This is the homepage of Peter Alexander. I am currently at Facebook working on AR/VR. Any opinions are my own.

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# Pythagorean Polygons - Part 1

It’s been a while since I solved any coding problems, so I’ve decided to re-solve a Project Euler problem that I looked at years ago: Pythagorean Polygons. Back then I used C++, but this time I’ll use the D programming language :-)

In summary, the problem is to find the number of distinct convex polygons with vertices on integer coordinates, integer length sides, and a perimeter less than or equal to 120. The name of the problem comes from the fact that the edges are the hypotenuses of Pythagorean triples.

The brute force method is quite simple:

- Generate all the Pythagorean triples (x, y, d) with d < 120.
- Do a depth first search on the integer lattice from (0, 0) to (0, 0), using the triples as your edges, ensuring that you always turn anti-clockwise (or always clockwise).

Generating the Pythagorean triples is easy enough:

It will also help to have them in anti-clockwise order.

Our triples are likely to contain several colinear edges. So we’ll need a way of avoiding those:

Now all that’s left is to do the depth-first search. I’m just going to use simple recursion to implement it, just to save code.

The `if`

-condition inside the loop is to protect against degenerate polygons with
two edges e.g. (0, 0), (60, 0), (0, 0), and the use of `next`

ensures that we
don’t use two colinear edges in a row, as this is disallowed in the problem.

Finally, we need to call it with the starting position:

If we change N to 30, we can test it using the given solutions in the problem.

Great. That matches with the solution and runs in under 2 seconds on my laptop.
Unforunately, trying it with N = 60 took just shy of 50 **minutes**.

The problem with this is that the branching factor of the search is equal to the number of triples (hundreds) and the depth exponent is large enough to make the running time impractical for larger N. We need a way to speed it up.

Let’s look at the signature of `dfs`

.

What’s the domain of each input?

`x`

and `y`

will be from -60 to 60.

`d`

will be between 0 and 120.

`i`

will be between 0 and the number of triples (448).

That gives us a maximum of 121 x 121 x 121 x 448 possible inputs. Since `dfs`

is a pure function, that means we can memoize the results and re-use them
for subsequent recursive calls. It’s a large number, but now the running time
is polynomial rather than exponential, so it will scale better.

Fortunately, D’s standard library, Phobos, provides a `memoize`

template
just for this purpose. With a couple of quick modifications to `dfs`

, we can
memoize it.

We just introduce an `alias`

for the memoized version of the function, and
change the recursive call to use that version instead. Internally, `memoize`

just maintains a hashtable of input tuples to outputs, so it’s quite fast.

With these changes, the N = 60 version is reduced from 50 minutes to 42 seconds. Unfortunately, N = 120 is still just out of reach. It will likely run in under an hour if you have enough memory, but we’d like it to run a lot faster than that.

In the next post, I’ll show how we can get N = 120 running in sub-second times.