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Transitive reduction

siiky

2023/03/04

There's this operation on graphs/relations called "transitive reduction" (I didn't learn its name until very recently). It can be used on a graph/relation to compute another (possibly smaller) graph/relation that has no redundant edges (assuming transitivity). And I've been thinking about how to do it for about two years (dam), because I needed it for some POSet things (Scheme § poset). Some weeks ago I was walking home, not thinking about anything in particular, and an algorithm just popped into my brain out of nowhere!

gemini://gemi.dev/cgi-bin/wp.cgi/view/en?Transitive_reduction

../wiki/transitive_reduction.gmi

https://git.sr.ht/~siiky/experiments/tree/main/item/transitive-reduction.scm

Scheme (§ poset)

The idea is so simple that I'm flabbergasted I didn't come up with it two years ago, when I was kinda obsessed. (Though I haven't proven it works, intuitively I think it does).

Let's say `a → b` means that node 'b' is directly reachable from node 'a' ("directly" means there are no intermediate nodes); and let's say `a →* b` means that node 'b' is reachable from node 'a', possibly through intermediate nodes (e.g. if `a → b → c`, we could say `a →* c`).

We'll call our graph G=(V, E), where V is the set of all nodes, and E is the relation `a →* b` (a, b ∈ V). We're looking to compute an E' from E that is the relation `a → b`.

And here it is at last: ∀a, c ∈ V: (a →* c ∧ ∃b ∈ V: b≠c ∧ `a → b` ∧ `b →* c`) ⇒ remove `a →* c` from E.

There's one caveat with this algorithm: it only works for acyclic graphs (aka DAGs, graphs with no cycles). That's not a problem for me (I wanted it for POSets after all; see § "Alternative definitions") so I didn't bother to think about the matter further, but beware.

gemini://gemi.dev/cgi-bin/wp.cgi/view/en?Directed_acyclic_graph

The implementation is also simple enough (see the ~siiky/experiments for previous versions):

(import (srfi 42))

(define (reachable? E s d)
  (memq d (alist-ref s E)))

(define (transitive-reduction E)
  (list-ec (:list s*sE E)
           (:let s (car s*sE))
           (:let sE (cdr s*sE))
           (cons s
                 (list-ec (:list d sE)
                          (if (not (any?-ec (:list c sE)
                                            (and (not (eq? c d))
                                                 (reachable? E c d)))))
                          d))))

Very important note: this implementation assumes that E is the transitive closure! It may not compute the correct result otherwise. I just made this choice to KISS: this way I don't have to recursively check reachability. When I apply it to the posets experiment I'll be sure to change that.

I like how it turned out. SRFI 42 made it pretty.

SRFI 42

A recursive `reachable?` could be something like this:

(define (reachable? E s d)
  (let ((sE (alist-ref s E)))
    (or (memq d sE)
        (any?-ec (:list c sE)
                 (reachable? E c d)))))