C-Plus-Plus/math/ncr_modulo_p.cpp

/**
 * @file
 * @brief This program aims at calculating nCr modulo p
 *
 */

#include <cassert>
#include <iostream>
#include <vector>

/** Finds the value of x, y such that a*x + b*y = gcd(a,b)
 *
 * @params[in] the numbers 'a', 'b' and address of 'x' and 'y' from above
 * equation
 * @returns the gcd of a and b
 */
int64_t gcdExtended(int64_t a, int64_t b, int64_t *x, int64_t *y) {
    if (a == 0) {
        *x = 0, *y = 1;
        return b;
    }

    int64_t x1 = 0, y1 = 0;
    int64_t gcd = gcdExtended(b % a, a, &x1, &y1);

    *x = y1 - (b / a) * x1;
    *y = x1;
    return gcd;
}

/** Find modular inverse of a with m i.e. a number x such that (a*x)%m = 1
 *
 * @params[in] the numbers 'a' and 'm' from above equation
 * @returns the modular inverse of a
 */
int64_t modInverse(int64_t a, int64_t m) {
    int64_t x = 0, y = 0;
    int64_t g = gcdExtended(a, m, &x, &y);
    if (g != 1) {  // modular inverse doesn't exist
        return -1;
    }
    else {
        int64_t res = (x % m + m) % m;
        return res;
    }
}

std::vector<int64_t> fac;

/** Find nCr % p
 *
 * @params[in] the numbers 'n', 'r' and 'p'
 * @returns the value nCr % p
 */
int64_t ncr(int64_t n, int64_t r, int64_t p) {
    // Base cases
    if (r > n) {
        return 0;
    }
    if (r == 1) {
        return n % p;
    }
    if (r == 0 || r == n){
        return 1;
    }
    // fac is a global array with fac[r] = (r! % p)
    int64_t denominator = modInverse(fac[r], p);
    if (denominator < 0) {  // modular inverse doesn't exist
        return -1;
    }
    denominator = (denominator * modInverse(fac[n - r], p)) % p;
    if (denominator < 0) {  // modular inverse doesn't exist
        return -1;
    }
    return (fac[n] * denominator) % p;
}

/**
 * @brief Main function
 * @returns 0 on exit
 */
int main() {
    // populate the fac array
    const int64_t size = 1e6 + 1;
    fac = std::vector<int64_t>(size);
    fac[0] = 1;
    const int64_t p = 1e9 + 7;
    for (int i = 1; i <= size; i++) {
        fac[i] = (fac[i - 1] * i) % p;
    }

    // test 6Ci for i=0 to 7
    for (int i = 0; i <= 7; i++) {
        std::cout << 6 << "C" << i << " = " << ncr(6, i, p) << "\n";
    }

    // (52323 C 26161) % (1e9 + 7) = 224944353
    assert(ncr(52323, 26161, p) == 224944353);
    std::cout << "Assertion passed, (52323 C 26161) % (1e9 + 7) = 224944353\n";
}