Finding pairs with product greater than sum

Question

Given as input, a sorted array of floats, I need to find the total number of pairs (i,j) such as A[i]*A[j]>=A[i]+A[j] for each i < j

Answer 1

Here's a binary search, O(n log n):

There's a breaking point for each number at A*B = A+B. You can reduce this to B = A / (A - 1). All numbers on one side or the other will fit it. It doesn't matter if there are negative numbers, etc.

• If A < 1, then all numbers <= B fit.

• If A > 1, then all numbers >= B fit.

• If A == 1, then there is no match(divide by zero).

(Wolfram Alpha link)

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So some pseudocode:

loop through i
    a = A[i]
    if(a == 1)
        continue
    if(a >= 2)
        count += A.length - i 
        continue

    j = binsearch(a / (a-1))

    if(j <= i)
        continue

    if(a < 1)
        count += j-i
    if(a > 1)
        count += A.length - j