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Re: PRNG/TRNG Cesaro's theorem

by danaj (Friar)
on Oct 08, 2017 at 08:13 UTC ( #1200939=note: print w/replies, xml ) Need Help??


in reply to PRNG/TRNG Cesaro's theorem

Of course you will want to do it differently, but here is something I wrote to play with this. I used the ntheory module because it has different RNGs as well as gcd and Pi.

#!/usr/bin/env perl use warnings; use strict; use ntheory ":all"; cesaro("drand48", sub { int(rand(1<<32)) } ); cesaro("ChaCha20", sub { irand64 } ); cesaro("/dev/urandom", sub { unpack("Q",random_bytes(8)) } ); sub cesaro { my($name, $rng) = @_; print "Using $name:\n"; for my $e (1..7) { my $n = 10**$e; my $t = 0; for (1..$n) { $t++ if gcd( $rng->(), $rng->() ) == 1; } printf "%8d %10.8f\n", $n, sqrt(6*$n/$t); } print " Pi ", Pi(), "\n\n"; }

I saved cut-n-paste by passing in the RNG code. One for Perl's default rand (which is drand48 on modern Perl), one for ntheory's CSPRNG, and one that gets data from /dev/urandom. You could replace that with something that got data from random.org. There are even a couple modules that do it automatically (Math::RandomOrg and Data::Entropy). Since we expect t/n ~ 6/Pi^2, a little algebra gives us Pi ~ sqrt(6n/t). I don't believe we can read anything into what the results might mean for the different RNG methods. The results will be different every time unless we seed (and the last can't be seeded since it is O/S collected entropy (long technical discussion of /dev/random vs. /dev/urandom vs. hardware could be had here)).

Using drand48: 10 2.73861279 100 3.06186218 1000 3.15964572 10000 3.14632429 100000 3.13988122 1000000 3.14035598 10000000 3.14143144 Pi 3.14159265358979 Using ChaCha20: 10 3.16227766 100 3.18896402 1000 3.17287158 10000 3.15675816 100000 3.14329189 1000000 3.13978836 10000000 3.14138726 Pi 3.14159265358979 Using /dev/urandom: 10 3.46410162 100 3.03821810 1000 3.11588476 10000 3.14217962 100000 3.13643021 1000000 3.14020372 10000000 3.14130744 Pi 3.14159265358979

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Re^2: PRNG/TRNG Cesaro's theorem
by marioroy (Parson) on Oct 09, 2017 at 01:30 UTC

    Hello brothers and sisters of the Monastery,

    Seeing danaj's post made me think of passing arguments to workers for a parallel demonstration. But first, I need to check if random numbers are unique between workers. They are not for non-threaded workers, irand64 and random_bytes.

    Here is the test code. Calling MCE::relay is a way to have workers display output orderly, starting with worker 1, 2, ..., 8. The init_relay value isn't used here, but the option tells MCE to load MCE::Relay and enable relay capability. Workers persist between each run.

    use strict; use warnings; use ntheory ":all"; use MCE::Flow; my ( $name, $rng ); my %rand = ( "drand48" => sub { int(rand(1 << 32)) }, "ChaCha20" => sub { irand64() }, "/dev/urandom" => sub { unpack("Q", random_bytes(8)) } ); MCE::Flow::init( max_workers => 8, init_relay => 0, user_begin => sub { $name = MCE->user_args()->[0]; $rng = $rand{ $name }; } ); sub func { MCE::relay { print MCE->wid(), ": ", $rng->(), "\n"; }; } for my $name ( "drand48", "ChaCha20", "/dev/urandom" ) { print "Usage $name:\n"; mce_flow { user_args => [$name] }, \&func; print "\n"; }

    Output.

    Usage drand48: 1: 3498494761 2: 2506930441 3: 1515366121 4: 523801801 5: 3827204777 6: 2835640457 7: 1844076137 8: 852511817 Usage ChaCha20: 1: 4471005142860083063 2: 4471005142860083063 3: 4471005142860083063 4: 4471005142860083063 5: 4471005142860083063 6: 4471005142860083063 7: 4471005142860083063 8: 4471005142860083063 Usage /dev/urandom: 1: 15746895497052787399 2: 15746895497052787399 3: 15746895497052787399 4: 15746895497052787399 5: 15746895497052787399 6: 15746895497052787399 7: 15746895497052787399 8: 15746895497052787399

    Later today will release MCE 1.831 and MCE::Shared 1.832 containing the fix.

    Usage drand48: 1: 600074529 2: 3903477505 3: 2911913185 4: 1920348865 5: 928784545 6: 4232187521 7: 3240623201 8: 2249058881 Usage ChaCha20: 1: 8740887910466299010 2: 12948789762855324085 3: 7574729187958724006 4: 14608687740989597345 5: 10145950054018120246 6: 11767641523694169551 7: 5811941457879652367 8: 2397613489984096139 Usage /dev/urandom: 1: 14391656456731294109 2: 2750708286643159769 3: 10844675827853246458 4: 7920672879166021322 5: 16939013845838223421 6: 9482848646152826462 7: 11535629003375292447 8: 6903845178044907896

    Once the release hits CPAN, I'd come back and post a parallel demonstration.

