codeworrior's blog

recurrence
By codeworrior, 13 years ago, In English
A recurrence is defined as
    F(x)=summation { F(y) : y is divisor of x and y!=x }
    F(1)=1
e.g. F(12)=F(1)+F(2)+F(3)+F(4)+F(6)=F(1)+F(2)+F(3)+F(1)+F(2)+F(1)+F(2)+F(3)=8

1 <  x  < 2^31
how can i solve this recurrence relation??
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Comments Comments (13)
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13 years ago, # |
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Either standard DP or memoized recursion.
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13 years ago, # |
← Rev. 2   Vote: I like it 0 Vote: I do not like it
Then, just prime factorize.
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13 years ago, # |
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Sorry, here is my post.

so you can write something like this:

#include <cstdio>
#include <map>
#include <math.h>
using namespace std;

map <int,int> m;
int n;

int rec ( int n )
{
    if ( m.find( n ) != m.end() )
        return m[n];
    int lim = sqrt(double(n));
    int res = 0;
    for ( int i = 1; i <= lim; ++i )
    {
        if ( n % i == 0 )
        {
            res += rec ( i );
            if ( i*i != n && i != 1 )
                res += rec ( n / i );
        }
    }
    m[n] = res;
    return res;
}

int main ()
{
    m[1]=1;
    scanf("%d", &n);
    printf("%d", rec(n) );
    return 0;
}
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  • 13 years ago, # ^ |
      Vote: I like it 0 Vote: I do not like it
    got it ACed..thanx...i never believed it could work.. i thought we have to work out some formula to get it ACed ..thanx..
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    • 13 years ago, # ^ |
        Vote: I like it 0 Vote: I do not like it

      Why not?

      We got about 1000 ( (10^9)^(1/3) ) divisors, make operations (find, add) with Map by log2(10^9) (it's about 30), and with 5000 test cases we got smthng like 1000*30*5000 = 150 000 000 operations (if there is no issue).

      You're welcome. :)

      By the way, where did you find that problem? (link)

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      • 13 years ago, # ^ |
        ← Rev. 2   Vote: I like it 0 Vote: I do not like it
        yes but we dont know what those divisors are..so we have to loop through the sqrt(n) values to get to those divisors...so it should be 46000*30*5000=6900000000(unless we find some way in which we can get divisor in O(n^1/3) if its possible then enlighten me :)  ..
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        • 13 years ago, # ^ |
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          Anyway my solutions passed. ;)
          And could you give a reference on a problem?
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13 years ago, # |
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Note that F(1) occurs d(n)-1 times. Using similar idea, F(p) should occur d(n/p)-1 times (or something similar).

So, at the end, the answer is something like sum(d(n/p)-1) + d(n)-1 where sum is take over all prime divisors of n. Obviously, each of d's here can be found in constant time given that we have a prime factorization for n.

Any more question?
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13 years ago, # |
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You are right; the conclusion I made was way too abrupt. I wonder if there exists a simple 'formula' to get this; my inclination is 'yes' but I have not worked it out in detail yet. I think it can be written as some expression of prime divisors of n. Any idea?
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13 years ago, # |
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i tried to get a formula for 2 days but could not get..i also think there could be some formula for it so i wasnt tying DP and in the end got AC using DP..but there should exist a formula and that would be a lot cooler way to do this problem..
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  • 13 years ago, # ^ |
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    Some things to keep in mind:

    1) F(p^i) = 2^(i-1).
    2) F(n) = F(p1^i1 ... pk^ik)

    Using this, you may be able to reduce the computation time a bit. Right now, even getting F(p^i * q^j) seems non-trivial.

    At this point, I think I would definitely use DP here, after putting 1) in place [even more so in competition-type problem].
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13 years ago, # |
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Have you tried generating some first values and then searching this sequence on oeis.org? I'm quite sure that you will find it there(if formula in closed form exists).
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