归并排序

归并排序

归并的排序的核心在于合并，递归到底一个数字自然有序

python

def merge(A, p, q, r):
    L = A[p:q+1]
    R = A[q+1:r+1]
    L.append(float("inf"))
    R.append(float("inf"))
    i = 0
    j = 0
    k = p
    while k <= r:
        if L[i] <= R[j]:
            A[k] = L[i]
            i = i + 1
            k = k + 1
        else:
            A[k] = R[j]
            j = j + 1
            k = k + 1
def merge_sort(A, p, r):
    if p < r:
        q = (p + r) // 2
        merge_sort(A,p,q)
        merge_sort(A,q+1,r)
        merge(A,p,q,r)

if __name__ == "__main__":
    A = [5, 2, 4, 7, 1, 3, 2, 6]
    print("排序前", A)
    merge_sort(A, 0, 7)
    print("排序后", A)

c++

#include <iostream>
#include <limits.h>
using namespace std;

void merge(int A[], int p, int q, int r)
{
    int n1 = q - p + 1;
    int n2 = r - q;
    int L[n1+1];
    int R[n2+1];
    L[n1] = INT_MAX;
    R[n2] = INT_MAX;
    for (int i = 0; i < n1; i++)
    {
        L[i] = A[p+i];
    }
    for (int i = 0; i < n2; i++)
    {
        R[i] = A[q+i+1];
    }
    int i = 0;
    int j = 0;
    for (int k = p; k <= r; k++)
    {
        if (L[i] <= R[j])
        {
            A[k] = L[i];
            i = i + 1;
        }
        else
        {
            A[k] = R[j];
            j = j + 1;
        }
    }
}

void merge_sort(int A[], int p, int r)
{
    if (p < r)
    {
        int q = (p + r) / 2;
        merge_sort(A, p, q);
        merge_sort(A, q+1, r);
        merge(A, p, q, r);
    }
}

int main()
{
    int A[] = {5, 2, 4, 7, 1, 3, 2, 6};
    cout << "排序前 ";
    for(int i = 0; i < 8; i++)
    {
        cout << A[i] << " ";
    }
    cout << endl;
    merge_sort(A, 0, 7);
    cout << "排序后 ";
    for(int i = 0; i < 8; i++)
    {
        cout << A[i] << " ";
    }
    return 0;
}

算法分析

合并的时间复杂度为 $\theta(n)$
归并时间复杂度 $T(n)=\begin{cases}
\theta(1) & n=1 \
2T(n/2) + \theta(n) & n>1
\end{cases}$
通过求解递归树，每一层为$cn$, 共$nlgn+1$层，故时间复杂度为$\theta(nlgn)$
在30个元素以上，归并排序由于插入排序`