Lazy lists

[spir]

On 02/23/2011 03:28 AM, bearophile wrote:
> A task from the RosettaCode site asks to generate a Sierpinski carpet like this, on given order:
>
>
> ###########################
> # ## ## ## ## ## ## ## ## #
> ###########################
> ###   ######   ######   ###
> # #   # ## #   # ## #   # #
> ###   ######   ######   ###
> ###########################
> # ## ## ## ## ## ## ## ## #
> ###########################
> #########         #########
> # ## ## #         # ## ## #
> #########         #########
> ###   ###         ###   ###
> # #   # #         # #   # #
> ###   ###         ###   ###
> #########         #########
> # ## ## #         # ## ## #
> #########         #########
> ###########################
> # ## ## ## ## ## ## ## ## #
> ###########################
> ###   ######   ######   ###
> # #   # ## #   # ## #   # #
> ###   ######   ######   ###
> ###########################
> # ## ## ## ## ## ## ## ## #
> ###########################
>
>
> This is a Scala implementation of a function that prints the carpet:
>
>
> def nextCarpet(carpet: List[String]): List[String] = (
>    carpet.map(x =>  x + x + x) :::
>    carpet.map(x =>  x + x.replace('#', ' ') + x) :::
>    carpet.map(x =>  x + x + x))
>
> def sierpinskiCarpets(n: Int) = (Iterator.iterate(List("#"))(nextCarpet) drop n next) foreach println
>
>
>
> A D version that uses arrays:
>
> import std.stdio, std.string, std.algorithm, std.array, std.range;
>
> string[] nextCarpet(string[] c) {
>      auto b = array(map!q{a ~ a ~ a}(c));
>      return b ~ array(map!q{a ~ a.replace("#"," ") ~ a}(c)) ~ b;
> }
>
> void main() {
>      auto c = recurrence!((a, n){ return nextCarpet(a[n-1]); })(["#"]);
>      writeln(array(take(c, 4)).back.join("\n"));
> }
>
>
> Few notes:
> - I don't know how to take just the 4th item of a lazy sequence. array(take(c, 4)).back is not good.
>
> - recurrence() is a bit overkill. A function like iterate() simplifies the code:
> auto c = iterate!nextCarpet(["#"], 4);
>
> - I don't see a simple way to create a lazy nextCarpet(), without those array(). The seed (["#"]) can't be an array, but even wrapping it with a lazy map!q{a}(["#"]) solves nothing. chain(map(chain...))) are all different types, so I think it can't work. The types in that Scala code are sound because it uses a lazy list type, that supports the ::: operator for concatenation, and List("#") to create the correctly typed seed. That carpet.map() returns a List[String]. So both the input and output of the Scala nextCarpet() are of the same type, List[String]. So such lazy list type template becomes really useful if you want to program in a lazy functional style.

Maybe we should reverse the point of view and start with Range. A base type for 
anything 'sequential' (iterable) is Range of E (element type). Then, an array 
E[] is a Range of E that just happens to have its elements pre-stored. Ditto 
for any other collection. And a slice, or any other kind of 'view' into a 
collection, is a Range of E that just happens to have direct access to its 
elements (in memory, via pointer; or via cursor, or whatever).

Denis
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