Fastest way to generate binomial coefficients

Question

I need to calculate combinations for a number.

What is the fastest way to calculate nCp where n>>p?

I need a fast way to generate binomial coefficients for a

Answer 1

I recently wrote a piece of code that needed to call for a binary coefficient about 10 million times. So I did a combination lookup-table/calculation approach that's still not too wasteful of memory. You might find it useful (and my code is in the public domain). The code is at

http://www.etceterology.com/fast-binomial-coefficients

It's been suggested that I inline the code here. A big honking lookup table seems like a waste, so here's the final function, and a Python script that generates the table:

extern long long bctable[]; /* See below */

long long binomial(int n, int k) {
    int i;
    long long b;
    assert(n >= 0 && k >= 0);

    if (0 == k || n == k) return 1LL;
    if (k > n) return 0LL;

    if (k > (n - k)) k = n - k;
    if (1 == k) return (long long)n;

    if (n <= 54 && k <= 54) {
        return bctable[(((n - 3) * (n - 3)) >> 2) + (k - 2)];
    }
    /* Last resort: actually calculate */
    b = 1LL;
    for (i = 1; i <= k; ++i) {
        b *= (n - (k - i));
        if (b < 0) return -1LL; /* Overflow */
        b /= i;
    }
    return b;
}

---

#!/usr/bin/env python3

import sys

class App(object):
    def __init__(self, max):
        self.table = [[0 for k in range(max + 1)] for n in range(max + 1)]
        self.max = max

    def build(self):
        for n in range(self.max + 1):
            for k in range(self.max + 1):
                if k == 0:      b = 1
                elif  k > n:    b = 0
                elif k == n:    b = 1
                elif k == 1:    b = n
                elif k > n-k:   b = self.table[n][n-k]
                else:
                    b = self.table[n-1][k] + self.table[n-1][k-1]
                self.table[n][k] = b

    def output(self, val):
        if val > 2**63: val = -1
        text = " {0}LL,".format(val)

        if self.column + len(text) > 76:
            print("\n   ", end = "")
            self.column = 3
        print(text, end = "")
        self.column += len(text)

    def dump(self):
        count = 0
        print("long long bctable[] = {", end="");

        self.column = 999
        for n in range(self.max + 1):
            for k in range(self.max + 1):
                if n < 4 or k < 2 or k > n-k:
                    continue
                self.output(self.table[n][k])
                count += 1
        print("\n}}; /* {0} Entries */".format(count));

    def run(self):
        self.build()
        self.dump()
        return 0

def main(args):
    return App(54).run()

if __name__ == "__main__":
    sys.exit(main(sys.argv))