十大排序算法精解与实战

堆排序的实现

堆排序基于二叉堆数据结构，通常使用数组实现。构建最大堆或最小堆后，通过反复提取堆顶元素完成排序。

构建最大堆的调整过程：

def heapify(arr, n, i):
    largest = i
    left = 2 * i + 1
    right = 2 * i + 2
    
    if left < n and arr[left] > arr[largest]:
        largest = left
        
    if right < n and arr[right] > arr[largest]:
        largest = right
        
    if largest != i:
        arr[i], arr[largest] = arr[largest], arr[i]
        heapify(arr, n, largest)

完整的堆排序算法：

def heap_sort(arr):
    n = len(arr)
    
    for i in range(n//2 - 1, -1, -1):
        heapify(arr, n, i)
    
    for i in range(n-1, 0, -1):
        arr[0], arr[i] = arr[i], arr[0]
        heapify(arr, i, 0)

计数排序的实现

计数排序适用于整数数据且范围不大的场景。统计每个元素出现次数后计算前缀和，确定元素最终位置。

实现步骤：

def counting_sort(arr):
    max_val = max(arr)
    count = [0] * (max_val + 1)
    
    for num in arr:
        count[num] += 1
    
    for i in range(1, len(count)):
        count[i] += count[i-1]
    
    output = [0] * len(arr)
    for num in reversed(arr):
        output[count[num]-1] = num
        count[num] -= 1
    
    return output

桶排序的实现

桶排序将数据分到有限数量的桶中，每个桶单独排序后合并。适用于均匀分布的数据。

典型实现：

def bucket_sort(arr, bucket_size=5):
    min_val, max_val = min(arr), max(arr)
    bucket_count = (max_val - min_val) // bucket_size + 1
    buckets = [[] for _ in range(bucket_count)]
    
    for num in arr:
        buckets[(num - min_val) // bucket_size].append(num)
    
    return [num for bucket in buckets for num in sorted(bucket)]

基数排序的实现

基数排序按数字的每位进行排序，从最低位到最高位依次处理。需要稳定的子排序算法（通常用计数排序）。

LSD（最低位优先）实现：

def radix_sort(arr):
    max_digits = len(str(max(arr)))
    
    for digit in range(max_digits):
        buckets = [[] for _ in range(10)]
        
        for num in arr:
            digit_val = (num // 10**digit) % 10
            buckets[digit_val].append(num)
        
        arr = [num for bucket in buckets for num in bucket]
    
    return arr

性能对比与适用场景

堆排序时间复杂度稳定为O(n log n)，适合需要保证最坏情况性能的场景。计数排序在数据范围较小时效率可达O(n+k)。桶排序在数据分布均匀时性能优异。基数排序对数字位数固定的数据表现良好，时间复杂度为O(nk)。

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