Fish Groups & Constraint Groups in AICs

Myth Jellies

Thought I would cross-post this here, because it contains another take on fish and constraint groups which might be of interest to coders.

Expanding the basic AIC building blocks - NxN Fish Groups

You can extend the concept of bilocated candidates into two dimensions by considering an NxN fish group.

Definitions for an NxN Fish Group

An NxN fish group consists of any intersection of N column houses with N row houses. A fish group pertains to only one particular digit. The particular digit does not need to be present as a candidate in any of the cells that form the NxN fish group.

An NxN fish group is considered true for a particular digit only if it contains exactly N true candidates for that digit. A fish group being true doesn't necessarily tell you where in the fish group that digit's candidates are true, but it does tell you that any like candidates that are outside of the fish group that share a row or column with the fish group must be false. Likewise, if the fish group is false, then at least one of the outside candidates that shares a row, and at least one of the outside candidates that shares a column with the fish group must be true.

NxN Fish Group - Identifying Weak Links

Any group of like candidates, not part of the fish group, that all share either a row or a column with the fish group is weakly linked to the fish group. You cannot have both the fish group and a candidate from the outside group be true.

If a fish group has no weak links in one of its dimensions (rows or columns) then the fish group must be true.

NxN Fish Group - Identifying & Using Strong Links

If a group of like candidates encompasses either all of the outside candidates sharing rows with the fish group, or all of the outside candidates sharing columns with the fish group; then that group is strongly linked to the fish group. Either the outside group or the fish group (or possibly both if they share candidates) must be true. If there is no intersection between the two groups, they are conjugate, and they can even be colored. If there are no candidates for the outside group in one dimension (rows or columns), then the fish group must be true and outside candidates sharing the other dimension may be removed.

Note that when a fish group is false, it may still contain a significant number of true candidates. Because of this, it is practically impossible to come up with cases where two large fish patterns are directly strongly linked.

AIC Notation for Fish Groups

To denote a fish group using AIC notation, use a slash in between the rows/columns in the location field. Changing the order of rows and columns so that the correct set of houses faces the candidates interacting with them is allowed. While this is not so important in the case of fish groups, it will help make things clearer in the case of the more generic Constraint Groups.

Examples of Fish Groups in AICs

Finned fish and finned sashimi fish are simple AICs of the form...

{Fish Group} = {Fin Group}

...which eliminates candidates weakly linked to both the fish and the fin. For example...

(1)c248/r159 = (1)r1c13 => is a finned swordfish with r23c2 <> 1

Kraken Fish are identical to longer AIC chains or nets with a fish group.

Here is the original finned fish example

Ruud · Site Admin

Thanks for your contribution, Myth Jellies. It is such a shame that posts like these receive so little response. To be honest, I had to read your post 3 or 4 times before it sank in.

Incorporating "almost fish" or ARCS (almost row-column subsets) in chains is something that I had in mind when I wrote my ARCS post in the Players forum.

It takes some time to understand the concept of "fish group" vs. "outside group". Most of the time, the outside group is(are) the fin(s), which many consider part of the fish and not something that lies outside the fish.

A simple way to incorporate Almost Fish into chains is this:

Consider N rows with candidates limited to N+1 columns. You can now create a group for each of these columns, containing the candidates that belong to this pattern.

Jean-Christophe · Posted: Sat May 12, 2007 8:41 am Post subject:

kyriako · Posted: Sun Jul 26, 2009 9:34 am Post subject: Kraken Fish in Nice Loops

Hello,

I have been programming a solver with Nice Loops which work with grouped strong links and
ALSes.
Code and exe:http://home.arcor.de/guentermeinel/download.html

Now I would like to add the "Kraken X-Wing" as a linking element.
As Myth Jellies pointed out, one can treat it as a strong link in the form of:
{Fish Group} = {Fin Group}
In the following example, this would mean that I can connect the Kraken via weak links
from/to the "<"-cells and from/to the ">"-cells:

hobiwan · Posted: Tue Jul 28, 2009 4:55 pm Post subject:

First things first: I have only skimmed over the preceeding posts, so the following is only an educated guess...

When constructing a chain or loop you often find candidates that are linked to more than one other candidate. You have to pick one of the possibilities to continue in the chain. Your '?' candidates are exactly that: They can be a '<' or a '>', you are free to choose what you like.

Let's take your second grid (?(<) means '?' taken as '<'):

daj95376 · Posted: Tue Jul 28, 2009 9:24 pm Post subject:

Nothing competes with a real example -- even if it's a larger fish.

kyriako · Posted: Wed Jul 29, 2009 7:06 pm Post subject: Finned X-Wing

Hello hobiwan,

thank you for your comment.
I had'nt seen that the "?" cells are elimination cells of the finned X-Wing.
Therefore, I think that theses cells should not(or even must not?)be used for linking the fish.
Regarding example 3:
in my oppinion it is a special case of the finned X-Wing. The condition for elimination is that a cell "sees" both strings of the X-Wing as well as the fin.
This means that all remaining cells in the box are elimination cells and the fin cannot be linked at all.
However, if this pattern has only one fin cell, linking is again possible:

hobiwan · Posted: Thu Jul 30, 2009 2:36 pm Post subject: Re: Finned X-Wing

kyriako · Posted: Thu Jul 30, 2009 6:33 pm Post subject:

Thank you hobiwan,
I see the example 3 from a practical point of view:
one method of checking the effect of a finned fish is to look at two possibilities:
1. one of the fin cells receives the candidate
2. one of base cells receives the candidate

If in both cases the candidate in a cell can be eliminated, then it is an elimination cell.

By the way, if there was no fin, the elimination cells would be:

hobiwan · Posted: Fri Jul 31, 2009 6:23 am Post subject:

kyriako · Posted: Sat Aug 01, 2009 6:14 pm Post subject:

Hello hobiwan,
interesting hint.
Thank you.

Kyriako.

daj95376 · Posted: Sat Aug 01, 2009 10:57 pm Post subject:

kyriako · Posted: Sun Aug 02, 2009 6:13 pm Post subject:

Hello daj95376,
shame on me, not having seen this simple thing.

Kyriako