Efficient Generation of Nth Row of Binomial Coefficient Modulo M

Efficient Generation of Nth Row of Binomial Coefficient Modulo M

Efficient Generation of Nth Row of Binomial Coefficient Modulo M

Introduction

Calculating the binomial coefficient (also known as the combinatorial coefficient) is a common task in mathematics and programming. However, when the modulo M is not prime, finding an efficient method becomes more challenging. In this article, we will explore a technique that allows us to efficiently generate the Nth row of the binomial coefficient modulo M, even when M is not necessarily prime. This technique utilizes prime factorization and a sieve algorithm to achieve efficient computation.

The Implementation

Let's take a look at the implementation of the technique:


from collections import Counter

N = 1 + 10 ** 6
v = [False] * N
spf = [1] * N

for i in range(2, N, 2):
    spf[i] = 2
  
for i in range(3, N, 2):
    if not v[i]:
        spf[i] = i
        j = i
        while j * i < N:
            if not v[j * i]:
                v[j * i] = True
                spf[j * i] = i
            j += 2

def facts(n, mod):
    res = [1] * (n + 1)
    ctr = Counter()
    for i in range(1, n + 1):
        a = n - i + 1
        b = i
        while a > 1:
            ctr[spf[a]] += 1
            a //= spf[a]
        while b > 1:
            ctr[spf[b]] -= 1
            b //= spf[b]
        for p in ctr:
            res[i] *= pow(p, ctr[p], mod)
            res[i] %= mod
    return res
     
print(*facts(1000, 18))

Explanation

The implementation consists of two parts: the sieve algorithm to find the smallest prime factors and the main function `facts()` to generate the Nth row of the binomial coefficient modulo M.

The sieve algorithm populates the `spf` (smallest prime factor) array, which stores the smallest prime factor for each number up to N. This is done by iterating through the numbers starting from 2, marking the multiples of each prime number, and assigning the smallest prime factor accordingly. This step enables efficient prime factorization later in the `facts()` function.

The `facts()` function takes two parameters: `n` and `mod`. It initializes a result array `res` with all elements set to 1. The function also maintains a counter `ctr` using the `Counter()` class from the `collections` module.

Inside the main loop, for each value of `i` from 1 to `n`, the function calculates the prime factorization of the numerator and denominator of the binomial coefficient using the `ctr` counter. It starts by initializing `a` with `n - i + 1` and `b` with `i`. Then, it iteratively divides `a` and `b` by their smallest prime factors (`spf`) until they are reduced to 1. During this process, it updates the `ctr` counter by incrementing the count for `spf[a]` and decrementing the count for `spf[b]`.

After obtaining the prime factorization, the function calculates the binomial coefficient modulo M by iterating through each prime factor (`p`) in the `ctr` counter. It uses the `pow()` function with three arguments: `p` as the base, `ctr[p]` as the exponent, and `mod` as the modulo. This efficiently computes `pow(p, ctr[p], mod)`.

Finally, the function multiplies the calculated value with the corresponding element in the `res` array and takes the modulo `mod` to ensure the result stays within the desired range. The `res` array stores the Nth row of the binomial coefficient modulo M, with each element representing a specific binomial coefficient value.

Conclusion

The technique presented in this article provides an efficient way to generate the Nth row of the binomial coefficient modulo M, even when M is not necessarily prime. By leveraging prime factorization and a sieve algorithm, we can compute the binomial coefficient values modulo M efficiently. This technique is valuable when dealing with large values of N and non-prime modulo values. Understanding and implementing this technique can enhance our ability to solve combinatorial problems involving binomial coefficients in various mathematical and programming scenarios.

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