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### Re: Triangle Numbers Revisited

by FoxtrotUniform (Prior)
 on Oct 13, 2004 at 23:54 UTC ( #399057=note: print w/replies, xml ) Need Help??

in reply to Triangle Numbers Revisited

Just off the top of my head:

In the inner while loop, it looks like you're testing each \$i, \$j to see if it's triangular (although I must confess, I don't really understand p_tri). Since we can enumerate triangle numbers (if Jobby is to be believed), why not iterate through those?

Edit: Are you sure you're getting correct solutions? It looks like \$prev starts out as a triangular number, but then you just decrement it. (Aside: if this algorithm gives you consistently correct solutions, then the greatest triangular number lower than n is always present in n's triangular decomposition. I think.) Man, I'm an idiot. Since when is \$prev a trinum?

Edit 2: Oh, I get it, p_tri returns the rank of the previous triangular number, not the number itself... although it returns zero when given a triangular number, which is weird but useful.

Edit 3: Here's the start of an implementation. It's not as fast as the code posted above (by about a factor of two, if Unix time is to be trusted), probably because it calculates a lot of trinums and makes a lot of function calls. I'm posting it more or less as a proof of concept.

Edit 4: Inlining the calls to &trinum results in slightly faster code than Limbic~Region's. Code updated, benchmarks added.

```#! /usr/bin/perl -w

use strict;

# trinum(n) returns the nth triangular number
sub trinum { my (\$n) = @_; return \$n * (\$n+1) * 0.5; }

# prev_trinum(n) returns the RANK OF the greatest triangular number le
+ss
#  than n.
# Code blatantly ripped off from Limbic~Region [id://399054]
sub prev_trinum
{
my \$num = shift;
my \$x = ( sqrt( 8 * \$num + 1 ) + 1 )/ 2;
my \$t = int \$x;
return \$t == \$x ? 0 : --\$t;
}

# trinum_decomp(n) tries to find a three-triangular-number decompositi
+on
#  of n.  Based on L~R's method from the post cited above, but
#  enumerates trinums rather than guessing.
sub trinum_decomp
{
my (\$n) = @_;

my \$prev = &prev_trinum(\$n);
return (\$n, 0, 0) unless \$prev;

while(\$prev) {
my \$triprev = (\$prev * \$prev + \$prev)/2;
my \$diff = \$n - \$triprev;
my @tail = &twonum_decomp(\$diff);
if(defined \$tail[0]) {
return (\$triprev, @tail);
}
\$prev--;
}
warn "Can't find trnum decomp for \$n\n";
return (-1, -1, -1); # ugly
}

# twonum_decomp(n) tries to find a two-triangular-number decomposition
#  of n.  If such a decomposition does not exist, returns undef.
sub twonum_decomp
{
my (\$n) = @_;

my \$prev = &prev_trinum(\$n);
return (\$n, 0) unless \$prev;

while(\$prev) {
my \$triprev = (\$prev * \$prev + \$prev)/2;
my \$i = 1; my \$tri_i = (\$i * \$i + \$i)/2;
do {
if(\$tri_i + \$triprev == \$n) {
return (\$tri_i, \$triprev);
}
\$i++; \$tri_i = (\$i * \$i + \$i)/2;
} while(\$triprev + \$tri_i <= \$n);
\$prev--;
}
return undef;
}

my \$target = \$ARGV[0] || 314159;
print join(',', &trinum_decomp(\$target));

__END__
mjolson@riga:~/devel/scratch
Wed Oct 13-18:38:42 583 >time ./trinum 987654321
987567903,14028,72390
real    0m0.089s
user    0m0.060s
sys     0m0.000s

mjolson@riga:~/devel/scratch
Wed Oct 13-18:18:25 578 >time ./limbic_trinum 987654321
987567903, 14028, 72390
real    0m0.106s
user    0m0.090s
sys     0m0.000s

--
Yours in pedantry,
F o x t r o t U n i f o r m

Replies are listed 'Best First'.
Re^2: Triangle Numbers Revisited
by Limbic~Region (Chancellor) on Oct 14, 2004 at 00:09 UTC
All,
In HTML comments, I have provided a spoiler explanation of how the code works.

Update: 2013-10-24 When I first wrote this node, we didn't have spoiler tags. The HTML comments remain but I have also added spoiler for those who don't want to view the source.

Cheers - L~R

Re^2: Triangle Numbers Revisited
by Limbic~Region (Chancellor) on Oct 14, 2004 at 02:34 UTC
FoxtrotUniform,
The trouble with selecting a single number to benchmark from is that it may favor one method over the other. With this in mind, I created another rudimentary benchmark that shows the two methods are fairly equal: I haven't spent any time thinking about optimizations, but if I come up with any tomorrow I will post it along with a "real" benchmark.

Cheers - L~R

The trouble with selecting a single number to benchmark from is that it may favor one method over the other.

Good catch. Another problem I ran into was that the load on my test system was varying (lab machine, someone was logged in remotely doing a bunch of matlab foolery), so I ended up getting quite different "real" time results with the same input... which is probably what a casual reader would check.

It would be interesting to know what kinds of inputs favour whose method, and (ideally) why.

--
Yours in pedantry,
F o x t r o t U n i f o r m

Re^2: Triangle Numbers Revisited
by Limbic~Region (Chancellor) on Oct 14, 2004 at 04:54 UTC
FoxtrotUniform,
I couldn't sleep as I had several optimizations pop into my head I just had to try. I played around with caching known triangular numbers as well as results of known non-triangular numbers. I only got a marginal speed increase which I guessed would be the case earlier in the CB. Moving away from caching, I tried a combination of my method and your method with dramatic results: From 1 minute to just over 3 seconds.

Cheers - L~R

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