heapsort

Question

我写了heapsort (Cormen)。算法排序正确，但复杂性高于预期。

void heap_sort(int tab[], int length)
{
    build_max_heap(tab, length);
    int heap_size = length;
    for (int i = length-1; i > 1; i--)
    {
        int tmp = tab[1];
        tab[1] = tab[i];
        tab[i] = tmp;
        heap_size--;
        max_heapify(tab, 1, heap_size);
    }
}

void max_heapify (int tab[], int i, int length)
{
    int largest;
    int l = i * 2;
    int r = i * 2 + 1;
    if (l < length and tab[l] > tab[i])
        largest = l;
    else
        largest = i;
    if (r < length and tab[r] > tab[largest])
        largest = r;
    if (largest != i)
    {
        int tmp = tab[i];
        tab[i] = tab[largest];
        tab[largest] = tmp;
        max_heapify(tab, largest, length);
    }
}

void build_max_heap(int tab[], int length)
{
    for (int i = length/2; i >= 1; i--)
        max_heapify(tab, i, length);
}

对于 rand() 生成的 15000000 个数字，它比使用 Shell 排序的排序持续时间更长：

void shell_sort (int tab[], int length)
{
    int x = 2;
    int q;
    do
    {
        x*=2;
        q=2*(length/x) + 1;
        for(int i = q, val, j; i < length; i++)
        {
            val = tab[i];
            for(j = i - q ; j >= 0 and tab[j] > val; j-=q)
            {
                tab[j + q] = tab[j];
            }
            tab[j + q] = val;
        }
    }while (q > 1);
}

测试：

HEAPSORT
Time for 1000000 elements: 0.336 s
Time for 2000000 elements: 0.732 s
Time for 3000000 elements: 1.142 s
Time for 4000000 elements: 1.595 s
Time for 5000000 elements: 2.034 s
Time for 6000000 elements: 2.513 s
Time for 7000000 elements: 3.023 s
Time for 8000000 elements: 3.51 s
Time for 9000000 elements: 4.02 s
Time for 10000000 elements: 4.558 s
Time for 11000000 elements: 5.095 s
Time for 12000000 elements: 5.595 s
Time for 13000000 elements: 6.183 s
Time for 14000000 elements: 6.7 s
Time for 15000000 elements: 7.367 s

SHELLSORT
Time for 1000000 elements: 0.343 s
Time for 2000000 elements: 0.779 s
Time for 3000000 elements: 1.182 s
Time for 4000000 elements: 1.654 s
Time for 5000000 elements: 2.218 s
Time for 6000000 elements: 2.672 s
Time for 7000000 elements: 3.34 s
Time for 8000000 elements: 3.778 s
Time for 9000000 elements: 4.297 s
Time for 10000000 elements: 4.903 s
Time for 11000000 elements: 4.872 s
Time for 12000000 elements: 5.514 s
Time for 13000000 elements: 6.29 s
Time for 14000000 elements: 6.994 s
Time for 15000000 elements: 7.121 s

我重复了很多次测试。算法有什么问题？

Answer 1

首先请注意，big-O 和原始性能有着复杂的关系。在堆排序的情况下，较差的内存局部性将导致它在计算机上的扩展性比 big-O 所建议的更差。相比之下，shell 排序只比 O(n log(n)) 稍微差一点，而且它的大多数传递具有不错的内存局部性。

因此，我对它们具有可比性并不感到惊讶。

就是说，您可以尝试将 max_heapify 从递归函数转换为循环。这可能会避免一定数量的堆栈操作。

heapsort - 实现的复杂性

heapsort - implementation's complexity

c++

sorting

algorithm

time-complexity