Sorting

Consider an array $A = [10, 7, 8, 19, 41, 35, 25, 31]$. Suppose the merge sort algorithm is executed on array $A$ to sort it in increasing order. The merge sort algorithm will carry out a total of 7 merge operations. A merge operation on sorted left array $L$ and sorted right array $R$ is said to be void if the output of the merge operation is the elements of array $L$ followed by the elements of array $R$. The number of void merge operations among these 7 merge operations is __________. (*answer in integer*)

Consider the following snippet of a C program. Assume that swap(&x, &y) exchanges the contents of x and y.

int main() {
      int array[] = {3, 5, 1, 4, 6, 2};
      int done = 0;
      int i;

      while (done == 0) {
            done = 1;
            for (i=0; i<=4; i++) {
                  if (array[i] < array[i+1]) {
                        swap(&array[i], &array[i+1]);
                        done = 0;
                  }
            }
            for (i=5; i>=1; i--) {
                  if (array[i] > array[i-1]) {
                        swap(&array[i], &array[i-1]);
                        done = 0;
                  }
            }
      }
      printf("%d",array[3]);
}

The output of the program is ________.

Assume that the algorithms considered here sort the input sequences in ascending order. If the input is already in ascending order, which of the following are TRUE? I. Quicksort runs in $\Theta(n^2)$ time II. Bubblesort runs in $\Theta(n^2)$ time III. Mergesort runs in $\Theta(n)$ time IV. Insertion sort runs in $\Theta(n)$ time

Suppose you are provided with the following function declaration in the C programming language.

int partition(int a[], int n);

The function treats the first element of a[] as a pivot, and rearranges the array so that all elements less than or equal to the pivot is in the left part of the array, and all elements greater than the pivot is in the right part. In addition, it moves the pivot so that the pivot is the last element of the left part. The return value is the number of elements in the left part. The following partially given function in the C programming language is used to find the $k^{th}$ smallest element in an array a[] of size n using the partition function. We assume $k \leq n$.

int kth_smallest(int a[], int n, int k)
{
    int left_end = partition(a,n);

    if ( left_end+1 == k ){
        return a[left_end];
    }

    if ( left_end+1 > k ){
        return kth_smallest( ____________ );
    } else {
        return kth_smallest( ____________ );
    }
}

The missing argument lists are respectively