Question

3. Use the randomized select algorithm based on partition to find the median of A 14,2, 12, 6, 13, 9, 15) 4. What is the worst case time complexity for the randomized select algo- rithm? What is its average case time? Why is its average case time much better than its worst case time?

please I need it urgent thanks algorithms please as soon as possible thanks in algorithms

Answer 1

3.

A={4, 2, 12, 6, 13, 9, 15}

Divide the list into sublists each of length five

A1={4, 2, 12, 6, 13} A2={9, 15}

Sort each list:

A1={2, 4, 6, 12, 13} A2={9, 15}

Then, get the median out of each list and put them in a list of medians, M:

M = {6, 12}

Pick the median from that list—since the length of the list is 2, and we determine the index of the median by the length of the list divided by two: we get   $\frac{2}{2} = 1$ , the index of the median is 1, and M[1] = 12

Use this as the pivot element and put all elements in A that are less than 76 to the left and all elements greater than 12 to the right:

A' = {4, 2, 6, 9, 12, 13, 15}

To find the third lowest score in A, find the index of 12, which is 4. How does 4 compare with 3? Since 4>3, we must recurse on the left half of the list A', which is {4, 2, 6, 9}.

9 is at index three.

So, 9 is the fourth smallest number of A.

4.

Worst Case Time Complexity - Selection algorithm in each step performs O(n) operations. But this 'n' keeps on reducing by half each time. There are totally log(N) steps. This makes it N + N/2 + N/4 + ... + 1 (log(N) times) = 2N = O(N)

Average Case Time Complexity - Average runtime is Θ(n).