Given an array of size N, find the smallest positive integer value that cannot be represented as sum of some elements from the array.
Example 1:
Input:
N = 6
array[] = {1, 10, 3, 11, 6, 15}
Output:
2
Explanation:
2 is the smallest integer value that cannot
be represented as sum of elements from the array.
how to approach this question
A Simple Solution is to start from value 1 and check all values one by one if they can sum to values in the given array. This solution is very inefficient as it reduces to subset sum problem which is a well known NP Complete Problem
We can solve this problem in O(n) time using a simple loop. Let the input array be arr[0…n-1]. We initialize the result as 1 (smallest possible outcome) and traverse the given array. Let the smallest element that cannot be represented by elements at indexes from 0 to (i-1) be ‘res’, there are following two possibilities when we consider element at index i:
1) We decide that ‘res’ is the final result : If arr[i] is greater than ‘res’, then we found the gap which is ‘res’ because the elements after arr[i] are also going to be greater than ‘res’.
2) The value of ‘res’ is incremented after considering arr[i] : The value of ‘res’ is incremented by arr[i] (why? If elements from 0 to (i-1) can represent 1 to ‘res-1’, then elements from 0 to i can represent from 1 to ‘res + arr[i] – 1’ be adding ‘arr[i]’ to all subsets that represent 1 to ‘res’)