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 Subject: Re: [OMPI devel] Reviewing MPI_Dims_create From: Christoph Niethammer (niethammer_at_[hidden]) Date: 2014-02-11 06:02:42 Hello, After rethinking Jeff's comments about caching prime numbers I came to the conclusion that we can omit the prime numbers at all and go directly for the factorization. :D We then only need at most log_2(INT_MAX) * sizeof(int) = 32 * 4 byte = 128 byte of memory for the factors. Computational costs reduce as well as the factorization itself is done by a loop with at most \sqrt(num) / 2 iterations - which is the same as in the original prime number detection loop. I think this is the cleanest way which reduces also source code size. ;) Find attache patch against the trunk. Best regards Christoph -- Christoph Niethammer High Performance Computing Center Stuttgart (HLRS) Nobelstrasse 19 70569 Stuttgart Tel: ++49(0)711-685-87203 email: niethammer_at_[hidden] http://www.hlrs.de/people/niethammer ----- UrsprÃ¼ngliche Mail ----- Von: "Andreas SchÃ¤fer" An: "Open MPI Developers" Gesendet: Dienstag, 11. Februar 2014 06:24:56 Betreff: Re: [OMPI devel] Reviewing MPI_Dims_create OK, so we were thinking the same thing :-) The optimization you mention below is the same I've used in my updated patch. On 02:29 Tue 11 Feb , Christoph Niethammer wrote: > As mentioned in my former mail I did not touch the factorization > code. > But to figure out if a number n is *not* a prime number it is sufficient to check up to \sqrt(n). > Proof: > let n = p*q with q > \sqrt{n} > --> p < \sqrt(n) > So we have already found factor p before reaching \sqrt(n) and by this n is no prime any more and no need for further checks. ;) > > > The mentioned factorization may indeed include one factor which is larger than \sqrt(n). :) > > Proof that at least one prime factor can be larger than \sqrt(n) example: > 6 = 2*3 > \sqrt(6) = 2.4494897427832... < 3 Q.E.D. > > > Proof that no more than one factor can be larger than \sqrt(n): > let n = \prod_{i=0}^K p_i with p_i \in N and K > 2 > and assume w.l.o.g. p_0 > \sqrt(n) and p_1 > \sqrt(n) > --> 1 > \prod_{i=2}^K p_i > which is a contradiction as all p_i \in N. Q.E.D. > > > So your idea is still applicable with not much effort and we only need prime factors up to sqrt(n) in the factorizer code for an additional optimization. :) > > First search all K' factors p_i < \sqrt(n). If then n \ne \prod_{i=0}^{K'} p_i we should be sure that p_{K'+1} = n / \prod_{i=0}^{K'} p_i is a prime. No complication with counts IMHO. I leave this without patch as it is already 2:30 in the morning. :P > > Regards > Christoph > > -- > > Christoph Niethammer > High Performance Computing Center Stuttgart (HLRS) > Nobelstrasse 19 > 70569 Stuttgart > > Tel: ++49(0)711-685-87203 > email: niethammer_at_[hidden] > http://www.hlrs.de/people/niethammer > > ----- UrsprÃ¼ngliche Mail ----- > Von: "Andreas SchÃ¤fer" > An: "Open MPI Developers" > Gesendet: Montag, 10. Februar 2014 23:24:24 > Betreff: Re: [OMPI devel] Reviewing MPI_Dims_create > > Christoph- > > your patch has the same problem as my original patch: indeed there may > be a prime factor p of n with p > sqrt(n). What's important is that > there may only be at most one. I've submitted an updated patch (see my > previous mail) which catches this special case. > > Best > -Andreas > > > On 19:30 Mon 10 Feb , Christoph Niethammer wrote: > > Hello, > > > > I noticed some effort in improving the scalability of > > MPI_Dims_create(int nnodes, int ndims, int dims[]) > > Unfortunately there were some issues with the first attempt (r30539 and r30540) which were reverted. > > > > So I decided to give it a short review based on r30606 > > https://svn.open-mpi.org/trac/ompi/browser/trunk/ompi/mpi/c/dims_create.c?rev=30606 > > > > > > 1.) freeprocs is initialized to be nnodes and the subsequent divisions of freeprocs have all positive integers