Dienstag, 21. Mai 2013

Parallel all permutations - non-recursive - GPU optimal

This post is a general one - not voxel related.

Some time ago I developed a a simple, fast and yet non-recursive method to directly compute the n'th permutation. Thats useful especially when working on the graphics card with CUDA for parallel evaluations. (it might have been published somewhere already - so if you know the reference, you can post it as a comment)

The algorithm is based on the following property. Lets say there are 6 characters in the string to permute. Then the result is 6! = 720 combinations. In the first column of the results, we get 720/6=120 lines of each character. This means if the index counts from 0 to 719, then the first character to be removed is computed as remove_index = index/120.

For the next column, 5 numbers are left. This means, that 120 = 5 * 24 lines of the same character - and so on. 

We the following list of removes:
column0: remove_index = (a/120)%5;
column1: remove_index = (a/24)%4;
column2: remove_index = (a/6)%3;
column3: remove_index = (a/2)%2;
column4: remove_index = (a/1)%1;

Code snipplet:

std::string default = "12345";
int perm=1, digits=default.size();
for (int i=1;i<=digits;perm*=i++);
for (int a=0;a
<perm;a++)
{
std::string avail=default;
for (int b=digits,div=perm;b>0; b--)
{
div/=b;
int index = (a/div)%b;
printf("%c", avail[index] );
//avail[index]=avail[b-1]; // non-lexigraphic but fast
avail.erase(index,1) ; // lexigraphically correct
}
printf("\n");
}
printf("permutations:%d\n",perm);

In the code, perm defines the Lehmer code (wiki)

Kommentare:

  1. Can you give us an example of a more practical use for your algorithm, please?

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  2. It has multiple applications that range from cryptography over group theory (math), randomization to the travelling salesman problem

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  3. Writing this just before going to bed (and thus implying an excuse for not really understanding your code snippet one bit atm :P). But I'd say it looks really, really brilliant!

    I made a similar one a while (well, years) ago. Not so elegant or short, but it does the same thing - outputs a particular permutation for a string directly (Or actually not a string per se, but a character position remapping array). Just input the string and which permutation nr. The last permutation is the input string in reverse.

    Code snippet (obviously I'm not as versed in C++ as you):


    // Make combination nr. by mapping letter positions to p[]
    // Combination nr. 1 is original string
    // last combination is original string backwards

    // nr = combination (or permutation) nr to return.
    // a = nr of characters in the string
    // p[x] = string remapper

    // p[x] = new x.th letter position from position p[x] in source string
    // ex: p[1] = 3 means new first character is the old 3rd character (from original string).

    void shuffle(long nr, int a, int p[])
    {
    int x = a;
    int s;
    int y;
    do // magic
    {
    s = a - x; // shuffle in reverse (more natural)
    p[s] = (int)((nr-1)/fac((long)x-1))+1;
    nr = nr - (fac(x-1) * (long)(p[s]-1));
    x--;
    } while (x>0);
    x=a-1;
    do // more magic
    {
    y = x - 1;
    do
    {
    if (p[y]<=p[x]) p[x] = p[x] + 1;
    y--;
    } while (y>=0);
    x--;
    } while (x>0);
    }

    (Probably the "shuffle" function should have been named "permute").

    Regards,
    raron

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  4. faster version here which also evaluates

    https://sites.google.com/site/cudapermutations/

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    1. Thank you for the reference. Does it have a lower complexity or is it just faster due to GPU?

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