Proof for Add'em Up

I tried to solve this problem:

What I came up: Find and delete two minimum integers. Insert sum of found integers. Repeat it until the size of the list become 1.

I think this greedy algorithm works. However, I’m not sure how to (mathematically) prove this algorithm.

Any suggestions? Thanks.

The number oft additions is constant for a given set of integers.
So I would try induction…?

1 Like

The evaluation of the sum can be represented by a full (each internal node has two children) binary tree whose leaves are the numbers of the initial list (Xi)i and the internal nodes represent the operations. Let Di be the depth of the leaf Xi in the evaluation tree. It is not hard to see that the cost of the evaluation is Sumi XiDi.

Let us prove that there exists a minimal-cost tree in which the two (or two of the) smallest values leaves are children of the same node (at the maximum depth). Indeed, consider a minimal-cost tree and any deepest internal node, its two children must be leaves, say of values a and b and of depth D maximum in the tree. Consider any other leaf c, say of depth Dc ≤ D. Swapping a (or b, whatever) and c in the tree generates a cost difference of a(Dc-D) + c(D-Dc) = (c-a)(D-Dc). But if c ≤ a, then (c-a)(D-Dc) ≤ 0 and this implies (c-a)(D-Dc) = 0 (otherwise we would have found a tree of a lesser cost). Consequently if a and b are not yet the two smallest values, we can swap these values with a and b while keeping the cost minimal, hence building an optimal tree satisfying the property.

This proves that an optimal strategy can always start by adding the two smallest values: This gives you the greedy choice strategy to follow. Pick the two smallest values, replace them by their sum and repeat on the new set of values.

NB: For a complete formal proof, look for Huffman Coding on the internet: replacing the frequencies/probabilities of the letters by the input numbers should lead to the same problem.