Introspective Sort
November 11, 2016
The sorting algorithm that we have been working up to in three previous exercises is introspective sort, or introsort, invented by David Musser in 1997 for the C++ Standard Library. Introsort is basically quicksort, with median-of-three partitioning and a switch to insertion sort when the partitions get small, but with a twist. The problem of quicksort is that some sequences have the property that most of the recursive calls don’t significantly reduce the size of the data to be sorted, causing a quadratic worst case. Introsort fixes that by switching to heapsort if the depth of recursion gets too large; since heapsort has guaranteed O(n log n) behavior, so does introsort. The changeover from quicksort to heapsort occurs after k * floor(log(length(A))) recursive calls to quicksort, where k is a tuning parameter, frequently set to 2, that can be used to adjust performance of the sorting algorithm.
Your task is to implement introsort. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.
Programming Praxis 
Is that a natural logarithm?
The
logfunction in Scheme, that I used in my program, is a natural logarithm to base e. Theoretically, the logarithm should be to base 2, since you are calculating the depth of recursion assuming a perfect split into two equal-size sub-arrays at each recursive call. In practice, you probably want to try many different values of k to find the optimum value for your circumstances; a value close to 1 means that you will be making many calls to heapsort, which is naturally slower than quicksort, but a value far from 1 means that you are continuing to make non-productive recursive calls rather than switching to heapsort.Fair enough, though this means that with k=2, we are doing heapsort quite a lot, even with random input (so I’m surprised that introsort seems to be faster, though that might just be noise).
I went back and looked at Musser’s paper. He uses 2 * floor(log2 n), but suggests testing to determine an empirically good value that produces good results with your environment. I’ve done a little bit of experimenting, but intend to do more.
Here’s a solution in C99.
The program output is included at the bottom of this post. It shows runtimes for various scenarios. Each experiment was conducted with 10 separate sorts, and the time reported is the aggregate time for all 10 sorts. Rows correspond to various array sizes.
Column 1: array size
Column 2: Random array quicksort
Column 3: Random array heapsort
Column 4: Random array introsort
Column 5: Killer array quicksort
Column 6: Killer array heapsort
Column 7: Killer array introsort
The killer arrays were generated using the ‘Median-Of-Three Killer Sequence’ procedure from an earlier problem.
For all experiments, quicksort includes the optimizations from earlier problems, 1) inline swap, 2) early cutoff to insertion sort, and 3) median-of-three pivot selection. These same optimizations were also used for introsort.
I increased the stack size to prevent stack overflows. Compiler optimizations were disabled.
#include <stddef.h> #include <stdio.h> #include <stdlib.h> #include <time.h> static inline void swap(int* a, int* b) { int tmp = *a; *a = *b; *b = tmp; } // reheapify downward from specified subroot static void reheapify(int* heap, int subroot, size_t n) { while (1) { int left = 2*subroot+1; if (left >= n) break; int right = 2*subroot+2; int candidate = left; if (right < n && heap[right] > heap[left]) candidate = right; if (heap[subroot] >= heap[candidate]) break; swap(&heap[subroot], &heap[candidate]); subroot = candidate; } } void heapsort(int* array, size_t n) { if (n <= 1) return; // heapify with Floyd's method, O(n) for (int i = (n-2)/2; i >= 0; --i) { reheapify(array, i, n); } // heapsort procedure: // 1) swap root (max) with the last element // 2) exclude last element from heap // 3) reheapify new root downward for (int i = n-1; i > 0; --i) { swap(&array[0], &array[i]); reheapify(array, 0, i); } } void insertionsort(int* array, size_t n) { for (int i = 1; i < n; ++i) { for (int j = i; j > 0 && array[j-1] > array[j]; --j) { swap(&array[j], &array[j-1]); } } } static int partition(int* array, size_t n) { int pivot = array[0]; // median-of-three if (n >= 3) { int sample[3]; sample[0] = array[0]; sample[1] = array[n/2]; sample[2] = array[n-1]; insertionsort(sample, 3); pivot = sample[1]; } int i = -1; int j = n; while (1) { do ++i; while (array[i] < pivot); do --j; while (array[j] > pivot); if (i >= j) return j; swap(&array[i], &array[j]); } } void quicksort(int* array, size_t n) { if (n < 2) return; // early cutoff to insertion sort if (n < 22) { insertionsort(array, n); return; } int p = partition(array, n); quicksort(array, p + 1); quicksort(&array[p + 1], n - p - 1); } // floor(log(n)). Assumes n > 0. static int ilog2(int n) { int output = 0; while (n >>= 1) ++output; return output; } static void introsort_internal(int* array, size_t n, size_t depth) { if (n < 2) return; // early cutoff to insertion sort if (n < 22) { insertionsort(array, n); return; } // early cutoff to heapsort if (depth > 2 * ilog2(n)) { heapsort(array, n); return; } int p = partition(array, n); introsort_internal(array, p + 1, depth + 1); introsort_internal(&array[p + 1], n - p - 1, depth + 1); } void introsort(int* array, size_t n) { introsort_internal(array, n, 0); } static void init_random_array(int* array, size_t n) { for (int i = 0; i < n; ++i) { array[i] = rand(); } } static void init_median_of_three_killer(int* array, size_t n) { // if n is odd, set the last element to n-1, and proceed // with n decremented by 1 if (n % 2 != 0) { array[n-1] = n; --n; } size_t m = n/2; for (int i = 0; i < m; ++i) { // first half of array (even indices) if (i % 2 == 0) array[i] = i + 1; // first half of array (odd indices) else array[i] = m + i + (m % 2 != 0 ? 