Quicksort linear time?

It is, on average, indeed O(N log N).

It's just that a graph of f(N) = N log(N) appears remarkably linear.

Graph it and see for yourself, or refer to the one below. This average time is what makes the algorithm so very clever:

The reason the slope looks linear is partly the slow change in Log(N) but the main reason is that the random numbers that populate the array are limited to [0-1,000,000). This causes the large arrays to mostly be filled with duplicates and as the qsort algorithm gets down to smaller groups, sorting becomes much faster. As the array size increases from, say 10,000,000 to 20,000,000 the duplicates double in count on average and hence the sorting tracks almost entirely linearly.

This can be seen from the following graph:

The orange and grey lines are the execution times of the unconstrained and constrained arrays. The yellow and blue lines are linear from 0 to the end points of two runs. One run has ints constrained to [0-1000000) from the original code. The other isunconstrained to the 2^31 positive ints. Note how much longer the unconstrained sort takes since sorting of the increasing groups of duplicates is extremely fast.

Here's the code modification that shows the unconstrained execution time has a distinct curve as one would expect from NLogN.

#include <iostream>
#include <fstream>
#include <ctime>
#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;
            // Note that as the array size grows beyond 1000000
            // this will produce increasing numbers of duplicates
            // which will shorten the time when the subsets get small
    
        array[c] = (rand() << 16) | (rand() << 1) | (rand() & 1);
            // Note that in this example/system, RAND_MAX==0x7fff
            // get a random positive int distributed in the set of positive, 32 bit ints
    }

    return array;
}

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

int main()
{
    auto x = RAND_MAX;
    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;
}