5.338. tree_range

DESCRIPTIONLINKSGRAPH
Origin

Derived from πšπš›πšŽπšŽ.

Constraint

πšπš›πšŽπšŽ_πš›πšŠπš—πšπšŽ(π™½πšƒπšπ™΄π™΄πš‚,𝚁,π™½π™Ύπ™³π™΄πš‚)

Arguments
π™½πšƒπšπ™΄π™΄πš‚πšπšŸπšŠπš›
πšπšπšŸπšŠπš›
π™½π™Ύπ™³π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš’πš—πšπšŽπš‘-πš’πš—πš,𝚜𝚞𝚌𝚌-πšπšŸπšŠπš›)
Restrictions
π™½πšƒπšπ™΄π™΄πš‚β‰₯0
𝚁β‰₯0
𝚁<|π™½π™Ύπ™³π™΄πš‚|
|π™½π™Ύπ™³π™΄πš‚|>0
πš›πšŽπššπšžπš’πš›πšŽπš(π™½π™Ύπ™³π™΄πš‚,[πš’πš—πšπšŽπš‘,𝚜𝚞𝚌𝚌])
π™½π™Ύπ™³π™΄πš‚.πš’πš—πšπšŽπš‘β‰₯1
π™½π™Ύπ™³π™΄πš‚.πš’πš—πšπšŽπš‘β‰€|π™½π™Ύπ™³π™΄πš‚|
πšπš’πšœπšπš’πš—πšŒπš(π™½π™Ύπ™³π™΄πš‚,πš’πš—πšπšŽπš‘)
π™½π™Ύπ™³π™΄πš‚.𝚜𝚞𝚌𝚌β‰₯1
π™½π™Ύπ™³π™΄πš‚.πšœπšžπšŒπšŒβ‰€|π™½π™Ύπ™³π™΄πš‚|
Purpose

Cover the digraph G described by the π™½π™Ύπ™³π™΄πš‚ collection with π™½πšƒπšπ™΄π™΄πš‚ trees in such a way that each vertex of G belongs to one distinct tree. 𝚁 is the difference between the longest and the shortest paths (from a leaf to a root) of the final graph.

Example
2,1,πš’πš—πšπšŽπš‘-1𝚜𝚞𝚌𝚌-1,πš’πš—πšπšŽπš‘-2𝚜𝚞𝚌𝚌-5,πš’πš—πšπšŽπš‘-3𝚜𝚞𝚌𝚌-5,πš’πš—πšπšŽπš‘-4𝚜𝚞𝚌𝚌-7,πš’πš—πšπšŽπš‘-5𝚜𝚞𝚌𝚌-1,πš’πš—πšπšŽπš‘-6𝚜𝚞𝚌𝚌-1,πš’πš—πšπšŽπš‘-7𝚜𝚞𝚌𝚌-7,πš’πš—πšπšŽπš‘-8𝚜𝚞𝚌𝚌-5

The πšπš›πšŽπšŽ_πš›πšŠπš—πšπšŽ constraint holds since the graph associated with the items of the π™½π™Ύπ™³π™΄πš‚ collection corresponds to two trees (i.e.,Β π™½πšƒπšπ™΄π™΄πš‚=2): each tree respectively involves the vertices {1,2,3,5,6,8} and {4,7}. Furthermore 𝚁=1 is set to the difference between the longest path (for instance 2β†’5β†’1) and the shortest path (for instance 4β†’7) from a leaf to a root. FigureΒ 5.338.1 provides the two trees associated with the example.

Figure 5.338.1. The two trees associated with the example
ctrs/tree_range1
Symmetry

Items of π™½π™Ύπ™³π™΄πš‚ are permutable.

