Question

Discrete math question

2. Consider to the following two algorithms procedure SortA(a1,a2, ..., an: a list of real numbers with n 2 2) 1, for j := 2 to n 2. i:= 1 3. while aj > ai 4. 5. m: 6. or k 0toj -i-1 7. i:-i+1 aj-k:aj-k-1 ai := m

Answer 1

comparisons are made in while loop.
This is an example of insertion sort.

number of comparisons for j=2 is 1
number of comparisons for j=3 is 2
number of comparisons for j=4 is 3
...
number of comparisons for j=n is n-1

total number of comparisons = 1+2+3+...+(n-1)
 $=\sum_{i=1}^{n-1}1$ 

$formula\;for\;sum\;of\;first\;n-1\;natural\;numbers\;is\;\frac{n(n-1)}{2}$

$so,\;total\;number\;of\;comparisons=\frac{n(n-1)}{2}$