Basically the code constructs an adjacency matrix where 1 means connected and 0 means disconnected (it's not related to distances in any way), and finding a transitive closure means to find all pairs of vertices that have some path between them.

@Bubbler I know what a graph is in terms of objects full of pointers to other objects, like a linked list on steroids, and in terms of shortest path algorithms. I don't know any of the maths behind them, or how one can be represented as an array

@TessellatingHeckler An adjacency matrix is a square matrix of zeroes and ones. Each row/column is associated with a vertex, and the element at (x,y) is 1 if vertex x and vertex y are connected with an edge, 0 otherwise.

If we have a graph of three vertices A, B, C where AB and BC are connected, its adjacency matrix is

  A B C
A 0 1 0
B 1 0 1
C 0 1 0

@Bubbler I was just getting to that in your wikipedia link; so it's simply giving every node a number, then doing that? That is surprisingly simple

@TessellatingHeckler Yes, exactly.

Given +.× computes element-wise × then sum, ∨.∧ computes element-wise "and" and then "any" of booleans. It is essentially solving "if we want to go from vertex a to vertex b, is there a two-step path a→c→b for some vertex c?"

Repeating the process gives pairs of vertices accessible within 3 steps, 4 steps, ... until we get no more updates. The result is a matrix showing which vertices are connected (within any distance). This is called "transitive closure".

@Bubbler Thank you! I think that makes sense, but will need to stare at it for a while and try it

@Bubbler thanks for explaining it so well. i added these links to the answer.

@TessellatingHeckler the or-dot-and trick is one of the most beautiful apl expressions ever, worth staring at :) it comes up relatively often in golfing

it trades a bit of (worst-case) computational complexity - O(n^3 log(n)) vs floyd-warshall's O(n^3) - for conciseness. despite the additional log(n) factor, in practice it's likely to be faster than floyd-warshall because it makes use of optimized matrix multiplication and bit booleans.

more examples of ∨.∧: 0 1

@ngn Wow; I am amazed

I'm trying to use / (replicate), but I think APL keeps thinking it's reduction. I've got f←⊢(/⍨)(1 2⍴⍨≢) working, but ideally it would be (1 2⍴⍨≢)/⊢. Also it's funny that \ works pretty much the same way

> I think APL keeps thinking it's reduction

It indeed does.

darn, is there a better way of working around this than what I've got?

@JoKing This is slightly shorter, but replicate just doesn't work well with trains in general.

huh neat. I assume the last ⍨ is acting on the whole thing?

Yes.

I was working on a dfn solution for the same bytecount {⍵/⍨1 2⍴⍨≢⍵}

Actually a dfn has the same length.

ninja'd

ninja'd

So ⊢(op1)(op2) is equivalent to (op1)∘(op2)⍨? That's good to know for the future

Is dyadic / just a subset of dyadic \?

@JoKing Not now, but in 18.0 you can always use ⊢⍤/ etc. to say "the function /"

@JoKing No, only / can remove cells.

@JoKing Shorter!

@Bubbler In dzaima/APL, ⌿ is a pure function: Try it online!

@Adám I guess "∊⍴¨ instead of function / for non-boolean vectors" deserves a section in this tip?

@Bubbler Yes, immediately before or after ∊⊆. You have my permission to edit it in if you can keep the style: "Task: Given a…", "A dfn solution may be…", "{main content}", "Note that this does not work for higher-rank arrays.".

@Adám OK, I'll try.

@Adám - Is there an "expect date" for 18.0?

@JeffZeitlin This summer.

@Adám - I'll be looking forward to it!

Me too.

Have you been having fun with that 5100 emulator? :)

@JeffZeitlin Only a little bit. I added it to the APL Wiki.

One of the things I'd like to do in my copious spare time would be to hack on that emulator so that one can save and change disks.