## 5.2. all_differ_from_at_least_k_pos

Origin

Inspired by [Frutos97].

Constraint

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi Ί},\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }\right)$

Type
 $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Arguments
 $\mathrm{\pi Ί}$ $\mathrm{\pi \pi \pi }$ $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi },\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi Ί}\beta ₯0$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi },\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi £\pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi },\mathrm{\pi \pi \pi }\right)$
Purpose

Enforce all pairs of distinct vectors of the $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ collection to differ from at least $\mathrm{\pi Ί}$ positions.

Example
$\left(\begin{array}{c}2,β©\begin{array}{c}\mathrm{\pi \pi \pi }-β©2,5,2,0βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©3,6,2,1βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©3,6,1,0βͺ\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi }_\mathrm{\pi \pi \pi }$ constraint holds since:

• The first and second vectors differ from 3 positions, which is greater than or equal to $\mathrm{\pi Ί}=2$.

• The first and third vectors differ from 3 positions, which is greater than or equal to $\mathrm{\pi Ί}=2$.

• The second and third vectors differ from 2 positions, which is greater than or equal to $\mathrm{\pi Ί}=2$.

Typical
 $\mathrm{\pi Ί}>0$ $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|>1$
Symmetries
• Items of $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ are permutable.

• Items of $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }.\mathrm{\pi \pi \pi }$ are permutable (same permutation used).

Keywords
Arc input(s)

$\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi }_\mathrm{\pi \pi \pi }$$\left(\mathrm{\pi Ί},\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|*|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|-|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|$

Graph class
 $\beta ’$$\mathrm{\pi ½\pi Ύ}_\mathrm{\pi »\pi Ύ\pi Ύ\pi Ώ}$ $\beta ’$$\mathrm{\pi \pi \pi Ό\pi Ό\pi ΄\pi \pi \pi Έ\pi ²}$

Graph model

The Arc constraint(s) slot uses the $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi }_\mathrm{\pi \pi \pi }$ constraint defined in this catalogue.

PartsΒ (A) andΒ (B) of FigureΒ 5.2.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the arcs of the final graph are stressed in bold. The previous constraint holds since exactly $3Β·\left(3-1\right)=6$ arc constraints hold.

Signature

Since we use the arc generator on the items of the $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ collection, the expression $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|Β·|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|-|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|$ corresponds to the maximum number of arcs of the final graph. Therefore we can rewrite the graph property $\mathrm{\pi \pi \pi \pi }$ $=$ $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|Β·|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|-|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|$ to $\mathrm{\pi \pi \pi \pi }$ $\beta ₯$ $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|Β·|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|-|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|$. This leads to simplify $\underset{Μ²}{\stackrel{Β―}{\mathrm{\pi \pi \pi \pi }}}$ to $\stackrel{Β―}{\mathrm{\pi \pi \pi \pi }}$.