Question

1. Give the big-O characterization of the following loops, in terms of parameter n, and justify your answer: a) for (int i=1; i<=n, i++) {for (int j=1; j<=n; j++) {a constant-time operation}} b) for (int i=1;i<=n, i++) {for (int i=1; j<=i; j++) {a constant-time operation}} c) for (int i=1;i<=n*n, i++) {for (int j=1; j<=n; j++) {a constant-time operation }} d) for (int i=1; i<=n*n, i++) {for (int j=1; j<=i; j++) {a constant-time operation }} e) for (int i=1; i<=n, i++) {for (int j=1; j<=n*n; j++) {a constant-time operation}}

Answer 1

a)
in the worst case
i iterates from 1 to n and for each value of i, j iterates from 1 to n
so i =O(n) j= O(n)
overall = O(n^2)
b)
in the worst case
i iterates from 1 to n and in worst case j iterates from 1 to n
so i =O(n) j= O(n)
overall = O(n^2)
c)
in the worst case
i iterates from 1 to n^2 and for each value of i, j iterates from 1 to n
so i =O(n^2) j= O(n)
overall = O(n^3)
d)
in the worst case
i iterates from 1 to n^2 and in worst case j iterates from 1 to n
so i =O(n^2) j= O(n)
overall = O(n^3)
e)
in the worst case
i iterates from 1 to n and for each value of i, j iterates from 1 to n^2
so i =O(n) j= O(n^2)
overall = O(n^3)