How many queues satisfy this property? Let Q(N, P, R) denote the
number of queues with this property. You want to find an expression
for Q(N, P, R) (given in terms of N , P, and R).
Hints:
1
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