Talk:AC-3 algorithm

I'm not sure the pseudocode is right: the arc-reduce(x,y) function only reduces the domain of x, therefore you also need to call arc-reduce(y,x) to reduce the domain of y and add (z,y) arcs to the worklist if it was indeed reduced. This is done once in the current implementation, as the code for initiating the worklist

worklist := { (x, y) | there exists a relation R2(x, y) or a relation R2(y, x) }

puts (x,y) and (y,x) in the worklist. However, suppose we have reduced the domain of x while checking (x,y), then later reduced the domain of y for the first time while checking (y,x) (hence, with the current implementation, the (x,y) arc is not put back in the worklist). Isn't it possible that this reduction of the domain of y implies a reduction of the domain of x ?

If so, you just need to change

worklist := worklist + { (z, x) | z != y and there exists a relation R2(x, z) or a relation R2(z, x) }

to

worklist := worklist + { (z, x) | there exists a relation R2(z, x) or a relation R2(z, x) }

If you're not sure either, that version works anyway.

Also, I'm wondering if directed constraints exist (a constraint R2(x,y) such that you need to check the domain of x if the domain of y has been reduced but not the opposite). If so, it would be better to change

worklist := { (x, y) | there exists a relation R2(x, y) or a relation R2(y, x) }

to

worklist := { (x, y) | there exists a relation R2(x, y) }

Thanks for your time.

Baha2490 (talk) 00:58, 6 March 2013 (UTC)[reply]

The pseudo code is right. The main benefit of AC3 over other weaker propagation algorithms is that, once you finish arc-reduce(x,y), you don't need to check anything of y nor add (y, x) to the worklist - everything that could be pruned in x because of y will have been checked and taken care of. Also pruning a value in x won't affect y, because a value is pruned from x only when no values in y are connected to it.

This works because constraints are 'not' directed - if there's a value Y in D(y) that supports a value X in D(x), then the the inverse is also true - the X value also supports the Y value. I'd say that the paragraph which describes the CSP as directed graphs is misleading - the consistency between arcs is directed (when x->y is arc-consistent, y->x may not be; this depends on the current state of the variable domains), but at a different level the definition of the constraints must be symmetric ( R(x,y)=R(y,x), and this depends only on the mathematical definition of the constraints, not on the algorithm current state ). Diego (talk) 10:15, 6 March 2013 (UTC)[reply]