diff options
| -rw-r--r-- | challenge-117/james-smith/README.md | 24 | ||||
| -rw-r--r-- | challenge-117/james-smith/blog.txt | 1 | ||||
| -rw-r--r-- | challenge-117/james-smith/perl/ch-2.pl | 7 |
3 files changed, 29 insertions, 3 deletions
diff --git a/challenge-117/james-smith/README.md b/challenge-117/james-smith/README.md index aa4a62c493..e17e38bcbc 100644 --- a/challenge-117/james-smith/README.md +++ b/challenge-117/james-smith/README.md @@ -53,7 +53,12 @@ sub triangle { } ``` -### Now the counts... Schroder numbers +Note we don't "collect" routes in a datastructure and then print them all at the end, but instead render directly from within the +function. For `$N` larger than `10` the memory requirements for storing this information increases significantly, so this code is limited purely by disk rather than memory. + +### Now the counts... Schröder numbers + +*It's amazing what you find out about when you google the answers you get!* Due to the "memory" storage issues we can change the problem to one of counting rather than listing... The first approach is to just convert the `triangle` method above to count - we introduce a cache @@ -72,8 +77,14 @@ sub schroder_cache_array { } ``` -But as we've said before recursion is a curse - so to remove the function -overhead of recursion we can re-write this like this: +But as we've said before recursion is a curse - but we notice that +``` + T0,m = 1 + Tn,0 = Tn-1,0 + Tn-1,1 + Tn,m = Tn-1,m + Tn-1,m+1 + Tn,m-1 +``` +We can use that to define each row of a triangle with `Sn` as the last +value. ```perl sub schroder_non_recursive { @@ -88,6 +99,13 @@ sub schroder_non_recursive { } ``` +We again use the row "flip" as we only need one row and the previous +one to get values... + +Googling for `2, 6, 22, 90, 394` came up with https://en.wikipedia.org/wiki/Schröder_number, a page +about Schröder numbers - which gives up the following faster (about twice as fast as above) solution - +as Scrhoder numbers can be written as a recurrence relation: + ```perl sub schroder_recurrence_rel { my( $size, @S ) = ( shift, 1, 2 ); diff --git a/challenge-117/james-smith/blog.txt b/challenge-117/james-smith/blog.txt new file mode 100644 index 0000000000..8d12b20b58 --- /dev/null +++ b/challenge-117/james-smith/blog.txt @@ -0,0 +1 @@ +https://github.com/drbaggy/perlweeklychallenge-club/blob/master/challenge-117/james-smith/ diff --git a/challenge-117/james-smith/perl/ch-2.pl b/challenge-117/james-smith/perl/ch-2.pl index 854c9244b2..248d8d2dfe 100644 --- a/challenge-117/james-smith/perl/ch-2.pl +++ b/challenge-117/james-smith/perl/ch-2.pl @@ -142,3 +142,10 @@ sub schroder_recurrence_rel { } return $S[ $size ]; } + +## +# 1, 2, 6, 22, 90, 394, +# 1806, 8558, 41586, 206098, 1037718, +# 5293446, 27297738, 142078746, 745387038, 3937603038, +# 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, +# 95149655201962, 518431875418926, 2832923350929742, 15521467648875090 |