Program Efficiency

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 thats where most work occurs
• Big-O notation
• O(f(n)) means work grows like about f(n)
• Constants dont matter

2
Big-O Example 1

• for(int i 0 i lt 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

• for(int i 0 i lt x.length i)
• for(int j 0 j lt x.length j)
• no loops here

• Its n work for each of n times thru outer loop
• So, this is O(n2)

4
Big-O Example 3

• for(int i 0 i lt x.length i)
• for(int j i 1 j lt 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

• for(int i 1 i lt n i)
• for(int j 1 j lt n j)
• for(int k n k gt 1 k--)
• no loops here

• n ? n ?n work, so this is O(n3)

7
Big-O Example 5

• for(int i 0 i lt n i)
• for(int j 0 j lt i i j)
• no loops here

• 02 12 22 n2 n(n1)(2n1)/6
• So, O(n3)

8
Big-O Example 6

• for(int i n i gt 0 i - 2)
• no loops here

• About n/2 iterations
• So, O(n)

9
Big-O Example 7

• for(int i 0 i lt n i)
• for(int j i j gt 0 j / 2)
• no loops here

• 0 1 2 2 3 3 4 4 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)