1 : 0); // second half of array array[m + i] = (i+1) * 2; } } double time_sort_algorithm( void (*sort_fn)(int*, size_t), void (*init_fn)(int*, size_t), size_t n, int loops) { double elapsed = 0; int* array = malloc(n*sizeof(int)); for (int i = 0; i < loops; ++i) { init_fn(array, n); clock_t tic = clock(); sort_fn(array, n); clock_t toc = clock(); elapsed += (double)(toc-tic) / CLOCKS_PER_SEC; } free(array); return elapsed; } int main(void) { srand(time(NULL)); int min_order = 10; int max_order = 18; int loops = 10; for (int order = min_order; order <= max_order; ++order) { int n = 1 << order; printf("%*d", 6, n); double time; // Random Array quicksort time = time_sort_algorithm(quicksort, init_random_array, n, loops); printf(" %0.3f", time); // Random Array heapsort time = time_sort_algorithm(heapsort, init_random_array, n, loops); printf(" %0.3f", time); // Random Array introsort time = time_sort_algorithm(introsort, init_random_array, n, loops); printf(" %0.3f", time); // Killer Array quicksort time = time_sort_algorithm(quicksort, init_median_of_three_killer, n, loops); printf(" %7.3f", time); // Killer Array heapsort time = time_sort_algorithm(heapsort, init_median_of_three_killer, n, loops); printf(" %0.3f", time); // Killer Array introsort time = time_sort_algorithm(introsort, init_median_of_three_killer, n, loops); printf(" %0.3f", time); printf("\n"); } return 0; }This updated main function includes column numbers in the output.
int main(void) { srand(time(NULL)); int min_order = 10; int max_order = 18; int loops = 10; printf("1: Array Size\n"); printf("2: Random Array Quicksort\n"); printf("3: Random Array Heapsort\n"); printf("4: Random Array Introsort\n"); printf("5: Killer Array Quicksort\n"); printf("6: Killer Array Heapsort\n"); printf("7: Killer Array Introsort\n\n"); printf(" 1 2 3 4 5 6 7\n"); for (int order = min_order; order <= max_order; ++order) { int n = 1 << order; // Array Size printf("%*d", 6, n); double time; // Random Array Quicksort time = time_sort_algorithm(quicksort, init_random_array, n, loops); printf(" %0.3f", time); // Random Array Heapsort time = time_sort_algorithm(heapsort, init_random_array, n, loops); printf(" %0.3f", time); // Random Array Introsort time = time_sort_algorithm(introsort, init_random_array, n, loops); printf(" %0.3f", time); // Killer Array Quicksort time = time_sort_algorithm(quicksort, init_median_of_three_killer, n, loops); printf(" %7.3f", time); // Killer Array Heapsort time = time_sort_algorithm(heapsort, init_median_of_three_killer, n, loops); printf(" %0.3f", time); // Killer Array Introsort time = time_sort_algorithm(introsort, init_median_of_three_killer, n, loops); printf(" %0.3f", time); printf("\n"); } return 0; }Output:
1: Array Size 2: Random Array Quicksort 3: Random Array Heapsort 4: Random Array Introsort 5: Killer Array Quicksort 6: Killer Array Heapsort 7: Killer Array Introsort 1 2 3 4 5 6 7 1024 0.015 0.000 0.000 0.000 0.016 0.000 2048 0.000 0.000 0.015 0.016 0.000 0.016 4096 0.000 0.015 0.000 0.094 0.016 0.000 8192 0.015 0.016 0.016 0.359 0.016 0.015 16384 0.031 0.047 0.032 1.453 0.031 0.062 32768 0.063 0.094 0.062 5.813 0.078 0.109 65536 0.125 0.235 0.156 23.047 0.156 0.234 131072 0.250 0.469 0.281 88.672 0.328 0.453 262144 0.470 0.937 0.609 354.890 0.797 0.984@Daniel: good stuff, but you want to be calculating the depth limit k*log(n) at the start and not at each recursive call.
@matthew, Thanks!
Here’s the updated code along with updated output.
static void introsort_internal(int* array, size_t n, int countdown) { if (n < 2) return; // early cutoff to insertion sort if (n < 22) { insertionsort(array, n); return; } // early cutoff to heapsort if (countdown < 0) { heapsort(array, n); return; } int p = partition(array, n); introsort_internal(array, p + 1, countdown - 1); introsort_internal(&array[p + 1], n - p - 1, countdown - 1); } void introsort(int* array, size_t n) { if (n < 2) return; introsort_internal(array, n, 2 * ilog2(n)); }Output:
1: Array Size 2: Random Array Quicksort 3: Random Array Heapsort 4: Random Array Introsort 5: Killer Array Quicksort 6: Killer Array Heapsort 7: Killer Array Introsort 1 2 3 4 5 6 7 1024 0.000 0.015 0.000 0.000 0.000 0.000 2048 0.000 0.000 0.000 0.016 0.000 0.015 4096 0.000 0.000 0.016 0.047 0.015 0.000 8192 0.016 0.015 0.016 0.297 0.016 0.015 16384 0.016 0.062 0.016 1.172 0.062 0.063 32768 0.062 0.094 0.047 4.953 0.063 0.093 65536 0.125 0.173 0.078 20.625 0.141 0.203 131072 0.233 0.422 0.188 84.719 0.359 0.406 262144 0.470 0.906 0.500 338.078 0.766 0.937