最小化阵列的相邻项目功能 [英] Minimize function in adjacent items of an array

问题描述

我有一个数组（改编）的元素，和函数（ F ）是需要2元和返回一个数字。

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

我需要阵列的排列，使得 F（ARR [我]，编曲[I + 1]）是尽可能少的每个我在改编。 （它应该循环，也就是说，它也应该尽量减少 F（ARR [arr.length - 1]，ARR [0]））

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]))

此外， F 的作品有点像的距离，所以 F（A，B）= = F（B，A）

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

我并不需要的最佳解决方案，如果它的效率太低，而是一个合理的工作很好，速度很快，因为我需要计算它们pretty的很多实时（我不知道该怎么<$的长度C $ C>改编是，但我认为这可能是一些围绕30）

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)

推荐答案

什么是使得f（ARR [我]，编曲[I + 1]）是尽可能少的每一个我在改编是什么意思？你想减少的之的？你希望尽量减少最大的那些？是否要最大限度地减少F（改编[0]，改编[1]），然后再是最大限度地减少这种所有解决方案中，挑选一个最小化F（改编[1]，ARR [2]）等，等对？

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?

如果你想减少的之的，这是的完全的在其全部一般性旅行商问题（当然，指标TSP，也许，如果你的F公司的确形成一个度量）。有聪明的优化，以天真的解决方案，它会给你确切的最佳以及有关运行在合理的时间n = 30;你可以使用其中的一个，或者说给你近似的启发之一。

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.

如果你想减少的最大的，这是一个简单的问题，虽然仍然NP难：你可以做的答案二进制搜索;对于特定的值d，绘制边缘的量有f对（X，Y）

如果你希望尽量减少它的 lexiocographically 的，这是微不足道的：挑对的最短距离，并把它作为ARR [0]，编曲[1]，然后选择ARR [2]这是最接近给Arr [1]，等等。

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.

根据在您的F（，）s是未来，这可能是比TSP更简单的问题;这将是对您有用提到这一点。

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.

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