Question

Given a list of n objects, write a function that outputs the minimum set of numbers that sum to at least K. FOLLOW UP: can you beat O(n ln n)?

The minimum set will be a set with 1 element. Don’t we just have to traverse the array and find an element i.e. >= K.

Otherwise for O(nlgn), I understand we have to first sort the array and then we can find pair or triplets which sum >=k.
What if we don’t find such a combination and have to go for bigger sets won’t this problem be same as N sum problem?

Answer 1

Here’s a linear algorithm that uses linear-time median finding as a subroutine:

Findsum(A, K) {
  Let n be the length of A.
  Let M be the median element of A, found in linear time.
  Let L be the elements of A less than M.
  Let U be the elements of A greater than M.
  Let E be the elements of A equal to M.
  If the sum of the elements in U is at least K,
    Return Findsum(U, K).
  Else, if the sum of the elements in U and E is at least K,
    Return U together with enough elements of E that the sum is at least K.
  Else,
    Return Findsum(L, K - sum(U) - sum(E)).
}

Each recursive call is done on a list at most half the size of A and all other steps take at most linear time, so this algorithm takes linear time overall.