Talk:Heronian triangles

The Python part is badly formatted and does not show up in the index, someone who knows wiki-formatting should fix it --Zorro1024 (talk) 14:21, 22 March 2015 (UTC)

Fixed. The problem was Smoe's R entry which was not terminated properly (and which should have gone after the python entry rather than before it). --Rdm (talk) 16:27, 22 March 2015 (UTC)

I wonder if a spark of static or noise entered the Python version at some point in its editing history ? On my system the current draft overgenerates triangles, giving a different output from that shown. (It seems to find 1383 rather than 517 triangles). If not an editing glitch then possibly an artefact of changing Python versions ? I am running Python 2.7.10 on OS X 10.11. Hout (talk) 00:12, 25 October 2015 (UTC)

The solution originally just worked for Python 3. I've added the necessary __future__ import. --Andreas Perstinger (talk) 07:00, 25 October 2015 (UTC)

Thanks – that was fast.

On the topic of imports, I wonder if it might make good pedagogic (and perhaps engineering) sense to drop the import of product from itertools, and let the list comprehension do the generation of cartesian products ?

The fact that list monads and list comprehensions yield cartesian products unassisted is one of their most interesting (and arguably central) properties, and perhaps we can demonstrate that more clearly by rewriting the first (generating) half of that comprehension as h = [(a, b, c) for a in range(1, last) for b in range(a, last) for c in range(b, last)

(where last is maxside+1)

Advantages:

1. The filtering happens earlier. Rather than first generating 8 million potential tuples and only then starting to filter, we immediately begin to filter in the inner for loop (or inner call to concat map) of the process which is generating the cartesian product, and we never create the full oversized set in the first place.
2. By defining b and c in terms of a and b, we immediately eliminate the 6646600 out of 8000000 cases which otherwise have to be filtered out by the if (a <= b <= c) condition, and that condition can now be dropped.
3. Apart from the probable space improvement, there seems (as it happens) to be a time improvement in the range of 50% (at least on this system with Python 2.7).

Hout (talk) 10:11, 25 October 2015 (UTC)

Well, I personally don't care about the pedagogical value of using a list comprehension unassisted instead of itertools.product. But I agree that generating the full cartesian product isn't necessary. There is also itertools.combinations_with_replacement which also generates the filtered sequence. Using the following timing tests (where heronian contains the code for the task)

<lang python>import timeit

setup = "import heronian, itertools; last = 201" prod = """[(a, b, c) for a,b,c in itertools.product(range(1, last), repeat=3)

          if a <= b <= c and a + b > c and heronian.gcd3(a, b, c) == 1 and 
          heronian.is_heronian(a, b, c)]"""

list_comp = """[(a, b, c) for a in range(1, last) for b in range(a, last) for c in range(b, last)

               if a + b > c and heronian.gcd3(a, b, c) == 1 and 
               heronian.is_heronian(a, b, c)]"""

comb = """[(a, b, c) for a,b,c in itertools.combinations_with_replacement(range(1, last), 3)

          if a + b > c and heronian.gcd3(a, b, c) == 1 and 
          heronian.is_heronian(a, b, c)]"""

print(timeit.timeit(stmt=prod, setup=setup, number=3)) print(timeit.timeit(stmt=list_comp, setup=setup, number=3)) print(timeit.timeit(stmt=comb, setup=setup, number=3))</lang>

I get the following results:

<lang bash>$ uname -a

Linux arch 4.2.3-1-ARCH #1 SMP PREEMPT Sat Oct 3 18:52:50 CEST 2015 x86_64 GNU/Linux $ python2 -V Python 2.7.10 $ python2 timetest.py 9.67713499069 6.35034918785 6.6238899231 $ python3 -V Python 3.5.0 $ python3 timetest.py 26.007190777992946 22.86392442500801 22.943329881993122</lang>

IMHO itertools.combinations_with_replacement is on a par with your solution. (And the functions in the itertools module won't waste any space because the return the items of the return sequence successively.)

PS: If you wonder, why running the timing code with Python 3 is so much slower: It looks like fractions.gcd performs really bad on Python 3. Using math.gcd instead (new in Python 3.5) I get the following results:

<lang bash>$ head -n 4 heronian.py

from __future__ import division, print_function from math import sqrt, gcd from itertools import product

$ python3 timetest.py 7.764596431006794 4.7238479950028704 4.8884705100063</lang>--Andreas Perstinger (talk) 12:39, 26 October 2015 (UTC)