快速排序线性时间？

Question

我正在对quicksort（来自c++ STL的qsort）算法进行分析，代码为：

#include <iostream>
#include <fstream>
#include <ctime>
#include <bits/stdc++.h>
#include <cstdlib>
#include <iomanip>

#define MIN_ARRAY 256000
#define MAX_ARRAY 1000000000
#define MAX_RUNS 100

using namespace std;

int* random_array(int size) {
    int* array = new int[size];

    for (int c = 0; c < size; c++) {
        array[c] = rand()*rand() % 1000000;
    }

    return array;
}

int compare(const void* a, const void* b) { 
    return (*(int*)a - *(int*)b); 
}

int main()
{
    ofstream fout;
    fout.open("data.csv");
    fout << "array size,";
    srand(time(NULL));
    int size;
    int counter = 1;

    std::clock_t start;
    double duration;

    for (size = MIN_ARRAY; size < MAX_ARRAY; size *= 2) {
        fout << size << ",";
    }
    fout << "\n";

    for (counter = 1; counter <= MAX_RUNS; counter++) {
        fout << "run " << counter << ",";
        for (size = MIN_ARRAY; size < MAX_ARRAY; size *= 2) {
            try {
                int* arr = random_array(size);

                start = std::clock();
                qsort(arr, size, sizeof(int), compare);
                duration = (std::clock() - start) / (double)CLOCKS_PER_SEC;

                //cout << "size: " << size << " duration: " << duration << '\n';
                fout << setprecision(15) << duration << ",";

                delete[] arr;
            }
            catch (bad_alloc) {
                cout << "bad alloc caught, size: " << size << "\n";
                fout << "bad alloc,";
            }

        }
        fout << "\n";
        cout << counter << "% done\n";
    }
    
    fout.close();
    return 0;
}

当我运行它时，数据完全线性地返回：

这到底是怎么回事？快速排序不是 O(nlogn) 吗？

这是使用的数组大小以及所有 100 次运行的每个大小的平均时间（以秒为单位）：

#include <iostream>
#include <fstream>
#include <ctime>
#include <bits/stdc++.h>
#include <cstdlib>
#include <iomanip>

#define MIN_ARRAY 256000
#define MAX_ARRAY 1000000000
#define MAX_RUNS 100

using namespace std;

int* random_array(int size) {
    int* array = new int[size];

    for (int c = 0; c < size; c++) {
        array[c] = rand()*rand() % 1000000;
    }

    return array;
}

int compare(const void* a, const void* b) { 
    return (*(int*)a - *(int*)b); 
}

int main()
{
    ofstream fout;
    fout.open("data.csv");
    fout << "array size,";
    srand(time(NULL));
    int size;
    int counter = 1;

    std::clock_t start;
    double duration;

    for (size = MIN_ARRAY; size < MAX_ARRAY; size *= 2) {
        fout << size << ",";
    }
    fout << "\n";

    for (counter = 1; counter <= MAX_RUNS; counter++) {
        fout << "run " << counter << ",";
        for (size = MIN_ARRAY; size < MAX_ARRAY; size *= 2) {
            try {
                int* arr = random_array(size);

                start = std::clock();
                qsort(arr, size, sizeof(int), compare);
                duration = (std::clock() - start) / (double)CLOCKS_PER_SEC;

                //cout << "size: " << size << " duration: " << duration << '\n';
                fout << setprecision(15) << duration << ",";

                delete[] arr;
            }
            catch (bad_alloc) {
                cout << "bad alloc caught, size: " << size << "\n";
                fout << "bad alloc,";
            }

        }
        fout << "\n";
        cout << counter << "% done\n";
    }
    
    fout.close();
    return 0;
}

Answer 1

平均而言，确实是 O(N log N)。

只是f(N) = N log(N)的图形看起来非常线性。

将其绘制成图表并亲自查看，或参考下面的内容。这个平均时间使得算法如此聪明：