Sorting algorithm analysis

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From the previous algorithms, qsort is the slowest at the worst cases and others are always $O(n logn)$. However, qsort is known to be fastest at the most of cases.

Algorithm Average speed Worst speed Extra memory usage
Qsort Fastest($O(n log n)$) Slowest($O(n^2)$) None($O(c)$)
Merge sort Normal($O(n log n)$) Normal($O(n log n)$) Auxlist($O(n)$)
Heap sort Normal($O(n log n)$) Normal($O(n log n)$) None($O(c)$) 
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#include <iostream>
#include <math.h>
#include <time.h>
#include <vector>
#include <algorithm>
#include <fstream>
#include <chrono>

void swap(int& a, int& b){
    int c = a;
    a = b;
    b = c;
}

void downHeap(int lst[], int length, int idx){
    if (2 * idx + 2 < length){
        if (lst[idx] < lst[2 * idx + 1]){
            if(lst[2 * idx + 1] < lst[2 * idx + 2]){
                swap(lst[idx], lst[2 * idx + 2]);
                downHeap(lst, length, 2 * idx + 2);
            }
            else{
                swap(lst[idx], lst[2 * idx + 1]);
                downHeap(lst, length, 2 * idx + 1);
            }
        }
        else if (lst[idx] < lst[2 * idx + 2]){
            swap(lst[idx], lst[2 * idx + 2]);
            downHeap(lst,length, 2 * idx + 2);
        }
    }
    else if (2 * idx + 1 < length){
        if (lst[idx] < lst[2 * idx + 1]){
            swap(lst[idx], lst[2 * idx + 1]);
            downHeap(lst, length, 2 * idx + 1);
        }
    }
}

void listToHeap(int lst[], int length){
    int height = 0;
    int sumE   = 0;
    while (sumE < length){
        height += 1;
        sumE   *= 2;
        sumE   += 1;
    }
    for(int i = height - 2; i > -1; i -= 1){
        for(int j = pow(2,i) -1; j < pow(2,i+1) -1; j++){
            downHeap(lst, length, j);
        }
    }
}

void heapSort(int lst[], int length){
    listToHeap(lst, length);
    for(int i = length-1; i > 0; --i){
        swap(lst[0], lst[i]);
        downHeap(lst, i, 0);
    }
}

int partition(int lst[], int fromIdx, int toIdx){
    int selectPoint = (toIdx + fromIdx)/2;
    swap(lst[fromIdx], lst[selectPoint]);
    int criteria = lst[fromIdx];
    int lIdx = fromIdx;
    int rIdx = toIdx - 1;
    while(lIdx < rIdx){
        while(lIdx < rIdx){
            if(lst[lIdx] > criteria)
                break;
            lIdx += 1;
        }
        if(lIdx == rIdx)
            break;
        while (lIdx < rIdx){
            if (lst[rIdx] <= criteria)
                break;
            rIdx -= 1;
        }
        swap(lst[lIdx], lst[rIdx]);
    }
    if (lst[lIdx] > lst[fromIdx])
        lIdx -= 1;
    swap(lst[lIdx], lst[fromIdx]);
    return lIdx;
}

void _qsort(int lst[], int fromIdx, int toIdx){
    if(toIdx - fromIdx <= 1)
        return;
    int mid = partition(lst, fromIdx, toIdx);
    _qsort(lst, fromIdx, mid);
    _qsort(lst, mid + 1, toIdx);
}
    
void qsort(int lst[], int length){
    _qsort(lst, 0, length);
}

void _mergeSort(int lst[], int fromIdx, int toIdx){
    if (toIdx - fromIdx <= 1)
        return;
    int mid = (toIdx + fromIdx)/2;
    _mergeSort(lst, fromIdx, mid);
    _mergeSort(lst, mid, toIdx);

    int auxList[toIdx - fromIdx];
    int idx1 = fromIdx;
    int idx2 = mid;
    for(int i = 0; i < (toIdx - fromIdx); ++i){
        if (idx2 >= toIdx || idx1 < mid && lst[idx1] <= lst[idx2]){
            auxList[i] = lst[idx1];
            idx1 += 1;
        }
        else if (idx1 >= mid || idx2 < toIdx && lst[idx2] < lst[idx1]){
            auxList[i] = lst[idx2];
            idx2 += 1;
        }
        else{
            std::cout << "Invalid situation" << std::endl;
            exit(1);
        }
    }
    for(int i = 0; i < (toIdx - fromIdx); ++i){
        lst[i + fromIdx] = auxList[i];
    }
}
        
void mergeSort(int lst[], int length){
    return _mergeSort(lst, 0, length);
}


int main(){
    using namespace std;

    srand(time(NULL));
    std::chrono::duration<double> mergeTime;
    std::chrono::duration<double> qTime;
    std::chrono::duration<double> heapTime;
    
    ofstream qDat("qSort.dat"), mDat("mergeSort.dat"), hDat("heapSort.dat");
    for(int i = 0; i < 1000; ++i){
        int length = i * 9 + 100;
        int lst1[length];
        int lst2[length];
        for(int j = 0; j < length; ++j)
            lst1[j] = rand()%10000;

        for(int j = 0; j < length; ++j)
            lst2[j] = lst1[j];
        auto from = std::chrono::system_clock::now();
        mergeSort(lst2, length);
        auto to = std::chrono::system_clock::now();
        mergeTime = (to - from);
        
        for(int j = 0; j < length; ++j)
            lst2[j] = lst1[j];
        from = std::chrono::system_clock::now();
        qsort(lst2, length);
        to = std::chrono::system_clock::now();
        qTime = (to - from);
        
        for(int j = 0; j < length; ++j)
            lst2[j] = lst1[j];
        from = std::chrono::system_clock::now();
        heapSort(lst2, length);
        to = std::chrono::system_clock::now();
        heapTime = (to - from);
        qDat << length << " " << qTime.count()     << "\n";
        hDat << length << " " << heapTime.count()  << "\n";
        mDat << length << " " << mergeTime.count() << "\n";
    }
    qDat.close();
    mDat.close();
    hDat.close();
}

One minor result is that worst-case of qsort is awefull enough to make interests. (Worst-cases for qsort was a list with elements with the same value)