QuickSelect算法理解

Question

我一直在研究各种教程和文章，讨论快速排序和快速选择，但我对它们的理解仍然不稳定。

鉴于这种代码结构，我需要能够掌握并解释quickselect的工作原理。

// return the kth smallest item
int quickSelect(int items[], int first, int last, int k) {
    int pivot = partition(items, first, last);
    if (k < pivot-first) {
        return quickSelect(items, first, pivot, k);
    } else if (k > pivot) {
        return quickSelect(items, pivot+1, last, k-pivot);
    } else {
        return items[k];
    }
}

我需要一些帮助来分解伪代码，虽然我没有提供分区功能代码，但我想了解在提供quickselect功能的情况下会做什么。

我知道quicksort如何运作，而不是快速选择。我刚刚提供的代码是一个关于如何格式化快速选择的例子。

编辑：更正的代码是

int quickSelect(int items[], int first, int last, int k) 
{
    int pivot = partition(items, first, last);
    if (k < pivot-first+1) 
    { //boundary was wrong
        return quickSelect(items, first, pivot, k);
    } 
    else if (k > pivot-first+1) 
    {//boundary was wrong
        return quickSelect(items, pivot+1, last, k-pivot);
    }
    else 
    {
        return items[pivot];//index was wrong
    }
}

Courtesty：@Haitao

Answer 1

快速选择的重要部分是分区。所以我先解释一下。

快速选择中的分区选择pivot（随机或第一个/最后一个元素）。然后它以一种方式重新排列列表，即所有小于pivot的元素都在枢轴的左侧，而其他元素在右侧。然后它返回pivot元素的索引。

现在我们在这里找到第k个最小的元素。分区案例后：

1. k == pivot。然后你已经找到了最小的第k个。这是因为分区的工作方式。确切k - 1个元素小于kth元素。
2. k < pivot。然后第k个最小值位于pivot的左侧。
3. k > pivot。然后第k个最小的位于枢轴的右侧。要找到它，你实际上必须在右边找到k-pivot最小的数字。

Answer 2

不过，您的代码有一些错误..

int quickSelect(int items[], int first, int last, int k) {
    int pivot = partition(items, first, last);
    if (k < pivot-first+1) { //boundary was wrong
        return quickSelect(items, first, pivot, k);
    } else if (k > pivot-first+1) {//boundary was wrong
        return quickSelect(items, pivot+1, last, k-pivot);
    } else {
        return items[pivot];//index was wrong
    }
}

Answer 3

分区非常简单：重新排列元素，使得小于选定数据透视表的数组在数组中的位数低于数据透视表，而大于数据透视表的数据位于数组中的较高索引处。

我在previous answer中讨论了其余内容。

Answer 4

int quickSelect(int A[], int l, int h,int k)
{
        int p = partition(A, l, h); 
        if(p==(k-1)) return A[p];
        else if(p>(k-1)) return quickSelect(A, l, p - 1,k);  
        else return quickSelect(A, p + 1, h,k);

}

//分区功能与QuickSort相同

Answer 5

int partition(vector<int> &vec, int left, int right, int pivotIndex)
{
        int pivot = vec[pivotIndex];
        int partitionIndex = left;

        swap(vec[pivotIndex],vec[right]);
        for(int i=left; i < right; i++) {
                if(vec[i]<pivot) {
                        swap(vec[i],vec[partitionIndex]);
                        partitionIndex++;
                }
        }
        swap(vec[partitionIndex], vec[right]);

        return partitionIndex;
}

int select(vector<int> &vec, int left, int right, int k)
{
        int pivotIndex;
        if (right == left) {
                return vec[left];
        }
        pivotIndex = left + rand() % (right-left);

        pivotIndex = partition(vec,left,right,pivotIndex);
        if (pivotIndex == k) {
                return vec[k];
        } else if(k<pivotIndex) {
                /*k is present on the left size of pivotIndex*/
                return partition(vec,left,pivotIndex-1, k);
        } else {
                /*k occurs on the right size of pivotIndex*/
                return partition(vec, pivotIndex+1, right, k);
        }
        return 0;
}

Answer 6

我正在阅读《 CLRS算法》一书以学习快速选择算法，我们可以轻松实现该算法。

package selection;    
import java.util.Random;

/**
 * This class will calculate and print Nth ascending order element
 * from an unsorted array in expected time complexity O(N), where N is the
 * number of elements in the array.
 * 
 * The important part of this algorithm the randomizedPartition() method.
 * 
 * @author kmandal
 *
 */
public class QuickSelect {
    public static void main(String[] args) {
        int[] A = { 7, 1, 2, 6, 0, 1, 96, -1, -100, 10000 };
        for (int i = 0; i < A.length; i++) {
            System.out.println("The " + i + "th ascending order element is "
                    + quickSelect(A, 0, A.length - 1, i));
        }
    }

    /**
     * Similar to Quick sort algorithm partitioning approach works, but after
     * that instead of recursively work on both halves here will be recursing
     * into desired half. This step ensure to the expected running time to be
     * O(N).
     * 
     * @param A
     * @param p
     * @param r
     * @param i
     * @return
     */
    private static int quickSelect(int[] A, int p, int r, int i) {
        if (p == r) {
            return A[p];
        }
        int partitionIndex = randomizedPartition(A, p, r);
        if (i == partitionIndex) {
            return A[i];
        } else if (i < partitionIndex) {// element is present in left side of
                                        // partition
            return quickSelect(A, p, partitionIndex - 1, i);
        } else {
            return quickSelect(A, partitionIndex + 1, r, i);// element is
                                                            // present in right
            // side of partition
        }
    }

    /**
     * 
     * Similar to Quick sort algorithm this method is randomly select pivot
     * element index. Then it swap the random pivot element index with right
     * most element. This random selection step is expecting to make the
     * partitioning balanced. Then in-place rearranging the array to make all
     * elements in the left side of the pivot element are less than pivot
     * element and the right side elements are equals or grater than the pivot
     * element. Finally return partition index.
     * 
     * @param A
     * @param p
     * @param r
     * @return
     */
    private static int randomizedPartition(int[] A, int p, int r) {
        int partitionIndex = p;
        int random = p + new Random().nextInt(r - p + 1);// select
                                                            // pseudo random
                                                            // element
        swap(A, random, r);// swap with right most element
        int pivot = A[r];// select the pivot element
        for (int i = p; i < A.length - 1; i++) {
            if (A[i] < pivot) {
                swap(A, i, partitionIndex);
                partitionIndex++;
            }
        }
        swap(A, partitionIndex, r);
        return partitionIndex;
    }

    /**
     * Swapping 2 elements in an array.
     * 
     * @param A
     * @param i
     * @param j
     */
    private static void swap(int[] A, int i, int j) {
        if (i != j && A[i] != A[j]) {
            int temp = A[i];
            A[i] = A[j];
            A[j] = temp;
        }
    }
}


Output:
The 0th ascending order element is -100
The 1th ascending order element is -1
The 2th ascending order element is 0
The 3th ascending order element is 1
The 4th ascending order element is 1
The 5th ascending order element is 2
The 6th ascending order element is 6
The 7th ascending order element is 7
The 8th ascending order element is 96
The 9th ascending order element is 10000

Answer 7

您可以使用3向分区找到quickSelect here。

代码是用Scala编写的。此外，我将在掌握主题的过程中添加更多算法和数据结构。

您还可以注意到，使用quickSelect实现了奇数个元素的数组的中值函数。

编辑：需要对阵列进行洗牌以保证线性平均运行时间