Question

嘿，我正在寻找一些帮助来找到一个算法，将一个正数组分成k部分，这样每个部分都有（大约）相同的总和......让我们说我们有

1,2,3,4,5,6,7,8,9 zh k = 3算法应该像这样分区1,2,3,4,5 | 6,7 | 8,9 元素的顺序无法改变...找到一个贪婪的算法很容易，但我正在寻找一个总是返回最佳解决方案的回溯版本......

Annyone有任何提示吗？

Answer 1

最佳解决方案是什么意思？我相信你的意思是最小化每个分区距离到最佳分区的总和。最佳分区是它的元素总和等于总和除以分区数的分区。

如果你不介意效率，那么这种粗略的方法对你来说已经足够了。我没有测试算法来检查它的正确性，所以要小心。

void FindPartitions(int[] numbers, int i, IList<int>[] partitions, int currentPartition, IList<int>[] bestSolution, ref int minDistance)
{
    if (i == numbers.Length)
    {
        int sum = numbers.Sum();
        int avg = sum / partitions.Length;
        int currentDistance = 0;
        foreach (var partition in partitions)
            currentDistance += Math.Abs(partition.Sum() - avg);
        if (currentDistance < minDistance)
        {
            minDistance = currentDistance;
            for (int j = 0; j < partitions.Length; j++)
                bestSolution[j] = new List<int>(partitions[j]);
        }
    }
    else
    {
        partitions[currentPartition].Add(numbers[i]);
        FindPartitions(numbers, i + 1, partitions, currentPartition, bestSolution, ref minDistance);
        partitions[currentPartition].RemoveAt(partitions[currentPartition].Count - 1);
        if (currentPartition < partitions.Length - 1)
            FindPartitions(numbers, i, partitions, currentPartition + 1, bestSolution, ref minDistance);
    }
}

Answer 2

这是一个不使用任何动态数据结构（如列表）的解决方案。它们完全没有必要，实际上会使算法慢得多。

设K为此处的分区数，N为数组中的元素数。

int start[K];

void register() {
   /* calculate the error between minimum and maximum value partitions */
   /* partition boundaries are start[0], start[1], start[2], ... */
   /* if lower error than previously stored, remember the best solution */
}

void rec(int s, int k) {
  if (k == K) register();
  for (int i = s; i < N; i++) {
    start[k] = i;
    rec(i + 1, k + 1);
  }
}

/* entry */
start[0] = 0;
rec(1, 1);
/* then output the best solution found at register() */

注意：这是一个O（n ^K）算法。它是次指数因为这是不等同于一般的NP完全分区问题，你在这里寻找线性数组的连续段而不是给定总集的任意子集。

Answer 3

这是Javascript中的递归算法。此函数返回将分配每个工作程序的总计。让我们说输入数组bookLoads是一个正数的数组，你想尽可能公平地分成k部分（让我们说k工作者）

这是一个工作小提琴： https://jsfiddle.net/missyalyssi/jhtk8vnc/3/

function fairWork(bookLoads, numWorkers = 0) {
    // recursive version

    // utilities
    var bestDifference = Infinity;
    var bestPartition = {};
    var initLoads = {};
    var workers = Array.from({length: numWorkers}, (val, idx) => {
      initLoads[idx] = 0;
      return idx;
    });
    var bookTotal = bookLoads.reduce((acc, curr) => {return acc + curr}, 0); 
    var average = bookTotal / bookLoads.length;

    // recursive function
    function partition(books = bookLoads, workers, loads={}) {

      // if only one worker give them all the books
      if (workers.length == 1 && books.length > 0) {
        var onlyWorker = workers[0];
        loads[onlyWorker] += books.reduce((acum, curr, idx, arr) => {
                               return acum + curr;
                             },0);
        books = [];
      }

      // base case
      if (books.length == 0) {
        var keys = Object.keys(loads);
        var total = 0;
        for (let key = 0; key < keys.length; key++) {
          // square so that difference shows 
          total += Math.pow(Math.abs(average - loads[keys[key]]), 2);
        }
        if (total < bestDifference) {
          bestDifference = total;
          bestPartition= loads;
        }
        return bestPartition;
      }

      var currBookLoad = books[0];

      // add book to curr worker 1
      var newWorker1Loads = Object.assign({}, loads);
      var worker1 = workers[0]; 
      newWorker1Loads[worker1] = newWorker1Loads[worker1] + currBookLoad || currBookLoad;
      partition(books.slice(1), workers, newWorker1Loads)

      // give to next worker
      var newNextWorkerLoads = Object.assign({}, loads);
      var worker2 = workers[1]; 
      newNextWorkerLoads[worker2] = newNextWorkerLoads[worker2] + currBookLoad || currBookLoad;
      partition(books.slice(1), workers.slice(1), newNextWorkerLoads)

      return bestPartition;
    }
    // function call
    return partition(bookLoads, workers, initLoads)
}
fairWork([3,1,2,3], 3)
//Result will be: Object {0: 3, 1: 3, 2: 3}
fairWork([1,2,3,4,5,6,7,8,9], 3)
//Result will be: Object {0: 15, 1: 13, 2: 17}

用于解决分区问题的递归回溯算法

3 个答案: