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modular_binomial_fast.sf
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#!/usr/bin/ruby
# Translated by: Trizen
# Date: 27 April 2022
# https://github.com/trizen
# Fast algorithm for computing the binomial coefficient modulo some integer m.
# The implementation is based on Lucas' Theorem and its generalization given in the paper
# Andrew Granville "The Arithmetic Properties of Binomial Coefficients", In Proceedings of
# the Organic Mathematics Workshop, Simon Fraser University, December 12-14, 1995.
# Translation of binomod.gp v1.5 by Max Alekseyev, with some minor optimizations.
# See also:
# https://home.gwu.edu/~maxal/gpscripts/
func factorial_without_prime(n,p,pk,from,res) {
return 1 if (n <= 1)
if (p > n) {
return factorialmod(n, pk)
}
# Simple implementation:
# return (1..n -> reduce {|a,b| (p `divides` b) ? a : mulmod(a, b, pk) })
if (*from == n) {
return *res
}
if (*from > n) {
*from = 0
*res = 1
}
var r = *res;
for v in ((*from + 1) .. n) {
(p `divides` v) || do {
r = mulmod(r, v, pk)
}
}
*res = r
*from = n
return r
}
func lucas_theorem(n,k,p) { # p is prime
var r = 1
while (k) {
var (nqp, np) = divmod(n,p)
var (kqp, kp) = divmod(k,p)
if (kp > np) { return 0 }
r = mulmod(r, divmod(
factorialmod(np, p),
factorialmod(kp, p) *
factorialmod(np-kp, p), p),
p)
(n,k) = (nqp,kqp)
}
return r
}
func modular_binomial(n,k,m) {
if (m.is_zero) { return NaN }
if (m.is_one) { return 0 }
if (k.is_neg) { k = (n - k) }
if (k.is_neg) { return 0 }
if (n.is_neg) { return (((-1)**k * __FUNC__(-n + k - 1, k, m)) % m) }
if (k > n) { return 0 }
if (k.is_zero || (k == n)) { return 1%m }
if (k.is_one || (k == n-1)) { return n%m }
if (n-k < k) { k = (n - k) }
var F = []
for p,q in (m.abs.factor_exp) {
if (q == 1) {
F << Mod(lucas_theorem(n,k,p), p)
next
}
var d = n.ilog(p)+1
var np = {|i| idiv(n, p**i) % p }.map(0 .. d)
var kp = {|i| idiv(k, p**i) % p }.map(0 .. d)
var e = []
for i in (0 .. d) {
e[i] = ((np[i] < (kp[i]+(i>0 ? e[i-1] : 0))) ? 1 : 0)
}
for i in (d-1 `downto` 0) {
e[i] += e[i+1]
}
if (e[0] >= q) {
F << Mod(0, p**q)
next
}
var rq = q-e[0]
var pq = p**q
var prq = p**rq
var N = {|i| idiv(n, p**i) % prq }.map(0 .. d)
var K = {|i| idiv(k, p**i) % prq }.map(0 .. d)
var R = {|i| idiv(n-k, p**i) % prq }.map(0 .. d)
var NKR = N.range.map {|i|
[N[i], K[i], R[i], N[i]+K[i]+R[i]]
}.sort_by { .tail }
N = NKR.map{ _[0] }
K = NKR.map{ _[1] }
R = NKR.map{ _[2] }
var acc = [1]
for k in (1 .. min(max(N+K+R), 1e3)) {
if (p `divides` k) {
acc.push(acc[-1])
} else {
acc.push(mulmod(acc[-1], k, prq))
}
}
var from = 0
var res = 1
var v = (p**e[0] * prod(0 .. d, {|j|
var pairs = []
var a = (acc[N[j]] \\ pairs.push([\a, N[j]]))
var b = (acc[K[j]] \\ pairs.push([\b, K[j]]))
var c = (acc[R[j]] \\ pairs.push([\c, R[j]]))
pairs.sort_by { .tail }.each_2d {|x,y|
*x = factorial_without_prime(y, p, prq, \from, \res)
}
Mod(a, pq) / (b * c)
}))
if ((p>2 || rq<3) && (q<=e.len)) {
v *= (-1)**e[rq-1]
}
F << v
}
F || return 0
chinese(F...).lift % m
}
say ("binomial(10^10, 10^5) mod 13! = ", modular_binomial(1e10, 1e5, 13!))
#
## Run some tests
#
assert_eq(modular_binomial(10, 2, 43), 2)
assert_eq(modular_binomial(10, 8, 43), 2)
assert_eq(modular_binomial(10, 2, 24), 21)
assert_eq(modular_binomial(10, 8, 24), 21)
assert_eq(modular_binomial(100, 42, -127), binomial(100, 42) % -127)
assert_eq(modular_binomial(12, 5, 100000), 792)
assert_eq(modular_binomial(16, 4, 100000), 1820)
assert_eq(modular_binomial(100, 50, 139), 71)
assert_eq(modular_binomial(1000, 10, 1243), 848)
assert_eq(modular_binomial(124, 42, 1234567), 395154)
assert_eq(modular_binomial(1e9, 1e4, 1234567), 833120)
assert_eq(modular_binomial(1e10, 1e5, 1234567), 589372)
assert_eq(modular_binomial(-1e10, 1e5, 4233330243), 2865877173)
assert_eq(modular_binomial(1e10, 1e4, 13!), 1845043200)
assert_eq(modular_binomial(1e10, 1e5, 13!), 1556755200)
assert_eq(modular_binomial(1e10, 1e6, 13!), 5748019200)
assert_eq(modular_binomial(-1e10, 1e4, 13!), 4151347200)
assert_eq(modular_binomial(-1e10, 1e5, 13!), 1037836800)
assert_eq(modular_binomial(-1e10, 1e6, 13!), 2075673600)
assert_eq(modular_binomial(3, 1, 9), binomial(3, 1) % 9)
assert_eq(modular_binomial(4, 1, 16), binomial(4, 1) % 16)
assert_eq(modular_binomial(1e9, 1e5, 43*97*503), 585492)
assert_eq(modular_binomial(1e9,1e6,5041689707), 15262431)
assert_eq(modular_binomial(1e7,1e5, 43**2 * 97**3 * 13**4), 1778017500428)
assert_eq(modular_binomial(1e7,1e5, 42**2 * 97**3 * 13**4), 10015143223176)
assert_eq(modular_binomial(1e9,1e5,12345678910), 4517333900)
assert_eq(modular_binomial(1e9,1e6,13**2 * 5**6), 2598375)
assert_eq(modular_binomial(1e10, 1e5, 1234567), 589372)
assert_eq(modular_binomial(1e5, 1e3, 43), binomial(1e5,1e3)%43)
assert_eq(modular_binomial(1e5, 1e3, 43*97), binomial(1e5,1e3) % (43*97))
assert_eq(modular_binomial(1e5, 1e3, 43*97*43), binomial(1e5,1e3) % (43*97*43))
assert_eq(modular_binomial(1e5, 1e3, 43*97*(5**5)), binomial(1e5,1e3) % (43*97*(5**5)))
assert_eq(modular_binomial(1e5,1e3, next_prime(1e4)**2), binomial(1e5, 1e3) % next_prime(1e4)**2)
assert_eq(modular_binomial(1e5,1e3, next_prime(1e4)), binomial(1e5, 1e3) % next_prime(1e4))
assert_eq(modular_binomial(1e6,1e3, next_prime(1e5)), binomial(1e6, 1e3) % next_prime(1e5))
assert_eq(modular_binomial(1e6,1e3, next_prime(1e7)), binomial(1e6, 1e3) % next_prime(1e7))
assert_eq(modular_binomial(1234567, 1e3, 20!), binomial(1234567, 1e3)%20!)
assert_eq(modular_binomial(1234567, 1e4, 20!), binomial(1234567, 1e4)%20!)
__END__
var upto = 10
for n in (-upto .. upto), k in (-upto .. upto), m in (-upto .. upto) {
if (m != 0) {
say "Testing: binomial(#{n}, #{k}, #{m})"
assert_eq(modular_binomial(n, k, m), binomial(n, k) % m)
}
}