calgorithmquicksortglibcquickselect

kth smallest number - quicksort faster than quickselect


I have implemented the following quickselect algorithm to achieve O(n) complexity for median selection (more generally kth smallest number):

static size_t partition(struct point **points_ptr, size_t points_size, size_t pivot_idx)
{
    const double pivot_value = points_ptr[pivot_idx]->distance;

    /* Move pivot to the end. */
    SWAP(points_ptr[pivot_idx], points_ptr[points_size - 1], struct point *);

    /* Perform the element moving. */
    size_t border_idx = 0;
    for (size_t i = 0; i < points_size - 1; ++i) {
            if (points_ptr[i]->distance < pivot_value) {
                    SWAP(points_ptr[border_idx], points_ptr[i], struct point *);
                    border_idx++;
            }
    }

    /* Move pivot to act as a border element. */
    SWAP(points_ptr[border_idx], points_ptr[points_size - 1], struct point *);

    return border_idx;
}

static struct point * qselect(struct point **points_ptr, size_t points_size, size_t k)
{
    const size_t pivot_idx = partition(points_ptr, points_size, rand() % points_size);

    if (k == pivot_idx) { //k lies on the same place as a pivot
            return points_ptr[pivot_idx];
    } else if (k < pivot_idx) { //k lies on the left of the pivot
            //points_ptr remains the same
            points_size = pivot_idx;
            //k remains the same
    } else { //k lies on the right of the pivot
            points_ptr += pivot_idx + 1;
            points_size -= pivot_idx + 1;
            k -= pivot_idx + 1;
    }

    return qselect(points_ptr, points_size, k);
}

Then I tried to compare it with a glibc's qsort() with O(nlog(n)) and was surprised by its superior performance. Here is the measurement code:

double wtime;
wtime = 0.0;
for (size_t i = 0; i < 1000; ++i) {
    qsort(points_ptr, points_size, sizeof (*points_ptr), compar_rand);
    wtime -= omp_get_wtime();
    qsort(points_ptr, points_size, sizeof (*points_ptr), compar_distance);
    wtime += omp_get_wtime();
}
printf("qsort took %f\n", wtime);

wtime = 0.0;
for (size_t i = 0; i < 1000; ++i) {
    qsort(points_ptr, points_size, sizeof (*points_ptr), compar_rand);
    wtime -= omp_get_wtime();
    qselect(points_ptr, points_size, points_size / 2);
    wtime += omp_get_wtime();
}
printf("qselect took %f\n", wtime);

with results similar to qsort took 0.280432, qselect took 8.516676 for an array of 10000 elements. Why is quicksort faster than quickselect?


Solution

  • Thanks for your suggestions guys, problem with my implementation of quickselect was that it exhibits its worst-case complexity O(n^2) for inputs that contain many repeated elements, which was my case. Glibc's qsort() (it uses mergesort by default) does not exhibit O(n^2) here.

    I have modified my partition() function to perform a basic 3-way partitioning and the median-of-three which works nicely for quickselect:

    /** \breif Quicksort's partition procedure.                                  
     *                                                                           
     * In linear time, partition a list into three parts: less than, greater than
     * and equals to the pivot, for example input 3 2 7 4 5 1 4 1 will be        
     * partitioned into 3 2 1 1 | 5 7 | 4 4 4 where 4 is the pivot.              
     * Modified version of the median-of-three strategy is implemented, it ends with
     * a median at the end of an array (this saves us one or two swaps).         
     */                                                                          
    static void partition(struct point **points_ptr, size_t points_size,
                          size_t *less_size, size_t *equal_size)
    {                                                                            
        /* Modified median-of-three and pivot selection. */                      
        struct point **first_ptr = points_ptr;                                   
        struct point **middle_ptr = points_ptr + (points_size / 2);              
        struct point **last_ptr = points_ptr + (points_size - 1);                
        if ((*first_ptr)->distance > (*last_ptr)->distance) {                    
            SWAP(*first_ptr, *last_ptr, struct point *);                         
        }                                                                        
        if ((*first_ptr)->distance > (*middle_ptr)->distance) {                  
            SWAP(*first_ptr, *middle_ptr, struct point *);                       
        }                                                                        
        if ((*last_ptr)->distance > (*middle_ptr)->distance) { //reversed        
            SWAP(*last_ptr, *middle_ptr, struct point *);                        
        }                                                                        
        const double pivot_value = (*last_ptr)->distance;                      
    
        /* Element swapping. */                                                  
        size_t greater_idx = 0;                                                  
        size_t equal_idx = points_size - 1;                                      
        size_t i = 0;                                                            
        while (i < equal_idx) {                                                  
            const double elem_value = points_ptr[i]->distance;                   
    
            if (elem_value < pivot_value) {                                      
                SWAP(points_ptr[greater_idx], points_ptr[i], struct point *);    
                greater_idx++;                                                   
                i++;                                                             
            } else if (elem_value == pivot_value) {                              
                equal_idx--;                                                     
                SWAP(points_ptr[i], points_ptr[equal_idx], struct point *);      
            } else { //elem_value > pivot_value                                  
                i++;                                                             
            }                                                                    
        }                                                                        
    
        *less_size = greater_idx;                                                
        *equal_size = points_size - equal_idx;                                   
    }
    
    /** A selection algorithm to find the kth smallest element in an unordered list.
     */                                                                          
    static struct point * qselect(struct point **points_ptr, size_t points_size,
                                  size_t k)
    {                                                                            
        size_t less_size;                                                        
        size_t equal_size;                                                       
    
        partition(points_ptr, points_size, &less_size, &equal_size);             
    
        if (k < less_size) { //k lies in the less-than-pivot partition           
            points_size = less_size;                                             
        } else if (k < less_size + equal_size) { //k lies in the equals-to-pivot partition
            return points_ptr[points_size - 1];                                  
        } else { //k lies in the greater-than-pivot partition                    
            points_ptr += less_size;                                             
            points_size -= less_size + equal_size;                               
            k -= less_size + equal_size;                                         
        }                                                                        
    
        return qselect(points_ptr, points_size, k);                              
    }
    

    Results are indeed linear and better than qsort() (I have used the Fisher-Yates shuffle as @IVlad have suggested, so the absolute qsort() times are worse):

    array size  qsort     qselect   speedup
    1000        0.044678  0.008671  5.152328
    5000        0.248413  0.045899  5.412160
    10000       0.551095  0.096064  5.736730
    20000       1.134857  0.191933  5.912773
    30000       2.169177  0.278726  7.782467