any approach recording paths will fail in general, because there is no guaranty for a unique or even small number of optimal solutions.

Just imagine a triangle with n=100 rows but only weight 1 in every cell, the number of optimal paths is 2**99 then and the weight is always 100.

But it's nonetheless possible to calculate the weight of the optimal path, and even to reconstruct one of those optimal path.

And this in at most n**2/2 (here ~5000) steps (time and space complexity)

see Re^2: Brute force vs algorithm (PWC # 100)

If "a" minimal path is wanted (and not all) it sort of looks the same :)

#!/usr/bin/perl

use strict; # https://perlmonks.org/?node_id=11128406
use warnings;
use List::Util qw( reduce );

local $_ = <<END;
            1
           2 4
          6 4 9
         5 1 7 2
END
my @d = map [ split ], split /\n/;

my %cache;

print path(0, 0), "\n";

sub path
  {
  my ($r, $c) = @_;
  $cache{"@_"} //= $r < $#d
    ? $d[$r][$c] . ' + ' .  reduce { eval $a <= eval $b ? $a : $b }
      path($r+1, $c), path($r+1, $c+1)
    : $d[$r][$c]
  }
[download]

Outputs:

1 + 2 + 4 + 1
[download]

Here, my take on this,(a variation from that solution)
• it calculates best cost and path for both test cases.
• for one "pathological" triangle with 100 rows with all cells 1
• for one "random" triangle with 100 rows and random entries

Our approaches are not that different, your dynamic "cache" is just my explicit "weight" triangle.

And you are needing more resources for recursion and hash.

But you could improve it by adding bound criteria (i.e. no need to calculate a subtree if it has no chance to be better than the current minimum).

edit

Though I'm not sure this will pay of, my algorithm is already O(m) with m #cells.

# PWC 100
use v5.12;
use warnings;
use JSON::PP;
use Test::More;
use List::Util qw/reduce min first/;
use Data::Dump qw/pp dd/;

# --------- Testing

while ( my $line = <DATA> ) {
    chomp $line;

    my ($json, $exp_cost, $exp_path) = split /\s+/, $line;
    my $triangle   = decode_json($json);
    $exp_path = decode_json($exp_path);

    my ($cost,$path) = compute($triangle);

    is( $cost, $exp_cost,
        "cost: $cost" );

    is_deeply( $path, $exp_path ,
               "path: ".pp $path);

}

# --- 100 rows element "1"
my $depth = 100;
my $pathological = [ map [ (1) x $_], 1..$depth ];

my ($cost,$path) = compute($pathological);

is( $cost, $depth,
    "pathologigal cost: $cost" );

# --- 100 rows random elements
my $random = [ map [ map {1+int rand 10} 1..$_ ], 1..$depth ];
($cost,$path) = compute($random);

say "random cost: $cost";
say "random path: ",pp $path;

done_testing;


# --------- Implementation

sub compute {
    my @triangle = @{(shift)};
    my @weight = map { [@$_] } @triangle; # deep copy


    for ( my $i = $#weight; $i > 0; $i--) {
        my $children = $weight[$i];
        my $parents  = $weight[$i-1];

        my $j;

        reduce {                        # --- by overlapping pairs
            $parents->[$j++] +=         # add to parent ...
              min $a,$b;                # ... minimal child
            $b                          # = next $a
        } @$children;
    }

    my $result = $weight[0][0];

    my $path = calc_path(\@triangle,\@weight,$result);

    return ($result,$path);
}


sub calc_path {
    my ($triangle, $weight, $diff) = @_;

    my @path;

    for my $row (@$triangle) {
        my ($i,$sum) = each @$weight;

        my $j = -1;
        first { $j++; $diff == $_ } @$sum;

        my $step = $row->[$j];
        push @path, $step;
        $diff -= $step;
    }

    return \@path;
}



__DATA__
[[1],[2,4],[6,4,9],[5,1,7,2]] 8   [1,2,4,1]
[[9],[1,6],[7,8,2],[5,8,2,3]] 19  [9,6,2,2]
[download]

C:/Perl_524/bin\perl.exe -w d:/tmp/pm/pwc100.pl 
ok 1 - cost: 8
ok 2 - path: [1, 2, 4, 1]
ok 3 - cost: 19
ok 4 - path: [9, 6, 2, 2]
ok 5 - pathologigal cost: 100
random cost: 305
random path: [
  10,
  3,
  2,
  6,
  3,
  5,
  2,
  2,
  2,
  1,
  6,
  1,
  8,
  3,
  1,
  6,
  1,
  3,
  1,
  1,
  1,
  2,
  2,
  3,
  3,
  4,
  1,
  4,
  3,
  1,
  7,
  1,
  1,
  1,
  2,
  2,
  5,
  1,
  6,
  1,
  6,
  2,
  1,
  8,
  5,
  3,
  1,
  3,
  2,
  1,
  2,
  5,
  3,
  5,
  1,
  4,
  8,
  9,
  6,
  5,
  1,
  3,
  4,
  2,
  3,
  5,
  7,
  2,
  3,
  1,
  3,
  2,
  1,
  5,
  5,
  4,
  4,
  1,
  2,
  4,
  2,
  3,
  1,
  4,
  2,
  6,
  1,
  2,
  3,
  2,
  3,
  1,
  1,
  2,
  2,
  4,
  2,
  1,
  2,
  1,
]
1..5
[download]

Cheers Rolf
_{(addicted to the Perl Programming Language :)
Wikisyntax for the Monastery}