## (solution) Please answer each question with as much detail as possible.

Consider the following functions for problems 1 and 2.

void selectionSort(int array)

{

sort(array, 0);

}

void sort(int array, int i)

{

if (i &lt; array.length - 1)

{

int j = smallest(array, i);

int temp = array[i];

array[i] = array[j];

array[j] = temp;

sort(array, i + 1);

}

}

int smallest(int array, int j)

{

if (j == array.length - 1)

return array.length - 1;

int k = smallest(array, j + 1);

return array[j] &lt; array[k] ? j : k;

} 1. Draw the recursion tree for selectionSort when it is called for an array of length 4.

Show the activations of selectionSort, sort and smallest in the tree.

2. Determine a formula that counts the numbers of nodes in the recursion tree. What is Big for execution time? Determine a formula that expresses the height of the tree. What is

the Big- for memory?

For problems 3 and 4, consider the following functions that implement the dequeue

operation for a priority queue that is implemented with a heap.

int pQueue;

int length;

int dequeue()

{

int node = 1;

int value = pQueue[--length];

int maxValue = pQueue[node];

int location = sift(node * 2, value);

pQueue[location] = value;

return maxValue;

}

int sift(int node, int value)

{ if (node &lt;= length)

{

if (node &lt; length &amp;&amp; pQueue[node] &lt; pQueue[node + 1])

node++;

if (value &lt; pQueue[node])

{

pQueue[node / 2] = pQueue[node];

return sift(node * 2, value);

}

}

return node / 2;

} 3. Write the recurrence equation and initial conditions that expresses the execution time cost

for the sift function in the above algorithm. Draw the recursion tree assuming that n = 8.

Assume that n represents the number of nodes in the subtree at each step of the sifting

operation.

4. Determine the critical exponent for the recurrence equation in problem 3. Apply the Little

Master Theorem to solve that equation specifically stating which part of the theorem is

applied to arrive at the solution.

Solution details:

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Sep 13, 2020

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