AmericanPsycho's blog

Compute binomial coefficients

By AmericanPsycho, history, 9 years ago, In English

Hi everyone.

Lately I've done some combinatorial problems. And I am wondering.
What methods are used to calculate the binomial coefficients?
What is the most effective?

I have used dynamic programming, although it is effective for many
queries and can be modulated with any number, the range is too limited.


When the number is a bit higher, I have used modular arithmetic
but I need to precalculate all factorials and is needed a prime number
as a module and really the range is not so large just 10 ^ 5 maybe.
If the problem have higher numbers, I'm lost

Can you tell me what techniques you use and when they can be used are?
I hope you can help me
Thank you

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9 years ago, hide # |
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Vote: I like it -20 Vote: I do not like it

Sorry, I didn't read your blog fully

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9 years ago, hide # |
 
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In nCr mod p if numbers are large(upto 10^18) and p is not a prime number. So in that case: p = a1^b1*a2^b2*a3^b3....an^bn. so in that case find nCr mod (a1^b1), nCr mod(a2^b2) till nCr mod(an^bn) using lucas theorem and further use CRT to get the end result. Problem link: https://www.codechef.com/MAY17/problems/SANDWICH

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9 years ago, hide # |
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Contest finished, I'm ready to give solution for your question:

You can find C(n, r) in

Unable to parse markup [type=CF_TEX]

time with O(n) precalculating. Read about modular inverse of number, precalculate factorials, then use formula: 
intt InverseEuler (intt n, intt MOD) {
    return powmod(n, MOD-2, MOD);  /// here powmod means binary exponentiation
}

intt C(intt n, intt r, intt MOD) {
    return (f[n]*((InverseEuler(f[r], MOD) * InverseEuler(f[n-r], MOD)) % MOD)) % MOD;
}
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    9 years ago, hide # ^ |
     
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    You can precalculate modular inverses in

    Unable to parse markup [type=CF_TEX]

    and you can find C(n, r) in

    Unable to parse markup [type=CF_TEX]

    .
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      9 years ago, hide # ^ |
       
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      Yes, but code will be O(nlogn) anyway.

      If there are queries, precalculating will be nice optimization.

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        9 years ago, hide # ^ |
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        Nope, it will be

        Unable to parse markup [type=CF_TEX]

        .
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          9 years ago, hide # ^ |
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          If you calculate inverses of factorials then how can you do in

          Unable to parse markup [type=CF_TEX]

          ? I know a way to calculate all modular inverses from 1 to Mod, in

          Unable to parse markup [type=CF_TEX]

          Then Pre-calculation will take O(n log MOD) too!

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    9 years ago, hide # ^ |
     
    Vote: I like it +7 Vote: I do not like it

    This is not what he's asking for :) he says that he already knows this.

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