在python中求解（m选择x）= n的高效算法

Question

Given

m : number of posters to design
n : total number of available colors

solve for

x : number of colors for each poster so that each poster has a unique combination of colors

subject to the equation

(n choose x) = m

I have coded the above problem in python and the source code is given below

factorial = []

def generateList(n):
    factorial.append(1)
    factorial.append(1)

    for i in range(2,n+1):
        factorial.append(i * factorial[i-1])

def calculateFac(n,i):
    return int((factorial[n]) / (factorial[i] * factorial[n-i]))

def ColorChoice(m,n):
    for i in range(1,int(n/2)+1):
        if m == calculateFac(n,i):
            return i
    return -1

def checkChoose(m,n):
    generateList(n)
    return ColorChoice(m,n)

print (checkChoose(35,7))

The above solution will only work for small integers but I need a solution to work for larger numbers, eg when n = 47129212243960.

Is there any efficient way to solve this?

Answer 1

Since (n choose x) == (n choose (nx)) , and it seems like you want to find the minimal x , we can search for x between 0 and n/2 . Also, for arbitrary n and m , it's possible no such x exists, but probably what you want is the minimal such x , if one exists, such that (n choose x) >= m , ie the smallest x guaranteeing you can make m unique colour combinations -- with that x you may even be able to make more than m unique colour combinations.

There's a simple O(n) solution, using the fact that (n choose (x+1)) / (n choose x) == (nx)/(x+1) , which you can see by expanding the "choose" expressions in terms of factorials, and cancelling things out.

def x(m,n):
    n_choose_x = 1
    for x in xrange(1, n/2 + 1):
        n_choose_x = n_choose_x * (n+1-x) / x
        if n_choose_x >= m:
            return x
    return -1

print(x(70,8))
print(x(71,8))
print(x(57,8))
print(x(56,8))
print(x(55,8))
print("")
print(x(9999999, 47129212243960))
print(x(99999999471292122439609999999, 47129212243960))
print(x(99999999947129212243960999999471292122439609999999, 47129212243960))

This prints:

4
-1
4
3
3

1
3
4