Merge Sort - 归并排序

核心：将两个有序对数组归并成一个更大的有序数组。通常做法为递归排序，并将两个不同的有序数组归并到第三个数组中。

先来看看动图，归并排序是一种典型的分治应用。

Python

#!/usr/bin/env python


class Sort:
    def mergeSort(self, alist):
        if len(alist) <= 1:
            return alist

        mid = len(alist) / 2
        left = self.mergeSort(alist[:mid])
        print("left = " + str(left))
        right = self.mergeSort(alist[mid:])
        print("right = " + str(right))
        return self.mergeSortedArray(left, right)

    #@param A and B: sorted integer array A and B.
    #@return: A new sorted integer array
    def mergeSortedArray(self, A, B):
        sortedArray = []
        l = 0
        r = 0
        while l < len(A) and r < len(B):
            if A[l] < B[r]:
                sortedArray.append(A[l])
                l += 1
            else:
                sortedArray.append(B[r])
                r += 1
        sortedArray += A[l:]
        sortedArray += B[r:]

        return sortedArray

unsortedArray = [6, 5, 3, 1, 8, 7, 2, 4]
merge_sort = Sort()
print(merge_sort.mergeSort(unsortedArray))

原地归并

Java

public class MergeSort {
    public static void main(String[] args) {
        int unsortedArray[] = new int[]{6, 5, 3, 1, 8, 7, 2, 4};
        mergeSort(unsortedArray);
        System.out.println("After sort: ");
        for (int item : unsortedArray) {
            System.out.print(item + " ");
        }
    }

    private static void merge(int[] array, int low, int mid, int high) {
        int[] helper = new int[array.length];
        // copy array to helper
        for (int k = low; k <= high; k++) {
            helper[k] = array[k];
        }
        // merge array[low...mid] and array[mid + 1...high]
        int i = low, j = mid + 1;
        for (int k = low; k <= high; k++) {
            // k means current location
            if (i > mid) {
            // no item in left part
                array[k] = helper[j];
                j++;
            } else if (j > high) {
            // no item in right part
                array[k] = helper[i];
                i++;
            } else if (helper[i] > helper[j]) {
            // get smaller item in the right side
                array[k] = helper[j];
                j++;
            } else {
            // get smaller item in the left side
                array[k] = helper[i];
                i++;
            }
        }
    }

    public static void sort(int[] array, int low, int high) {
        if (high <= low) return;
        int mid = low + (high - low) / 2;
        sort(array, low, mid);
        sort(array, mid + 1, high);
        merge(array, low, mid, high);
        for (int item : array) {
            System.out.print(item + " ");
        }
        System.out.println();
    }

    public static void mergeSort(int[] array) {
        sort(array, 0, array.length - 1);
    }
}

时间复杂度为 $O(N \log N)$, 使用了等长的辅助数组，空间复杂度为 $O(N)$。

Reference

• Mergesort - Robert Sedgewick 的大作，非常清晰。