Data Structures Exam 1 - Big O Notation

Polynomial Time v Non-Polynomial Time

-If something runs in Polynomial time the Big O notation is n^x (e.g. n, n^2, n^3, etc.) -If something runs in Non-Polynomial time the Big O notation is NOT of the form n^x.

Deterministic vs. Non-Deterministic

-Polynomial time is deterministic -Non-Polynomial time is non-deterministic

0! = ?

1

True or False. If: f(n) = 3n^2 + 10 n log n Then O(n) = n log(n)

False

True or False. If: f(n) = n log n + n/2 Then O(n) = n

False

Order of O(n) functions (listed in increasing order with fastest/smallest first)

O(1): Constant O(lg n) O(n): Linear O(n lg n) O(n^2): quadratic O(n^3): cubic O(2^n): exponential O(n!): factorial

Big-O Notation of: f(n) = n^2+100n+log(n)+1000

O(f(n))=n^2

Big-O Notation of: for (i = 1; i < n; i++) { for (j = 1; j < n; j++) { //statements } }

O(n)=n^2

Finding Big-O of nested for/while loops (to n).

cn^x, where x is the number of loops and and c is the number of constant time operations contained within the innermost loop.

Big-O Notation of: for (i = sum = 0; i < n; i++) sum += a[i];

f(n)=2+2n O(f(n))=2n=n

Big-O Notation of: a = 1 b = 3 c = 3a

f(n)=3 O(n)=1

Big-O Notation of: for (i=0; i < n; i++) { for (j = 0; i < n; j++) { // statements } } for (k = 0; k < n; k++) { // statements }

f(n)=n^2+n O(n)=n^2