(Homework Solution): +) | 100% 1- The time to convert an array, with priorities stored at subseripts 1

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+) | 100% 1- The time to convert an array, with priorities stored at subseripts 1 through n, to a minheap is in: A. e() B. o(logn) C.ol e performing mergesort on n items is: А.Ollogn) B.o(m + n) C.o(n) D, o(nlogn) 3. Which of the following is not true? 4. The cost function for the optimal matrix multiplhcation problem is: isksj lske isksi 5. The function π + 3n-log n is in which set? 6. f(n)-nlgn is in all of the following sets, except 7. Which statement is comect regarding the wweighted and weighted aetivity seheduling problems? A. O(log n) B.0(log(n)) C. Ω(n) D, dif) A. Both are casily solyed using a greedy techmique D. Weighted is solred using a greedy technique. wmw eighted is solved by dyaamie programming B. Unweighted is solved using a greedy technique, weighted is solved by dynamie programming C. Both require dynanuc programming S. What is the valne of 2

1. D.Θ(nlogn)

Explanation: The algorithm to build a min heap is given below.

```BUILD-HEAP(A)
heapsize := size(A);
for i := floor(heapsize/2) downto 1
do HEAPIFY(A, i);
end for
END```

A quick look over the above algorithm suggests that the running time is Θ(nlogn), since each call to Heapify costs Θ(log n) and Build-Heap makes Θ(n) such calls.

2. C. Θ(n)

Explanation:

We know that call MergeSort (log n) times. Since each recursive step is one half the length of n.

Since we know that merge sort is O(n log n) could stop here as MergeSort is called log n times, the merge must be called n times. But we can also reason that we must subdivide the n items until each input consists of one element. Clearly we must merge n such one item lists to arrive at a final out consisting of n items.

3. B. nlogn belongs to Ω(n2)

Explanation: The omega definition is given below.

```Ω (g(n)) = {f(n): there exist positive constants c and
n0 such that 0 <= cg(n) <= f(n) for
all n >= n0}.```

Clearly, for any positive n value, nlogn <= n2. This means that 0 <= c*n2 < nlogn is not satisfied.

4. c. C(i, j) = min i<=k<j {C(i, k) + C(k+1, j) + pi-1pkpj}

Explanation:

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