Question

Given an integer x and a sorted array a of N distinct integers, design a linear-time algorithm to determine if there exists two distinct indices i and j such that a[i] + a[j] == x

Answer 1

This is type of Subset sum problem

Here is my solution. I don’t know if it was known earlier or not. Imagine 3D plot of function of two variables i and j:

sum(i,j) = a[i]+a[j]

For every i there is such j that a[i]+a[j] is closest to x. All these (i,j) pairs form closest-to-x line. We just need to walk along this line and look for a[i]+a[j] == x:

 int i = 0; 
 int j = lower_bound(a.begin(), a.end(), x)  -  a.begin(); 
 while (j >= 0 && j < a.size()  &&  i < a.size())  {  
    int sum = a[i]+a[j]; 
    if (sum == x)   { 
        cout << "found: " << i << " " << j << endl;  
        return;
    }
    if (sum > x)    j--;
    else            i++;
    if (i > j) break;
 }  
 cout << " not found\n";

Complexity: O(n)