算法排序：归并排序

一、归并排序基本思想

归并排序（MERGE-SORT）是利用归并的思想实现的排序方法，该算法采用经典的分治（divide-and-conquer）策略（分治法将问题分(divide)成一些小的问题然后递归求解，而治(conquer)的阶段则将分的阶段得到的各答案"修补"在一起，即分而治之)。

分而治之

分解：将列表越分越小，直至分成一个元素。
一个元素是有序的。
合并：将两个有序列表归并，列表越来越大。

 　　可以看到这种结构很像一棵完全二叉树，本文的归并排序我们采用递归去实现（也可采用迭代的方式去实现）。分阶段可以理解为就是递归拆分子序列的过程，递归深度为log₂n。

二、合并相邻有序子序列过程

三、归并排序代码

1、一次归并代码

def merge(li, low, mid, high):
    i = low    
	j = mid + 1    
	ltmp = []    
	while i <= mid and j <= high:        
		if li[i] <= li[j]:            
			ltmp.append(li[i])            
			i += 1        
		else:            
			ltmp.append(li[j])            
			j += 1    
	while i <= mid:        
		ltmp.append(li[i])        
		i += 1    
	while j <= high:        
		ltmp.append(li[j])        
		j += 1    
	li[low:high + 1] = ltmp

2、归并排序

def _merge_sort(li，low，high)：
	if low < high：#至少两个元素
		mid = (low + high) // 2
		_merge_sort(li，low，mid)
		_merge_sort(li,mid+1,high)
		#merage(li,low,mid,high)
		print(li[low:high+1])

四、归并排序过程打印

1、归并排序归并过程

1、归并排序之归并

1、实现代码

def merge(li, low, mid, high):
    i = low
    j = mid + 1
    ltmp = []
    while i<=mid and j<=high: # 只要左右两边都有数
        if li[i] < li[j]:
            ltmp.append(li[i])
            i += 1
        else:
            ltmp.append(li[j])
            j += 1
    # while执行完，肯定有一部分没数了
    while i <= mid:
        ltmp.append(li[i])
        i += 1
    while j <= high:
        ltmp.append(li[j])
        j += 1
    li[low:high+1] = ltmp

li = [2,4,5,7,1,3,6,8]
merge(li, 0, 3, 7)
print(li)

2、测试结果

[1, 2, 3, 4, 5, 6, 7, 8]

2、一次的完整归并

1、实现代码

def merge(li, low, mid, high):
    i = low
    j = mid + 1
    ltmp = []
    while i<=mid and j<=high: # 只要左右两边都有数
        if li[i] < li[j]:
            ltmp.append(li[i])
            i += 1
        else:
            ltmp.append(li[j])
            j += 1
    # while执行完，肯定有一部分没数了
    while i <= mid:
        ltmp.append(li[i])
        i += 1
    while j <= high:
        ltmp.append(li[j])
        j += 1
    li[low:high+1] = ltmp

# li = [2,4,5,7,1,3,6,8]
# merge(li, 0, 3, 7)
# print(li)

def merge_sort(li, low, high):
    if low < high: #至少有两个元素，递归
        mid = (low + high) //2
        merge_sort(li, low, mid)
        merge_sort(li, mid+1, high)
        merge(li, low, mid, high)


li = list(range(20))
import random
random.shuffle(li)
print(li)
merge_sort(li, 0, len(li)-1)
print(li)

2、测试结果

[1, 9, 10, 4, 0, 11, 18, 5, 7, 8, 17, 15, 19, 2, 14, 16, 6, 12, 13, 3]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

2、打印归并过程

1、实现代码

def merge(li, low, mid, high):
    i = low
    j = mid + 1
    ltmp = []
    while i<=mid and j<=high: # 只要左右两边都有数
        if li[i] < li[j]:
            ltmp.append(li[i])
            i += 1
        else:
            ltmp.append(li[j])
            j += 1
    # while执行完，肯定有一部分没数了
    while i <= mid:
        ltmp.append(li[i])
        i += 1
    while j <= high:
        ltmp.append(li[j])
        j += 1
    li[low:high+1] = ltmp


def merge_sort(li, low, high):
    if low < high: #至少有两个元素，递归
        mid = (low + high) //2
        merge_sort(li, low, mid)
        merge_sort(li, mid+1, high)
        merge(li, low, mid, high)
        print(li[low:high+1])


li = list(range(10))
import random
random.shuffle(li)
print(li)
merge_sort(li, 0, len(li)-1)
print(li)

2、range(20)测试结果

[10, 6, 4, 18, 0, 12, 5, 15, 14, 17, 7, 8, 11, 13, 2, 1, 19, 3, 9, 16]
[6, 10]
[4, 6, 10]
[0, 18]
[0, 4, 6, 10, 18]
[5, 12]
[5, 12, 15]
[14, 17]
[5, 12, 14, 15, 17]
[0, 4, 5, 6, 10, 12, 14, 15, 17, 18]
[7, 8]
[7, 8, 11]
[2, 13]
[2, 7, 8, 11, 13]
[1, 19]
[1, 3, 19]
[9, 16]
[1, 3, 9, 16, 19]
[1, 2, 3, 7, 8, 9, 11, 13, 16, 19]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

3、range(10)测试结果

[3, 1, 9, 5, 8, 7, 6, 4, 0, 2]
[1, 3]
[1, 3, 9]
[5, 8]
[1, 3, 5, 8, 9]
[6, 7]
[4, 6, 7]
[0, 2]
[0, 2, 4, 6, 7]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

五、归并排序性能分析

六、快速排序、堆排序、归并排序-小结