The actual code of divide and conquer and merge sort

In computer science, divide and conquer is a very important algorithm design paradigm.

It solves the original problem by decomposing a complex problem into several smaller sub-problems, solving these sub-problems separately, and then merging their results.

Merge Sort is a classic example of using divide and conquer.

What is merge sort?.

Merge sort is a comparison-based sorting algorithm that uses a divide-and-conquer strategy for sorting.

It divides the array into two sub-arrays, sorts the two sub-arrays separately, and then merges them into an ordered array.

This process proceeds recursively until each subarray contains only one element.

Merge sort steps.

1. # Decomposition #: Divide the array to be sorted into two sub-arrays from the middle.

2. # Solve #: Recursively merge and sort two subarrays.

3. # Merge #: Combine two sorted subarrays into an ordered array.

Code implementation.

The following is the merge sort code implemented in Python:

def merge_sort(arr):
    """
    归并排序的主函数，接受一个列表作为输入，返回排序后的列表。

    """
    if len(arr) <= 1:
        return arr

    # 找到数组的中间位置
    mid = len(arr) // 2

    # 递归地对左半部分和右半部分进行排序
    left_half = merge_sort(arr[:mid])
    right_half = merge_sort(arr[mid:])

    # 合并两个已排序的部分
    return merge(left_half, right_half)

def merge(left, right):
    """
    合并两个已排序的列表，返回一个新的有序列表。

    """
    sorted_list = []
    left_index, right_index = 0, 0

    # 比较两个列表的元素，按顺序添加到sorted_list中
    while left_index < len(left) and right_index < len(right):
        if left[left_index] < right[right_index]:
            sorted_list.append(left[left_index])
            left_index += 1
        else:
            sorted_list.append(right[right_index])
            right_index += 1

    # 如果左半部分还有剩余元素，添加到sorted_list中
    while left_index < len(left):
        sorted_list.append(left[left_index])
        left_index += 1

    # 如果右半部分还有剩余元素，添加到sorted_list中
    while right_index < len(right):
        sorted_list.append(right[right_index])
        right_index += 1

    return sorted_list

Time complexity analysis.

The time complexity of merge sort can be analyzed by recursive tree.

Assuming that the length of the array is n, each decomposition operation requires O (log n) times, and each merge operation requires O (n) times.

Therefore, the total time complexity is: \[ T(n) = O(n \log n) \] This time complexity is optimal because any comparison-based sorting algorithm requires at least O (n log n) comparisons in the worst case.

Optimization in practical applications.

Although the time complexity of merge sort is already optimal, in practical applications, we can further optimize performance through some techniques.

E.g: 1. # Reduce memory allocation #: During the merge process, if you can in-place merge (in-place merge), you can reduce the overhead of memory allocation and replication.

However, this will increase the complexity of the code.

2. # Use iteration instead of recursion #: Recursive calls consume stack space and may cause stack overflow for very large arrays.

This problem can be avoided by using an explicit stack to simulate the recursive process.

3. # Decimal Array Optimization #: For very small arrays, insertion sort may be faster than merge sort.

Therefore, it is possible to switch to insertion sort when recursion reaches a certain depth.

Example application.

Suppose we have a log file that contains a lot of data and needs to be sorted for subsequent processing.

We can use merge sort to do this efficiently.

Here is a simple example:

import random

# 生成一个包含随机数的列表
data = [random.randint(0, 1000) for _ in range(10000)]

# 使用归并排序对数据进行排序
sorted_data = merge_sort(data)

# 打印前10个排序后的数据以验证结果
print(sorted_data[:10])

In this example, we generate a list of 10000 random integers and sort them using merge sort.

Finally, we print out the first 10 sorted data to verify the results.

Summarize.

Merge sorting is a classic and efficient sorting algorithm that decomposes problems into smaller sub-problems through divide and conquer, and then gradually solves these sub-problems, and finally merges them into a complete solution.

Although its time complexity has reached the optimal O (n log n), in practical application scenarios, we can still further improve its performance through various optimization methods.

Hope this article can help you better understand and apply the merge sort algorithm.