5.71. connect_points

DESCRIPTIONLINKSGRAPH
Origin

N.Β Beldiceanu

Constraint

πšŒπš˜πš—πš—πšŽπšŒπš_πš™πš˜πš’πš—πšπšœ(πš‚π™Έπš‰π™΄1,πš‚π™Έπš‰π™΄2,πš‚π™Έπš‰π™΄3,π™½π™Άπšπ™Ύπš„π™Ώ,π™Ώπ™Ύπ™Έπ™½πšƒπš‚)

Arguments
πš‚π™Έπš‰π™΄1πš’πš—πš
πš‚π™Έπš‰π™΄2πš’πš—πš
πš‚π™Έπš‰π™΄3πš’πš—πš
π™½π™Άπšπ™Ύπš„π™ΏπšπšŸπšŠπš›
π™Ώπ™Ύπ™Έπ™½πšƒπš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš™-πšπšŸπšŠπš›)
Restrictions
πš‚π™Έπš‰π™΄1>0
πš‚π™Έπš‰π™΄2>0
πš‚π™Έπš‰π™΄3>0
π™½π™Άπšπ™Ύπš„π™Ώβ‰₯0
π™½π™Άπšπ™Ύπš„π™Ώβ‰€|π™Ώπ™Ύπ™Έπ™½πšƒπš‚|
πš‚π™Έπš‰π™΄1*πš‚π™Έπš‰π™΄2*πš‚π™Έπš‰π™΄3=|π™Ώπ™Ύπ™Έπ™½πšƒπš‚|
πš›πšŽπššπšžπš’πš›πšŽπš(π™Ώπ™Ύπ™Έπ™½πšƒπš‚,πš™)
Purpose

On a 3-dimensional grid of variables, number of groups, where a group consists of a connected set of variables that all have a same value distinct from 0.

Example
8,4,2,2,πš™-0,πš™-0,πš™-1,πš™-1,πš™-0,πš™-2,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-1,πš™-0,πš™-2,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-1,πš™-1,πš™-1,πš™-1,πš™-1,πš™-0,πš™-2,πš™-0,πš™-1,πš™-0,πš™-2,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-0,πš™-2,πš™-0,πš™-0,πš™-0,πš™-2,πš™-2,πš™-2,πš™-2,πš™-2,πš™-0,πš™-0,πš™-0,πš™-2,πš™-0,πš™-0,πš™-0,πš™-2,πš™-0,πš™-0

FigureΒ 5.71.1 corresponds to the solution where we describe separately each layer of the grid. The πšŒπš˜πš—πš—πšŽπšŒπš_πš™πš˜πš’πš—πšπšœ constraint holds since we have two groups (π™½π™Άπšπ™Ύπš„π™Ώ=2): a first one for the variables of the π™Ώπ™Ύπ™Έπ™½πšƒπš‚ collection assigned to value 1, and a second one for the variables assigned to value 2.

Figure 5.71.1. The two layers of the solution
ctrs/connect_points1
Typical
πš‚π™Έπš‰π™΄1>1
πš‚π™Έπš‰π™΄2>1
π™½π™Άπšπ™Ύπš„π™Ώ>0
π™½π™Άπšπ™Ύπš„π™Ώ<|π™Ώπ™Ύπ™Έπ™½πšƒπš‚|
|π™Ώπ™Ύπ™Έπ™½πšƒπš‚|>3
Symmetry

All occurrences of two distinct values of π™Ώπ™Ύπ™Έπ™½πšƒπš‚.πš™ that are both different from 0 can be swapped; all occurrences of a value of π™Ώπ™Ύπ™Έπ™½πšƒπš‚.πš™ that is different from 0 can be renamed to any unused value that is also different from 0.

Usage

Wiring problemsΒ [Simonis90],Β [Zhou96].

Keywords

characteristic of a constraint: joker value.

final graph structure: strongly connected component, symmetric.

geometry: geometrical constraint.

problems: channel routing.

Arc input(s)

π™Ώπ™Ύπ™Έπ™½πšƒπš‚

Arc generator
𝐺𝑅𝐼𝐷([πš‚π™Έπš‰π™΄1,πš‚π™Έπš‰π™΄2,πš‚π™Έπš‰π™΄3])β†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš™πš˜πš’πš—πšπšœ1,πš™πš˜πš’πš—πšπšœ2)

Arc arity
Arc constraint(s)
β€’ πš™πš˜πš’πš—πšπšœ1.πš™β‰ 0
β€’ πš™πš˜πš’πš—πšπšœ1.πš™=πš™πš˜πš’πš—πšπšœ2.πš™
Graph property(ies)
𝐍𝐒𝐂𝐂=π™½π™Άπšπ™Ύπš„π™Ώ

Graph class
πš‚πšˆπ™Όπ™Όπ™΄πšƒπšπ™Έπ™²

Graph model

FigureΒ 5.71.2 gives the initial graph constructed by the 𝐺𝑅𝐼𝐷 arc generator associated with the Example slot.

Figure 5.71.2. Graph generated by 𝐺𝑅𝐼𝐷([8,4,2])
ctrs/connect_points2