## 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.

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### 8 Responses to “Introspective Sort”

1. matthew said

Is that a natural logarithm?

2. programmingpraxis said

The `log` function 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.

3. matthew said

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).

4. programmingpraxis said

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.

5. Daniel said

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;
}
```
```  1024 0.000 0.000 0.000   0.000 0.000 0.000
2048 0.016 0.000 0.000   0.031 0.000 0.000
4096 0.016 0.000 0.000   0.078 0.015 0.016
8192 0.000 0.015 0.000   0.344 0.016 0.015
16384 0.032 0.031 0.031   1.234 0.032 0.047
32768 0.062 0.078 0.047   4.938 0.093 0.078
65536 0.094 0.203 0.126  20.765 0.171 0.235
131072 0.234 0.485 0.296  86.234 0.360 0.469
262144 0.516 0.890 0.610 349.516 0.718 1.000
```
6. Daniel said

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
```
7. matthew said

@Daniel: good stuff, but you want to be calculating the depth limit k*log(n) at the start and not at each recursive call.

8. Daniel said

@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
```