Greetings,

I am pleased to provide the final work. Computing **collatz_seq (top 20)** for 1e9 requires ~ 3.8 GiB of available memory. Try 5e8 for lesser memory consumption.

This is a parallel demonstration using MCE::Flow and File::Map for caching. The 2nd example uses Inline::C for counting the number of steps. Unlike the other examples, workers working on chunks 2 and higher require results from prior chunks. This is accounted for. Obviously the worker processing chunk 1 doesn't need prior results. Once the workers being processing (i.e. after the mapped cache creation), it doesn't take long before full CPU utilization kicks in.

**Q.** Why does this work, especially when subsequent chunks need results from prior chunks?

**A.** The magic lies with using a smaller chunk size set to 2500. The initial ramp up is a one time occurrence. One cool thing about MCE is that input IO is sequential. This applies to number sequences as well. A worker obtaining chunk 1 begins processing immediately (there's no wasting time). The worker obtaining the next chunk begins processing but may need to pause a little here and there. Eventually, the chunks as far as timing goes (starting) are spread out where workers need to pause left often. This can be seen by looking at the CPU utilization. The Power of Randomness kicks in at some point with CPU utilization near 100% until completion.

Credit to PerlMonks choroba, Laurent_R, 1nickt, rjt, and vr. See this thread. Worthy mention goes to Leon Timmermans, author of File::Map. Wow!

Credit for the caching technique used here is based on the caching demonstration by iM71, a response in that thread. My first attempt at parallelization failed. I tried again by maximizing on MCE's strengths described above. It's surreal :)

Credit for reducing the number of loop iterations was from watching Notation and compressed dynamics, one minute into the video (i.e. the T(x) notation).

Below, the minimum and maximum argument (size) is 1e6 and 1e9 respectively. The two scripts will set to limit quietly if exceeded.

See also this thread for computing the longest Collatz.

**Cache miss update:**

Cache miss is less than 1%. Therefore, it is faster to compute for $n than waiting for the result.

**Final update:**

`#!/usr/bin/env perl
use strict;
use warnings;
use File::Map qw/map_anonymous unmap/;
use Time::HiRes qw/usleep/;
use MCE::Flow;
use MCE::Candy;
my $size = shift || 1e6;
$size = 1e6 if $size < 1e6; # minimum
$size = 1e9 if $size > 1e9; # maximum
map_anonymous my $cache, $size * 2 + 2, 'shared';
# fill cache with zeroes
substr($cache, 0, $size * 2 + 2, pack('s', 0) x ( $size + 1 ));
# local to workers and the manager process
my @seqs;
sub collatz_seq {
my ( $chunk_id, $seq_beg, $seq_end ) = @_;
my ( $n, $steps, $tmp );
for my $input ( $seq_beg..$seq_end ) {
$n = $input, $steps = 0;
$n % 2 ? ( $steps += 2, $n = (3 * $n + 1) >> 1 )
: ( $steps += 1, $n = $n >> 1 )
while $n != 1 && $n >= $input;
$tmp = unpack('s', substr($cache, $n * 2, 2));
# another worker with a lesser chunk_id is not yet
# completed processing $n, so pause a little
# if ( $tmp == 0 && $chunk_id > 1 ) {
# do {
# usleep 100;
# $tmp = unpack('s', substr($cache, $n * 2, 2));
# } while ( $tmp == 0 );
# }
# do this instead (faster): compute $n if cache miss
$tmp = _collatz($n) if $tmp == 0 && $chunk_id > 1;
substr($cache, $input * 2, 2, pack('s', $steps += $tmp));
push @seqs, [ $input, $steps + 1 ] if $steps > 400;
}
}
sub _collatz {
my ( $input ) = @_;
my ( $n, $steps ) = ( $input, 0 );
$n % 2 ? ( $steps += 2, $n = (3 * $n + 1) >> 1 )
: ( $steps += 1, $n = $n >> 1 )
while $n != 1 && $n >= $input;
my $tmp = unpack('s', substr($cache, $n * 2, 2));
$tmp = _collatz($n) if $tmp == 0;
substr($cache, $input * 2, 2, pack('s', $steps += $tmp));
return $steps
}
my $chunk_size;
$chunk_size = int( $size / MCE::Util::get_ncpu() / 80 + 1 );
$chunk_size += 1 if $chunk_size % 2;
MCE::Flow->init(
max_workers => MCE::Util::get_ncpu(),
chunk_size => $chunk_size, # specify 2500 if pausing above
bounds_only => 1,
gather => MCE::Candy::out_iter_array(\@seqs),
);
mce_flow_s sub {
my ( $mce, $chunk_ref, $chunk_id ) = @_;
collatz_seq($chunk_id, @{ $chunk_ref });
@seqs > 20
? MCE->gather($chunk_id, ( sort { $b->[1] <=> $a->[1] } @seqs
+)[ 0..19 ])
: MCE->gather($chunk_id, @seqs);
@seqs = ();
}, 2, $size;
MCE::Flow->finish; unmap $cache;
@seqs = ( sort { $b->[1] <=> $a->[1]} @seqs )[ 0..19 ];
printf "Collatz(%5d) has sequence length of %3d steps\n", @$_
for @seqs;
`

