Sorting Algorithm Analysis

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Among the previous algorithms, quick sort performs worst in the worst case, while the others are always $O(n \log n)$. However, quick sort is known to be the fastest in practice for most inputs.

Algorithm Average speed Worst speed Extra memory usage
Quick sort Fastest ($O(n \log n)$) Slowest ($O(n^2)$) None ($O(1)$)
Merge sort $O(n \log n)$ $O(n \log n)$ Auxiliary array ($O(n)$)
Heap sort $O(n \log n)$ $O(n \log n)$ None ($O(1)$) 
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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 notable result is that the worst case of quick sort is severe enough to be interesting. (The worst case for quick sort occurs when all elements have the same value.)