Question

I’ve come across this problem in a programming contest site and been trying different things for a few days but none of them seem to be efficient enough.

Here is the question: You are given a large array of integers and a number k. The goal is to divide the array into subarrays each containing no more than k elements, such that the sum of all the elements in all the sub arrays is maximal. Another condition is that none of these sub arrays can be adjacent to each other. In other words, we have to drop a few terms from the original array.

Its been bugging me for a while and would like to hear your perspective on approaching this problem.

Answer 1

Dynamic programming should do the trick. Short explanation why:

The key property of a problem susceptible to dynamic programming is that the optimal solution to the problem (here: the whole array) can always be expressed as composition of two optimal solutions to subproblems (here: two subarrays.) Not every split needs to have this property – it is sufficient for one such split to exist for any optimal solution.

Clearly if you split the optimal solution between arrays (on an element that has been dropped), then the subsolutions are optimal within both subarrays.

The algorithm:

Try every element of the array in turn as the splitting element, looking for the one that yields the best result. Solve the problem recursively for both parts of the array (the recursion stops when the subarray is no longer than k). Memoize solutions to avoid exponential time (the recursion will obviously try the same subarray many times.)