Writing $$$O(N^2)$$$ is enough. (as it includes both $$$O(NlogN)$$$ and $$$O(N)$$$)

Given an array of $$$N$$$ integers, is it possible to partition them into two subsets $$$S_1$$$ and $$$S_2$$$ such that the sum of the elements in $$$S_1$$$ equals the sum of the elements in $$$S_2$$$? This is a well-known NP-complete problem (https://en.wikipedia.org/wiki/Partition_problem), meaning that we highly doubt that a polynomial-time solution for it exists.

However, we can reduce this to your problem. Suppose we had a polynomial-time algorithm which solved your problem. Set $$$K=\sum a_i/2$$$, run your algorithm, then check if $$$X = 2$$$; this solves the partition problem in polynomial time.

Thus, if you have found a polynomial-time solution to your problem, then you would have also found a polynomial time solution for the partition problem, and solved one of the Millennium Problems.

So, yeah, I doubt an $$$\mathcal{O}(n^2)$$$ algorithm exists.