Minimize function in adjacent items of an array

Question

I have an array (arr) of elements, and a function (f) that takes 2 elements and returns a number.

I need a permutation of the array, such that f(arr[i], arr[i+1]) is as little as possible for each i in arr. (and it should loop, ie. it should also minimize f(arr[arr.length - 1], arr[0]))

Also, f works sort of like a distance, so f(a,b) == f(b,a)

I don't need the optimum solution if it's too inefficient, but one that works reasonable well and is fast since I need to calculate them pretty much in realtime (I don't know what to length of arr is, but I think it could be something around 30)

Answer 1

What does "such that f(arr[i], arr[i+1]) is as little as possible for each i in arr" mean? Do you want minimize the sum? Do you want to minimize the largest of those? Do you want to minimize f(arr[0],arr[1]) first, then among all solutions that minimize this, pick the one that minimizes f(arr[1],arr[2]), etc., and so on?

If you want to minimize the sum, this is exactly the Traveling Salesman Problem in its full generality (well, "metric TSP", maybe, if your f's indeed form a metric). There are clever optimizations to the naive solution that will give you the exact optimum and run in reasonable time for about n=30; you could use one of those, or one of the heuristics that give you approximations.

If you want to minimize the maximum, it is a simpler problem although still NP-hard: you can do binary search on the answer; for a particular value d, draw edges for pairs which have f(x,y)

If you want to minimize it lexiocographically, it's trivial: pick the pair with the shortest distance and put it as arr[0],arr[1], then pick arr[2] that is closest to arr[1], and so on.

Depending on where your f(,)s are coming from, this might be a much easier problem than TSP; it would be useful for you to mention that as well.

Answer 2

You're not entirely clear what you're optimizing - the sum of the f(a[i],a[i+1]) values, the max of them, or something else?

In any event, with your speed limitations, greedy is probably your best bet - pick an element to make a[0] (it doesn't matter which due to the wraparound), then choose each successive element a[i+1] to be the one that minimizes f(a[i],a[i+1]).

That's going to be O(n^2), but with 30 items, unless this is in an inner loop or something that will be fine. If your f() really is associative and commutative, then you might be able to do it in O(n log n). Clearly no faster by reduction to sorting.

Answer 3

I don't think the problem is well-defined in this form:

Let's instead define n fcns g_i : Perms -> Reals

g_i(p) = f(a^p[i], a^p[i+1]), and wrap around when i+1 > n

To say you want to minimize f over all permutations really implies you can pick a value of i and minimize g_i over all permutations, but for any p which minimizes g_i, a related but different permatation minimizes g_j (just conjugate the permutation). So therefore it makes no sense to speak minimizing f over permutations for each i.

Answer 4

Unless we know something more about the structure of f(x,y) this is an NP-hard problem. Given a graph G and any vertices x,y let f(x,y) be 1 if there is no edge and 0 if there is an edge. What the problem asks is an ordering of the vertices so that the maximum f(arr[i],arr[i+1]) value is minimized. Since for this function it can only be 0 or 1, returning a 0 is equivalent to finding a Hamiltonian path in G and 1 is saying that no such path exists.

The function would have to have some sort of structure that disallows this example for it to be tractable.