**Count steps via Inline C:**

`#!/usr/bin/env perl
use strict;
use warnings;
use File::Map qw/map_anonymous unmap/;
use Time::HiRes qw/usleep/;
use MCE::Flow;
use MCE::Candy;
use Inline C => Config => CCFLAGSEX => '-O2 -fomit-frame-pointer';
use Inline C => <<'END_OF_C_CODE';
#include <stdint.h>
void num_steps_c( SV* _n, SV* _s )
{
uint64_t n, input;
int steps = 0;
n = input = SvUV(_n);
while ( n != 1 && n >= input ) {
n % 2 ? ( steps += 2, n = (3 * n + 1) >> 1 )
: ( steps += 1, n = n >> 1 );
}
sv_setuv(_n, n);
sv_setiv(_s, steps);
return;
}
END_OF_C_CODE
my $size = shift || 1e6;
$size = 1e6 if $size < 1e6; # minimum
$size = 1e9 if $size > 1e9; # maximum
map_anonymous my $cache, $size * 2 + 2, 'shared';
# fill cache with zeroes
substr($cache, 0, $size * 2 + 2, pack('s', 0) x ( $size + 1 ));
# local to workers and the manager process
my @seqs;
sub collatz_seq {
my ( $chunk_id, $seq_beg, $seq_end ) = @_;
my ( $n, $steps, $tmp );
for my $input ( $seq_beg..$seq_end ) {
num_steps_c($n = $input, $steps);
$tmp = unpack('s', substr($cache, $n * 2, 2));
# another worker with a lesser chunk_id is not yet
# completed processing $n, so pause a little
# if ( $tmp == 0 && $chunk_id > 1 ) {
# do {
# usleep 100;
# $tmp = unpack('s', substr($cache, $n * 2, 2));
# } while ( $tmp == 0 );
# }
# do this instead (faster): compute $n if cache miss
$tmp = _collatz($n) if $tmp == 0 && $chunk_id > 1;
substr($cache, $input * 2, 2, pack('s', $steps += $tmp));
push @seqs, [ $input, $steps + 1 ] if $steps > 400;
}
}
sub _collatz {
my ( $input ) = @_;
num_steps_c( my $n = $input, my $steps );
my $tmp = unpack('s', substr($cache, $n * 2, 2));
$tmp = _collatz($n) if $tmp == 0;
substr($cache, $input * 2, 2, pack('s', $steps += $tmp));
return $steps
}
my $chunk_size;
$chunk_size = int( $size / MCE::Util::get_ncpu() / 80 + 1 );
$chunk_size += 1 if $chunk_size % 2;
MCE::Flow->init(
max_workers => MCE::Util::get_ncpu(),
chunk_size => $chunk_size, # specify 2500 if pausing above
bounds_only => 1,
gather => MCE::Candy::out_iter_array(\@seqs),
);
mce_flow_s sub {
my ( $mce, $chunk_ref, $chunk_id ) = @_;
collatz_seq($chunk_id, @{ $chunk_ref });
@seqs > 20
? MCE->gather($chunk_id, ( sort { $b->[1] <=> $a->[1] } @seqs
+)[ 0..19 ])
: MCE->gather($chunk_id, @seqs);
@seqs = ();
}, 2, $size;
MCE::Flow->finish; unmap $cache;
@seqs = ( sort { $b->[1] <=> $a->[1]} @seqs )[ 0..19 ];
printf "Collatz(%5d) has sequence length of %3d steps\n", @$_
for @seqs;
`

**Results:** Unix time (i.e. time perl script.pl 1e9).

`1e9, 32 cores
collatz_seq_before 0m34.211s
collatz_seq_final 0m26.340s
collatz_seq_inline_c 0m15.673s
Collatz(670617279) has sequence length of 987 steps
Collatz(848749995) has sequence length of 977 steps
Collatz(954843745) has sequence length of 972 steps
Collatz(954843751) has sequence length of 972 steps
Collatz(716132809) has sequence length of 969 steps
Collatz(537099606) has sequence length of 966 steps
Collatz(537099607) has sequence length of 966 steps
Collatz(268549803) has sequence length of 965 steps
Collatz(805649409) has sequence length of 964 steps
Collatz(805649410) has sequence length of 964 steps
Collatz(805649411) has sequence length of 964 steps
Collatz(805649415) has sequence length of 964 steps
Collatz(402824705) has sequence length of 963 steps
Collatz(604237057) has sequence length of 961 steps
Collatz(604237058) has sequence length of 961 steps
Collatz(604237059) has sequence length of 961 steps
Collatz(302118529) has sequence length of 960 steps
Collatz(906355586) has sequence length of 959 steps
Collatz(906355587) has sequence length of 959 steps
Collatz(906355588) has sequence length of 959 steps
`

Regards, Mario

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