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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

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Re^2: Triangle Numbers Revisited
by Limbic~Region (Chancellor) on Oct 14, 2004 at 00:09 UTC
    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
    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
    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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