高效排序算法全解析
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。算法分为两个主要部分：构建堆和排序。
构建最大堆的过程从最后一个非叶子节点开始，向上调整每个子树使其满足堆性质。调整方法是将当前节点与较大子节点交换，递归向下调整。
排序阶段将堆顶元素（最大值）与末尾元素交换，缩小堆范围并重新调整堆。重复此过程直到堆大小为1。
时间复杂度为O(n log n)，空间复杂度为O(1)。适用于大规模数据排序，但不稳定。
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)
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[i], arr[0] = arr[0], arr[i]
heapify(arr, i, 0)
计数排序的实现
计数排序是非比较排序算法，适用于整数数据且范围不大的情况。算法统计每个元素的出现次数，然后计算前缀和确定元素位置。
初始化计数数组时，大小为max_val - min_val +1。统计频率后，前缀和数组表示元素在输出数组中的最后位置。反向填充保证稳定性。
时间复杂度为O(n+k)，k是数据范围。空间复杂度为O(n+k)。当k远小于n时效率很高。
def counting_sort(arr):
max_val = max(arr)
min_val = min(arr)
range_size = max_val - min_val +1
count = [0] * range_size
output = [0] * len(arr)
for num in arr:
count[num - min_val] +=1
for i in range(1, len(count)):
count[i] += count[i-1]
for num in reversed(arr):
output[count[num - min_val] -1] = num
count[num - min_val] -=1
return output
桶排序的实现
桶排序将数据分散到有限数量的桶中，每个桶单独排序后合并。桶的数量和范围需要根据数据分布合理选择。
初始化桶时通常使用数组或链表存储。数据分配到桶后，可采用其他排序算法对每个桶排序。最后按顺序合并所有桶。
时间复杂度取决于桶的数量和内部排序算法，平均为O(n+k)。空间复杂度为O(n)。适用于数据均匀分布的情况。
def bucket_sort(arr, bucket_size=5):
min_val = min(arr)
max_val = 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)
sorted_arr = []
for bucket in buckets:
sorted_arr.extend(sorted(bucket))
return sorted_arr
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。可采用计数排序作为子程序进行稳定排序。
获取最大数字确定排序轮数。每轮根据当前位的数字分配元素，然后收集回原数组。需要处理负数时需特殊处理符号位。
时间复杂度为O(d(n+k))，d是最大数字的位数。空间复杂度为O(n+k)。适用于整数或字符串排序。
def counting_sort_for_radix(arr, exp):
n = len(arr)
output = [0] * n
count = [0] *10
for num in arr:
index = (num // exp) %10
count[index] +=1
for i in range(1, 10):
count[i] += count[i-1]
i = n-1
while i >=0:
index = (arr[i] // exp) %10
output[count[index] -1] = arr[i]
count[index] -=1
i -=1
for i in range(n):
arr[i] = output[i]
def radix_sort(arr):
max_val = max(arr)
exp =1
while max_val // exp >0:
counting_sort_for_radix(arr, exp)
exp *=10
算法比较与选择
堆排序适合内存受限场景，原地排序但不稳定。计数排序在数据范围小时效率极高。桶排序需要数据均匀分布。基数排序适合位数较少的整数排序。
实际应用中，数据特征决定算法选择。小范围整数优先考虑计数排序，均匀分布数据可用桶排序，通用场景堆排序表现良好，固定位数数据基数排序最优。
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