Disclaimer

I do not know if this technique has another name. I came up with it to solve specific types of problems but it probably has been done by many others before.

The situation

Assume we are given a sequence of numbers a_{1},a_{2},...,a_{n-1},a_{n} and we have a function f that is well defined for all sequences of numbers with the property f(b_{1},...,b_{m}) = f(f(b_{1},...,b_{m-1}),b_{m}) for any sequence b and the range of f has size K, which is relatively small. Our goal is to rearrange the numbers a_{1},..,a_{n} so that when the sequence is entered into f in this order the value of f is optimised (minimised or maximised).

The solution

If we try to brute force all permutations of the sequence there are n! options which is usually too slow. However what we can do is use dynamic programming and for each sequence ending in a_{j} with m terms store all possible values of f. Since we know the range of f is K at each step we are storing at most K terms.

Cost analysis

To calculate all possible values of f for a sequence ending in a_{j} with m+1 terms given we have calculated all possible values for f with sequence with m terms we need at most (n-1)K calculations. Doing this for each term to calculate all possible values for f with a sequence of m+1 terms gives at most n(n-1)K calculations. Doing it n times since we need sequences of length n gives at most n^2(n-1)K. Finally to find the optimised value requires at most nK more calculations. So at worst we have n^2(n-1)K+nK calculations.

Noticing the situation

The nature of the function is what determines whether this situation occurs. If the function is based on bitwise operations and the values of the sequence are small there is a good chance that this technique applies. Also if the function is based on arithmetic modulo N where N is small this technique might also apply.