Question

Here is an interesting programming puzzle I came across . Given an array of positive integers, and a number K. We need to find pairs(a,b) from the array such that a % b = K.

I have a naive O(n^2) solution to this where we can check for all pairs such that a%b=k. Works but inefficient. We can certainly do better than this can’t we ? Any efficient algorithms for the same? Oh and it’s NOT homework.

Answer 1

Sort your array and binary search or keep a hash table with the count of each value in your array.

For a number x, we can find the largest y such that x mod y = K as y = x - K. Binary search for this y or look it up in your hash and increment your count accordingly.

Now, this isn’t necessarily the only value that will work. For example, 8 mod 6 = 8 mod 3 = 2. We have:

x mod y = K => x = q*y + K =>
            => x = q(x - K) + K =>
            => x = 1(x - K) + K =>
            => x = 2(x - K)/2 + K =>
            => ...

This means you will have to test all divisors of y as well. You can find the divisors in O(sqrt y), giving you a total complexity of O(n log n sqrt(max_value)) if using binary search and O(n sqrt(max_value)) with a hash table (recommended especially if your numbers aren’t very large).