1. Program Efficiency Interested in “order of magnitude”
That is, how the work grows as n grows
Where n is the “problem size”
Usually we just look at the loops
Since that’s where most work occurs
Big-O notation
O(f(n)) means work grows like about f(n)
Constants don’t matter

2. Big-O Example 1 We say this is O(n)
for(int i = 0; i < x.length; i++) {
no loops here }
We say this is O(n)
Since it takes about x.length “work”
Depends on problem “size”
Size is always given as n

3. Big-O Example 2 It’s n “work” for each of n times thru outer loop
for(int i = 0; i < x.length; i++) {
for(int j = 0; j < x.length; j++) {
no loops here }
It’s n “work” for each of n times thru outer loop
So, this is O(n2)

4. Big-O Example 3 When i = 0, we have n - 1 “work”
for(int i = 0; i < x.length; i++) {
for(int j = i + 1; j < x.length; j++) {
no loops here }
When i = 0, we have n - 1 “work”
When i = 1, we have n - 2 “work”, etc.
So, (n - 1)+(n-2)+ … + 1 = n(n-1)/2, or O(n2)

5. Program Efficiency Big-O notation
O(1) constant time
O(n) linear time
O(n log n) log linear
O(n2) quadratic
O(n3) cubic
O(2n) exponential
O(n!) factorial
Polynomials are good, exponentiala are bad

6. Big-O Example 4 n  n n “work”, so this is O(n3)
for(int i = 1; i <= n; i++) {
for(int j = 1; j <= n; j++) {
for(int k = n; k >= 1; k--) {
no loops here }
n  n n “work”, so this is O(n3)

7. Big-O Example 5 02 + 12 + 22 + … + n2 = n(n+1)(2n+1)/6 So, O(n3)
for(int i = 0; i <= n; i++) {
for(int j = 0; j < i * i; j++) {
no loops here }
… + n2 = n(n+1)(2n+1)/6
So, O(n3)

8. Big-O Example 6 About n/2 iterations So, O(n)
for(int i = n; i >= 0; i -= 2) {
no loops here }
About n/2 iterations
So, O(n)

9. Big-O Example 7 0 + 1 + 2 + 2 + 3 + 3 + 4 + 4 + … + log n + log n
for(int i = 0; i < n; i++) {
for(int j = i; j > 0; j /= 2) {
no loops here }
… + log n + log n
Adds up to about (log n)((log n) + 1)
Where “log” is to the base 2
This is O((log n)2)