Week1-9Order-of-Growth classifications

Common Order-of-Growth classifications

Small set of functions

$1, logN, N, NlogN, N^2, N^3, ab^N$

linearithmic and linear scale with the input size!!

Binary Search

Implementation

public static int binarySearch( int[] a, int key )
{
    int lo = 0,
        hi = a.length - 1;

   while( lo <= hi )
   {
       int mid = ( hi + lo ) / 2;

       if( key < a[mid] )
       {
            hi = mid - 1;
       }
       if( key > a[mid] )
       {
           lo = mid + 1;
       } 
       else return mid;
    }
    return -1;
}

Mathematical Analysis

Proposition

BS uses at most $1 + logN$ compares to search in a sorted array of size $N$

Def

$T(N) = \#$ compares to BS in a sorted subarray of size $\leq N$

BS in recurrence: $T(N) \leq T(\frac{N}{2}) + 1$ for $N > 1$ with $T(1) = 1$. Iterate and we will get the proof.

$T(N) \leq logN + 1$

An $N^2logN$ algorithm for 3-SUM

1. Sort the N distinct numbers
2. For each pair of numbers a[i] and a[j], binary search for -(a[i] + a[j])

Guiding Principle

Typically, better order of growth = faster in practice