Question

There are two integer sequences A[] and B[] of length N,both unsorted.

Requirement: through the swapping of elements between A[] and B[]( can randomly exchange, not with same index), make the difference between {the sum of all elements in A[]} and {the sum of all elements in B[]} to be minimum.

PS: actually,it is an interview question I encountered.

Many thanks

Answer 1

This is going to be NP-hard! I believe you can do a reduction from Subset Sum to this.

As per BlueRaja/polygene’s comments, I will try to provide a full reduction from Subset Sum.

Here is a reduction:

Subset Sum problem: Given integers x₁, x₂, …, x_n, is there some non-empty subset which sums to zero?

Our problem: Given two integer arrays of size k, find the minimum possible difference of the sum of the two arrays, assuming we can shuffle around the integers in the arrays, treating both arrays as one array.

Say we had a polynomial time algo for our problem.

Say now you are given integers T = {x₁,x₂, …,x_n} (multiset)

Let S_i = x₁ + x₂ + …+ x_n + x_i.

Let T_i = {x₁, x₂, …, x_i-1, x_i+1, …, x_n } ( = T – x_i)

Define

A_i = Array formed using T_i

B_i = [S_i, 0, …, 0] (i.e one element is S_i and rest are zeroes).

Let m_i = the min difference found by our problem for arrays A_i and B_i

(we run our problem n times).

Claim: Some non-empty subset of T sums to zero if and only if, there is some i, for which m_i = 0.

Proof: (wlog) say x₁ + x₂ + .. + x_k = 0

Then

A = [x_k+1, …, x_n, 0, …0]

B = [x₂, x₃, …, x_k, S₁, 0, ..0]

gives the minimum difference m₁ to be |x₂ + .. + x_k + (x₁ + … + x_n) + x₁ – (x_k+1 + .. + x_n)| = |2(x₁+ x₂ + .. x_k)| = 0.

Similarly the if part can be proved.

In fact, this actually also follows (more easily) from Partition too: just create new array with all zeroes.

Hoepfully I haven’t made any mistakes.