Re^4: Challenge: Another Infinite Lazy List

[tlm]

in reply to Re^3: Challenge: Another Infinite Lazy List
in thread Challenge: Another Infinite Lazy List

You're right. I misread your original post. What threw me off was the mention of streams. In the solution you present (which is indeed entirely different from merging lists), using lazy lists seems forced to me, since their properties don't get you anything that you couldn't get very easily and directly with grep and the '..' operator:

use strict;
# use Stream;
use Math::Pari qw(gcd);
 
# We can test all the factors at once using the greatest common diviso
+r algorithm.
my @factors = @ARGV;
my $test_number = 1;
$test_number *= $_ for @factors;
die "Must provide a list of non-zero factors\n" unless $test_number > 
+1;
 
sub has_common_factor { gcd($_[0], $test_number) != 1 }
 
# my $integers = Stream::tabulate( sub { $_[0] }, 1);
# my $challenge = $integers->filter(\&has_common_factor);
# $challenge->show(50);

print "@{[grep has_common_factor($_), 1..50]}\n";

Thanks for pointing out the Streams module. It does some very nice things which I think I should incorporate in my stab at implementing lazy lists. It doesn't use memoization, however, which I think is absolutely essential to a workable implementation of lazy lists, at least in Perl5. What's most curious about this is that Dominus is the author of the Memoize module! Then again, the module is not available from CPAN. I suspect that both these facts have to do with realizing that this sort of object requires a little more syntactic support from Perl to be worth doing.

---

Update: Oh, yes. There's memoization in Stream.pm, all right; it's just done very discreetly (and elegantly, IMO):

sub tail {
  my $t = $_[0]{t};
  if (ref $t eq CODE) {        # It is a promise
    $_[0]{t} = &$t;  # <--- there!
  }
  $_[0]{t};
}

---

BTW the execution time of my merged-lists solution is not growing anywhere nearly as fast as the formula you gave (not that it matters a whole lot, but I give the stats below if anyone's interested). Still, the gcd-based solution is noticeably faster; whether this is the result of the fact that most of its heavy lifting (the gcd calculation) is done by a C routine, or something more fundamental is not yet clear to me.

OK, here are the stats for the merged-lists solution. All the runs were performed with arguments consisting of a prefix of the list primes from 2 to 37, with N=2000 throughout. The formula mentioned below is the one given by tall man here.

 R:  number of primes in the input
c1:  number of times merge was called
c2:  number of times a < comparison was executed in merge
c3:  time factor predicted by formula
r1:  c1/c3
r2:  c2/c3

R    c1    c2    c3   r1    r2
 2  2002  3002  1667 1.201 1.801
 3  3276  5095  4133 0.793 1.233
 4  4227  6821  7057 0.599 0.967
 5  4989  8203 10137 0.492 0.809
 6  5673  9534 13440 0.422 0.709
 7  6303 10744 16834 0.374 0.638
 8  6921 11966 20377 0.340 0.587
 9  7499 13098 23983 0.313 0.546
10  8015 14108 27602 0.290 0.511
11  8538 15149 31314 0.273 0.484
12  9025 16107 35040 0.258 0.460

[tall_man]

There are a couple of advantages of the lazy list form over grep.

1) You can get an exact number of values out of the sequence without having to know the maximum number range in advance. For example, your sample code produces only 36 values, not the expected 50. You would have to feed it a range up to 273 to get a full 50.

2) You could extract and print values in a loop without building a large result array (a map in void context could do the same, I suppose).

My estimates for timing were rough back-of-the-envelope calculations, so I'm not surprised that they are a bit off. There are a number of things I didn't take into account, such as the overlaps where more than one prime is a factor, and the fact that more numbers are eliminated with more factors, so you have to search bigger integers to finish the list. Thanks for your timing tests.

Math::Pari is a highly optimized math engine, so I'm not too surpised that single gcd checks beat multiple merges. But if you start with primes at 101 and above instead, the merge algorithm will win.