# ETOOBUSY ðŸš€ minimal blogging for the impatient

# PWC121 - The Travelling Salesman

**TL;DR**

On with TASK #2 from the Perl Weekly Challenge #121. Enjoy!

# The challenge

You are given a NxN matrix containing the distances between N cities.

Write a script to find a round trip of minimum length visiting all N cities exactly once and returning to the start.

Example`Matrix: [0, 5, 2, 7] [5, 0, 5, 3] [3, 1, 0, 6] [4, 5, 4, 0] Output: length = 10 tour = (0 2 1 3 0)`

BONUS 1: For a given number N, create a random NxN distance matrix and find a solution for this matrix.

BONUS 2: Find a solution for a random matrix of size 15x15 or 20x20

# The questions

As it often happens, some questions are moreâ€¦ *assumptions*.

As weâ€™re talking about distances between cities, the assumption is that they are greater than zero.

The NxN matrix needs not be symmetric, as also the example shows. This is consistent with one-way routes.

Itâ€™s a bit more difficult to understand whether the NxN matrix respects
some *triangular constraint*. I mean, we are assuming that each city is
*directly* connected to any other city, and also that this connection
might be *worse* than passing through other cities. This is not an issue
with the main problem, more for the bonus of generating a random
arrangement.

Last, regarding the second bonusâ€¦ is it 15x15 or 20x20?!?

# The solution

The first solution, that is still good for low values of $N$ (say up to
10 or so) is to just *brute force* the problem. With N cities, we fix
one of them (letâ€™s call it $0$) and generate all possible permutations
of the other ones (letâ€™s call them with integers from $1$ up to $N -
1$). This gives us all possible paths by starting from $0$, going
through the permutation, and going back to $0$.

Hereâ€™s the example in Raku:

```
sub tsp-bf ($dist-from-to) {
my $n = $dist-from-to.elems;
my ($best-distance, $best-path);
for permutations(1 .. $n - 1) -> $perm {
my $from = 0;
my $sum = 0;
for |$perm, 0 -> $to {
$sum += $dist-from-to[$from][$to];
$from = $to;
}
($best-distance, $best-path) = ($sum, $perm)
if ! defined($best-distance) || $sum < $best-distance;
}
return ($best-distance, (0, |$best-path, 0));
}
```

The `$dist-from-to`

is assumed to be a bi-dimensional array whose first
level contains each row of the input matrix. The permutations
built-in comes handy here, allowing us to concentrate on the problem.

This is easily translated in Perl, leveraging our old friend Iterator-based implementation of Permutations:

```
sub tsp_bf ($dist_from_to) {
my $n = $dist_from_to->@*;
my ($best_distance, $best_path);
my $pit = permutations_iterator(items => [1 .. $n - 1]);
while (my @perm = $pit->()) {
my ($from, $sum) = (0, 0);
for my $to (@perm, 0) {
$sum += $dist_from_to->[$from][$to];
$from = $to;
}
($best_distance, $best_path) = ($sum, [@perm])
if !defined($best_distance) || $sum < $best_distance;
} ## end while (my @perm = $pit->(...))
return ($best_distance, [0, $best_path->@*, 0]);
} ## end sub tsp_bf ($dist_from_to)
```

To address the first bonus, we just generate a random matrix with all positive values, except on the diagonal (the distance of a city from itself is 0). For simplicity weâ€™re restricting to integers here, but itâ€™s not a constraint.

```
sub generate_randoms ($n) {
my @retval;
for my $i (0 .. $n - 1) {
my @row = map { 1 + int(rand 13) } 1 .. $n;
$row[$i] = 0;
push @retval, \@row;
}
return \@retval;
}
```

The underlying model is a bitâ€¦ crude, maybe it would be better to just put points on a rectangle and calculate their mutual distances. Anyway, as an input for the problem it should be fine.

To address the other bonus, a different algorithm is needed. I took a
look at the Wikipedia page on the Travelling salesman problem and it
does not give too muchâ€¦ except that *branch and bound* solutions
should be better.

Anyway, in this case I settled for the Heldâ€“Karp algorithm because:

- it has a much better complexity than brute-force
- weâ€™re limited to 20x20 anyway, so we donâ€™t need anything
*too aggresive*.

The page on the algorithm is a bit confusing, because:

- it has a good textual explanation of the algorithm;
- it has an example that is more or less in line with the explanation;
- it has some pseudocode that I canâ€™t relate 1-to-1 with the two above.

For this reason, I changed it to be in line with the explanation and the example, resulting in this:

```
function algorithm TSP (G, n) is
for e := 2 to n do
g({}, e) := d(1, e)
end for
for s := 2 to nâˆ’1 do
for all S âŠ† {2, . . . , n}, |S| = s do
for all e âˆˆ S do
g(S\{e}, e) := min_{m âˆˆ S\{e}} [g(S\{m, e}, m) + d(m, e)]
# & keep best predecessor (yielding the minimum)
end for
end for
end for
opt := min_{k = 2 to n} [g({2, 3, ..., n}\{k}, k) + d(k, 1)]
return opt
end function
```

I will not put the implementations in Raku and Perl here, because theyâ€™re a bit long and are more or less an implementation of the algorithm above (with the small addition that allows to get the optimal path out of the calculation). They are available in the Perl Weekly Challenge repository, here and here.

The Perl implementation is a bit *lower level*, using only arrays,
while the Raku implementation uses sets etc. Iâ€™m not sure, though,
this is an advantage, at least computationally-wise, because the
Perl solution is way faster.

Both solutions have an exponential behaviour where adding one city more or less means a factor of about $2.4$ on the time required to calculate it all. The 20x20 example case is solved by Perl in about 2 and a half minutes, while an extrapolation for Raku would require a handful of hours. I think that the Raku implementation can be Raku-ized much more, though, yielding an easily parallelizable algorithm that might take advantage of multiple cores to speed up the execution.

As hinted by the Wikipedia page, the memory requirements can be a bit
taxing as the number of cities grows. I think it should be possible to
remove *lower level* values of hash `%g`

as we move up the ladder,
although the `%p`

hash tracking the predecessor still has to be kept
until the end of the computation.

All in all itâ€™s beenâ€¦ very interesting!