求解划分问题的递归回溯算法

Question

嘿，我正在寻求一些帮助来找到一种算法，该算法将正数数组分为 k 个部分，以便每个部分具有（大约）相同的总和......假设我们有

1,2,3,4 ,5,6,7,8,9 en k=3 那么算法应该像这样划分它 1,2,3,4,5|6,7|8,9 元素的顺序不能改变...找到一个贪婪算法很容易，但我正在寻找一个总是返回最佳解决方案的回溯版本...

有人有任何提示吗？

Answer 1

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

令 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。它是次指数的，因为这不相当于一般的 NP 完全划分问题，在这里您正在寻找线性数组的连续段而不是给定总集的任意子集。

Answer 2

你所说的最优解是什么意思？我相信你的意思是最小化每个分区到最佳分区的距离之和。最佳分区是其元素之和等于总和除分区数的分区。

如果您不介意效率，那么也许这种粗略的方法对您来说已经足够了。我还没有测试该算法来检查其正确性，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 3

这是 Javascript 中的递归算法。此函数返回每个工人将分配的总数。假设输入数组 bookLoads 是一个正数数组，您希望将其尽可能公平地划分为 k 个部分（假设在 k 个工作人员中）

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}