as divisor. > > So IMHO it would make more sense to check if nnodes > 0 in the MPI_PARAM_CHECK section at the begin instead of the following (see patch 0001): > > > > 99 if (freeprocs < 1) { > > 100 return OMPI_ERRHANDLER_INVOKE(MPI_COMM_WORLD, MPI_ERR_DIMS, > > 101 FUNC_NAME); > > 102 } > > > > > > 2.) I rewrote the algorithm stopping at sqrt(n) in getprimes(int num, int *nprimes, int **pprimes) > > which makes mathematically more sens (as the largest prime factor of any number n cannot exceed \sqrt{n}) - and should produce the right result. ;) > > (see patch 0002) > > Here the improvements: > > > > module load mpi/openmpi/trunk-gnu.4.7.3 > > $./mpi-dims-old 1000000 > > time used for MPI_Dims_create(1000000, 3, {}): 8.104007 > > module swap mpi/openmpi/trunk-gnu.4.7.3 mpi/openmpi/trunk-gnu.4.7.3-testing > >$ ./mpi-dims-new 1000000 > > time used for MPI_Dims_create(1000000, 3, {}): 0.060400 > > > > > > 3.) Memory allocation for the list of prime numbers may be reduced up to a factor of ~6 for 1mio nodes using the result from Dusart 1999 [1]: > > \pi(x) < x/ln(x)(1+1.2762/ln(x)) for x > 1 > > Unfortunately this saves us only 1.6 MB per process for 1mio nodes as reported by tcmalloc/pprof on a test program - but it may sum up with fatter nodes. :P > > > > $pprof --base=$PWD/primes-old.0001.heap a.out primes-new.0002.heap > > (pprof) top > > Total: -1.6 MB > > 0.3 -18.8% -18.8% 0.3 -18.8% getprimes2 > > 0.0 -0.0% -18.8% -1.6 100.0% __libc_start_main > > 0.0 -0.0% -18.8% -1.6 100.0% main > > -1.9 118.8% 100.0% -1.9 118.8% getprimes > > > > Find attached patch for it in 0003. > > > > > > If there are no issues I would like to commit this to trunk for further testing (+cmr for 1.7.5?) end of this week. > > > > Best regards > > Christoph > > > > [1] http://www.ams.org/journals/mcom/1999-68-225/S0025-5718-99-01037-6/home.html > > > > > > > > -- > > > > Christoph Niethammer > > High Performance Computing Center Stuttgart (HLRS) > > Nobelstrasse 19 > > 70569 Stuttgart > > > > Tel: ++49(0)711-685-87203 > > email: niethammer_at_[hidden] > > http://www.hlrs.de/people/niethammer > > > From e3292b90cac42fad80ed27a555419002ed61ab66 Mon Sep 17 00:00:00 2001 > > From: Christoph Niethammer > > Date: Mon, 10 Feb 2014 16:44:03 +0100 > > Subject: [PATCH 1/3] Move parameter check into appropriate code section at the > > begin. > > > > --- > > ompi/mpi/c/dims_create.c | 11 ++++++----- > > 1 file changed, 6 insertions(+), 5 deletions(-) > > > > diff --git a/ompi/mpi/c/dims_create.c b/ompi/mpi/c/dims_create.c > > index d2c3858..3d0792f 100644 > > --- a/ompi/mpi/c/dims_create.c > > +++ b/ompi/mpi/c/dims_create.c > > @@ -71,6 +71,11 @@ int MPI_Dims_create(int nnodes, int ndims, int dims[]) > > return OMPI_ERRHANDLER_INVOKE (MPI_COMM_WORLD, > > MPI_ERR_DIMS, FUNC_NAME); > > } > > + > > + if (1 > nnodes) { > > + return OMPI_ERRHANDLER_INVOKE (MPI_COMM_WORLD, > > + MPI_ERR_DIMS, FUNC_NAME); > > + } > > } > > > > /* Get # of free-to-be-assigned processes and # of free dimensions */ > > @@ -95,11 +100,7 @@ int MPI_Dims_create(int nnodes, int ndims, int dims[]) > > FUNC_NAME); > > } > > > > - if (freeprocs < 1) { > > - return OMPI_ERRHANDLER_INVOKE(MPI_COMM_WORLD, MPI_ERR_DIMS, > > - FUNC_NAME); > > - } > > - else if (freeprocs == 1) { > > + if (freeprocs == 1) { > > for (i = 0; i < ndims; ++i, ++dims) { > > if (*dims == 0) { > > *dims = 1; > > -- > > 1.8.3.2 > > > > > From bc862c47ef8d581a8f6735c51983d6c9eeb95dfd Mon Sep 17 00:00:00 2001 > > From: Christoph Niethammer > > Date: Mon, 10 Feb 2014 18:50:51 +0100 > > Subject: [PATCH 2/3] Speeding up detection of prime numbers using the fact > > that the largest prime factor of any number n cannot exceed \sqrt{n}. > > > > --- > > ompi/mpi/c/dims_create.c | 9 ++++++--- > > 1 file changed, 6 insertions(+), 3 deletions(-) > > > > diff --git a/ompi/mpi/c/dims_create.c b/ompi/mpi/c/dims_create.c > > index 3d0792f..1c1c381 100644 > > --- a/ompi/mpi/c/dims_create.c > > +++ b/ompi/mpi/c/dims_create.c > > @@ -5,7 +5,7 @@ > > * Copyright (c) 2004-2005 The University of Tennessee and The University > > * of Tennessee Research Foundation. All rights > > * reserved. > > - * Copyright (c) 2004-2005 High Performance Computing Center Stuttgart, > > + * Copyright (c) 2004-2014 High Performance Computing Center Stuttgart, > > * University of Stuttgart. All rights reserved. > > * Copyright (c) 2004-2005 The Regents of the University of California. > > * All rights reserved. > > @@ -20,6 +20,8 @@ > > > > #include "ompi_config.h" > > > > +#include > > + > > #include "ompi/mpi/c/bindings.h" > > #include "ompi/runtime/params.h" > > #include "ompi/communicator/communicator.h" > > @@ -293,13 +295,14 @@ getprimes(int num, int *pnprime, int **pprimes) { > > primes[i++] = 2; > > > > for (n = 3; n <= num; n += 2) { > > - for (j = 1; j < i; ++j) { > > + double nsqrt = sqrt((double) n); > > + for(j = 0; (double) primes[j] <= nsqrt; j++) { > > if ((n % primes[j]) == 0) { > > break; > > } > > } > > > > - if (j == i) { > > + if (primes[j] > nsqrt) { > > if (i >= size) { > > return MPI_ERR_DIMS; > > } > > -- > > 1.8.3.2 > > > > > From a012206cfbf7b8b22cef4cc5b811384e5e3ac32c Mon Sep 17 00:00:00 2001 > > From: Christoph Niethammer > > Date: Mon, 10 Feb 2014 19:02:03 +0100 > > Subject: [PATCH 3/3] Reduce memory usage by a better approximation for the > > upper limit of the number of primes. > > > > --- > > ompi/mpi/c/dims_create.c | 12 ++++++++++-- > > 1 file changed, 10 insertions(+), 2 deletions(-) > > > > diff --git a/ompi/mpi/c/dims_create.c b/ompi/mpi/c/dims_create.c > > index 1c1c381..8dd3144 100644 > > --- a/ompi/mpi/c/dims_create.c > > +++ b/ompi/mpi/c/dims_create.c > > @@ -281,9 +281,17 @@ getprimes(int num, int *pnprime, int **pprimes) { > > int n; > > int size; > > int *primes; > > + double lognum; > > > > - /* Allocate the array of primes */ > > - size = (num / 2) + 1; > > + /* Allocate the array of primes > > + * see Pierre Dusart, Math. Comp. 68 (1999), no. 225, 411???415.*/ > > + lognum = log(num); > > + if(num > 1) { > > + size = ceil(num/lognum * (1+1.2762/lognum)); > > + } > > + else { > > + size = 1; > > + } > > primes = (int *) malloc((unsigned) size * sizeof(int)); > > if (NULL == primes) { > > return MPI_ERR_NO_MEM; > > -- > > 1.8.3.2 > > > > > _______________________________________________ > > devel mailing list > > devel_at_[hidden] > > http://www.open-mpi.org/mailman/listinfo.cgi/devel > > > -- > ========================================================== > Andreas SchÃ¤fer > HPC and Grid Computing > Chair of Computer Science 3 > Friedrich-Alexander-UniversitÃ¤t Erlangen-NÃ¼rnberg, Germany > +49 9131 85-27910 > PGP/GPG key via keyserver > http://www.libgeodecomp.org > ========================================================== > > (\___/) > (+'.'+) > (")_(") > This is Bunny. Copy and paste Bunny into your > signature to help him gain world domination! > > _______________________________________________ > devel mailing list > devel_at_[hidden] > http://www.open-mpi.org/mailman/listinfo.cgi/devel > _______________________________________________ > devel mailing list > devel_at_[hidden] > http://www.open-mpi.org/mailman/listinfo.cgi/devel -- ========================================================== Andreas SchÃ¤fer HPC and Grid Computing Chair of Computer Science 3 Friedrich-Alexander-UniversitÃ¤t Erlangen-NÃ¼rnberg, Germany +49 9131 85-27910 PGP/GPG key via keyserver http://www.libgeodecomp.org ========================================================== (\___/) (+'.'+) (")_(") This is Bunny. Copy and paste Bunny into your signature to help him gain world domination! _______________________________________________ devel mailing list devel_at_[hidden] http://www.open-mpi.org/mailman/listinfo.cgi/devel  text/x-patch attachment: 0001-Omit-usage-of-pre-calculated-prime-numbers-and-facto.patch