    Regards, Mario

      MCE 1.831 and MCE::Shared 1.832 have been released containing the fix. What follows is the parallel demonstration for danaj's example. For sequence of numbers, the MCE bounds_only option is handy. Workers receive the begin and end values only.

      use strict; use warnings; use ntheory 0.67 ":all"; use MCE::Flow 1.831; my ( $name, $rng ); my %rand = ( "drand48" => sub { int(rand(1 << 32)) }, "ChaCha20" => sub { irand64() }, "/dev/urandom" => sub { unpack("Q", random_bytes(8)) } ); # Workers receive [ begin, end ] values. MCE::Flow::init( max_workers => MCE::Util::get_ncpu(), chunk_size => 10000, bounds_only => 1, user_begin => sub { $name = MCE->user_args()->[0]; $rng = $rand{ $name }; } ); sub func { my ( $beg_seq, $end_seq ) = @{ $_ }; my ( $t ) = ( 0 ); for ( $beg_seq .. $end_seq ) { $t++ if gcd( $rng->(), $rng->() ) == 1; } MCE->gather($t); } # The user_args option is how to pass arguments. # Workers persist between each run. sub cesaro { my ( $name ) = @_; print "Usage $name:\n"; for my $e ( 1..7 ) { my $n = 10 ** $e; my @ret = mce_flow_s { user_args => [$name] }, \&func, 0, $n - 1; my $t = 0; $t += $_ for @ret; printf "%8d %0.8f\n", $n, sqrt(6 * $n / $t); } printf "%9s %s\n\n", "Pi", Pi(); } for ( "drand48", "ChaCha20", "/dev/urandom" ) { cesaro($_); }

      Output.

      Usage drand48: 10 3.46410162 100 3.11085508 1000 3.12347524 10000 3.15335577 100000 3.14122349 1000000 3.14092649 10000000 3.14209456 Pi 3.14159265358979 Usage ChaCha20: 10 3.46410162 100 3.08606700 1000 3.11588476 10000 3.14606477 100000 3.14461263 1000000 3.14453748 10000000 3.14269499 Pi 3.14159265358979 Usage /dev/urandom: 10 3.16227766 100 3.13625024 1000 3.24727816 10000 3.13959750 100000 3.14238646 1000000 3.14247180 10000000 3.14023908 Pi 3.14159265358979

      The parallel code scales linearly. It runs about 4x faster on a machine with 4 "real" cores. A little faster with extra hyper-threads.

      Regards, Mario

        The OP mentioned (here) on marking down the number pairs whenever gcd(x, y) is equal to 1.

        Here, workers write the number pairs to a shared STDERR file handle. Locking is handled automatically by the shared-manager. For best performance, do not have workers write each pair individually to the shared file handle. Instead, have workers append to a local variable. Writing once per chunk/segment relieves pressure on the shared-manager process, which must keep up with many workers.

        To run, redirect STDERR to a file: perl demo.pl 2> pairs.txt

        use strict; use warnings; use ntheory 0.67 ":all"; use MCE::Flow 1.831; use MCE::Shared; mce_open my $err_fh, '>>', \*STDERR; my ( $name, $rng ); my %rand = ( "drand48" => sub { int(rand(1 << 32)) }, "ChaCha20" => sub { irand64() }, "/dev/urandom" => sub { unpack("Q", random_bytes(8)) } ); # Workers receive [ begin, end ] values. MCE::Flow::init( max_workers => MCE::Util::get_ncpu(), chunk_size => 10000, bounds_only => 1, user_begin => sub { $name = MCE->user_args()->[0]; $rng = $rand{ $name }; } ); sub func { my ( $beg_seq, $end_seq ) = @{ $_ }; my ( $t, $pairs, $r1, $r2 ) = ( 0, '' ); for ( $beg_seq .. $end_seq ) { ( $r1, $r2 ) = ( $rng->(), $rng->() ); if ( gcd($r1, $r2) == 1 ) { $pairs .= sprintf("%20s %20s\n", $r1, $r2); $t++; } } print $err_fh $pairs; MCE->gather($t); } # The user_args option is how to pass arguments. sub cesaro { my ( $name ) = @_; print "Usage $name:\n"; my $n = 10 ** 5; my @ret = mce_flow_s { user_args => [$name] }, \&func, 0, $n - 1; my $t = 0; $t += $_ for @ret; printf "%8d %0.8f\n", $n, sqrt(6 * $n / $t); printf "%9s %s\n\n", "Pi", Pi(); } cesaro("/dev/urandom");

        stdout

        Usage /dev/urandom: 100000 3.14168852 Pi 3.14159265358979

        stderr

        3287478503828099458 17020534165422626509 12874128583739175743 15858105624010204897 12114767169191945777 1381808520787800067 6217235583185614040 13880267056100467759 6095107227201871385 14484880859615821612 6023474899938562107 6188664627301334099 2843055836738994412 3086747352859159329 10461621510749606095 302140866635797242 11160050491040241465 8409335412227119072 3718439936437935345 5121716120103428116 ...

        Regards, Mario

Re^2: PRNG/TRNG Cesaro's theorem
by Anonymous Monk on Oct 08, 2017 at 13:45 UTC
    Notice that the number of correct digits of π is roughly half of the number of digits in $n, because this is basically a random walk process.

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