Asymptotic Analysis

Big-oh notation

definition  Let f(n) and g(n) be functions defined for nonnegative integers.  We say that f(n) is O(g(n)) if there is some integer n₀ and some constant C such that for all n > n₀,  | f(n) | <= Cg(n).

example  Let f(n) = 3n + 7.  Show that f(n) is O(n).

example  Let f(n) = 3n³ + 2n² + n + 1.  Show that f(n) is O(n³).



Big-oh notation is used to classify algorithms based on their running times.  

O(1) < O(log n) < O(n) < O(n log n) < O(n²) < O(n³) < ... < O(2ⁿ)

What are the big-oh running times for the sum and size programs we looked at earlier?

Some rules

If f(n) is O(c g(n)) for some constant c, then f(n) is O(g(n)).

If f₁(n) is O(g₁(n)) and f₂(n) is O(g₂(n)), then f₁(n)+f₂(n) is O(max(g₁(n), g₂(n))).

Some examples

The following program uses a selection sort to sort an array of integers:

void sort(int[] array){
    int n = array.length;

    for(pick = 0; pick<n-1; pick++){

// find the smallest element among array[pick], array[pick+1],...,array[n-1]
    imin = pick;
    for(i=pick+1;i<n;i++)
        if(array[i]<array[imin])
            imin=i;

// swap the smallest element with the pick element
    temp = array[pick];
    array[pick] = array[imin];
    array[imin] = temp;

    } // end of for(pick) loop

}

Analyze its running time.


The following program returns the nth number in the Fibonacci sequence  1,1,2,3,5,8,13,21,....

int fibonacci(int n){
  if(n==1 || n==2)
    return 1;
  else
    return fibonacci(n-1) + fibonacci(n-2);
}

Analyze its running time.

Here's an alternative way to compute the same values:

int fibonacci(int n,int f1,int f2){
  if(n==1)
    return f1;
  else if(n==2)
    return f2;
  else
    return fibonacci(n-1,f2,f1+f2);
}

Analyze its running time.