# Urgent Mathematics project (2 Hours)

If f(n)=2n2+n+12 and g(n)=n2

, please show f(n)=Θ(g(n)). (Hint: you need to

find c1, c2, and no to make c1g(n) ≤ f(n) ≤ c2g(n)) (20 points)

2. What are the big O notations of best case, worst case, and average case of

Merge Sort? (15 points)

3. (1) The Insertion Sort below can output an ascending array. Please modify

the pseudocode to make it output a descending array. (10 points)

(2)Please give an array with 5 elements, which is the worst case of the

modified insertion sort code. (5 points)

(3)Please give an array with 5 elements, which is the best case of the

modified insertion sort code. (5 points)

4. What is the big O notation of the following program (please show detailed

steps for full credits):

for (int i=1; i<n; i*=2)

for (int j=i; j<n; j++)

{//other constant time statements ;}

(30 points)

5. If a modified merge sort method splits the input array into three equal-sized

subarrays, what is the recurrence relation T(n) of this merge sort method?

(15 points)

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