Step by Step solution about How to resolve the algorithm Stirling numbers of the first kind step by step in the C programming language

Overview:

This C program computes and prints the first kind of Stirling numbers, which count the number of ways to partition a set of n elements into k non-empty subsets.

Data Structures:

• stirling_cache: A structure representing a cache of precomputed Stirling numbers. It consists of:
  • max field: Maximum value of n for which Stirling numbers are cached.
  • values field: Pointer to an array of integers storing the cached values.

Functions:

• stirling_number1: Computes the unsigned Stirling number of the first kind for the given values of n and k using recursion and the cache.
• stirling_cache_create: Creates a cache for Stirling numbers up to a specified maximum value. It allocates memory for the cache and precomputes the values.
• stirling_cache_destroy: Frees the memory allocated for the Stirling number cache.
• print_stirling_numbers: Prints a table of Stirling numbers of the first kind up to the specified maximum value.
• main: Entry point of the program. It creates a cache, computes and prints Stirling numbers for a specified maximum value, and then destroys the cache.

Algorithm:

The program uses the following formula to compute Stirling numbers:

S(n, k) = S(n - 1, k - 1) + (n - 1) * S(n - 1, k)

where S(n, k) is the Stirling number of the first kind for n and k.

Execution Flow:

1. The program creates a Stirling number cache up to a specified maximum value.
2. It computes and prints a table of Stirling numbers of the first kind for all values of n and k up to the maximum value.
3. Finally, the program releases the memory allocated for the cache.

#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>

typedef struct stirling_cache_tag {
    int max;
    int* values;
} stirling_cache;

int stirling_number1(stirling_cache* sc, int n, int k) {
    if (k == 0)
        return n == 0 ? 1 : 0;
    if (n > sc->max || k > n)
        return 0;
    return sc->values[n*(n-1)/2 + k - 1];
}

bool stirling_cache_create(stirling_cache* sc, int max) {
    int* values = calloc(max * (max + 1)/2, sizeof(int));
    if (values == NULL)
        return false;
    sc->max = max;
    sc->values = values;
    for (int n = 1; n <= max; ++n) {
        for (int k = 1; k <= n; ++k) {
            int s1 = stirling_number1(sc, n - 1, k - 1);
            int s2 = stirling_number1(sc, n - 1, k);
            values[n*(n-1)/2 + k - 1] = s1 + s2 * (n - 1);
        }
    }
    return true;
}

void stirling_cache_destroy(stirling_cache* sc) {
    free(sc->values);
    sc->values = NULL;
}

void print_stirling_numbers(stirling_cache* sc, int max) {
    printf("Unsigned Stirling numbers of the first kind:\nn/k");
    for (int k = 0; k <= max; ++k)
        printf(k == 0 ? "%2d" : "%10d", k);
    printf("\n");
    for (int n = 0; n <= max; ++n) {
        printf("%2d ", n);
        for (int k = 0; k <= n; ++k)
            printf(k == 0 ? "%2d" : "%10d", stirling_number1(sc, n, k));
        printf("\n");
    }
}

int main() {
    stirling_cache sc = { 0 };
    const int max = 12;
    if (!stirling_cache_create(&sc, max)) {
        fprintf(stderr, "Out of memory\n");
        return 1;
    }
    print_stirling_numbers(&sc, max);
    stirling_cache_destroy(&sc);
    return 0;
}