Reformulation

By introducing a distance variable D i , an occurrence variable O i and a leave variable L i (1≀i≀|π™½π™Ύπ™³π™΄πš‚|) for each item i of the π™½π™Ύπ™³π™΄πš‚ collection, where:

  • D i represents the number of vertices from i to the root of the corresponding tree,

  • O i gives the number of occurrences of value i within variables π™½π™Ύπ™³π™΄πš‚[1].𝚜𝚞𝚌𝚌,π™½π™Ύπ™³π™΄πš‚[2].𝚜𝚞𝚌𝚌,...,π™½π™Ύπ™³π™΄πš‚[n].𝚜𝚞𝚌𝚌,

  • L i is set to 1 if item i corresponds to a leave (i.e.,Β O i >0) and 0 otherwise,

the πšπš›πšŽπšŽ_πš›πšŠπš—πšπšŽ(π™½πšƒπšπ™΄π™΄πš‚,𝚁,π™½π™Ύπ™³π™΄πš‚) constraint can be expressed in term of a conjunction of one πšπš›πšŽπšŽ constraint, |π™½π™Ύπ™³π™΄πš‚| πšŽπš•πšŽπš–πšŽπš—πš constraints, |π™½π™Ύπ™³π™΄πš‚| linear constraints, one πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’ constraint, |π™½π™Ύπ™³π™΄πš‚| reified constraints, one πš˜πš™πšŽπš—_πš–πš’πš—πš’πš–πšžπš–, one πš–πšŠπš‘πš’πš–πšžπš– and one linear constraint, where:

  • The πšπš›πšŽπšŽ constraint models the fact that we have a forest of π™½πšƒπšπ™΄π™΄πš‚ trees.

  • Each πšŽπš•πšŽπš–πšŽπš—πš constraint provides the link between the attribute 𝚜𝚞𝚌𝚌 of the i-th item and the distance variable D π™½π™Ύπ™³π™΄πš‚[i].𝚜𝚞𝚌𝚌 associated with item π™½π™Ύπ™³π™΄πš‚[i].𝚜𝚞𝚌𝚌.

  • Each linear constraint associated with the i-th item states that the difference between the distance variable D i and the distance variable D π™½π™Ύπ™³π™΄πš‚[i].𝚜𝚞𝚌𝚌 is equal to 1.

  • The πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’ constraint provides the number of occurrences O i of value i (1≀i≀|π™½π™Ύπ™³π™΄πš‚|) within variables π™½π™Ύπ™³π™΄πš‚[1].𝚜𝚞𝚌𝚌,π™½π™Ύπ™³π™΄πš‚[2].𝚜𝚞𝚌𝚌,...,π™½π™Ύπ™³π™΄πš‚[|π™½π™Ύπ™³π™΄πš‚|].𝚜𝚞𝚌𝚌. Note that, when O i is equal to 0, the corresponding i-th item is a leave of one of the π™½πšƒπšπ™΄π™΄πš‚ trees.

  • Each reified constraint of the form L i ⇔O i >0 makes the link between the i-th occurrence variable O i and the i-th leave variable L i .

  • The πš˜πš™πšŽπš—_πš–πš’πš—πš’πš–πšžπš– constraint computes the minimum distance 𝙼𝙸𝙽 from the leaves to the corresponding roots. The leave variable L i is used in order to select only the distance variables corresponding to leaves.

  • The πš–πšŠπš‘πš’πš–πšžπš– constraint computes the maximum distance π™Όπ™°πš‡ from the vertices to the roots. Since the maximum is achieved by a leave we do not need to focus just on the leaves as it was the case for the minimum distance 𝙼𝙸𝙽.

  • The linear constraint π™Όπ™°πš‡-𝙼𝙸𝙽=𝚁 links together argument 𝚁 to the minimum and maximum distances.

With respect to the Example slot we get the following conjunction of constraints:

Β Β Β πšπš›πšŽπšŽ(2,βŒ©πš’πš—πšπšŽπš‘-1 𝚜𝚞𝚌𝚌-1, πš’πš—πšπšŽπš‘-2 𝚜𝚞𝚌𝚌-5,

Β Β Β Β Β Β Β Β Β Β Β Β πš’πš—πšπšŽπš‘-3 𝚜𝚞𝚌𝚌-5, πš’πš—πšπšŽπš‘-4 𝚜𝚞𝚌𝚌-7,

Β Β Β Β Β Β Β Β Β Β Β Β πš’πš—πšπšŽπš‘-5 𝚜𝚞𝚌𝚌-1, πš’πš—πšπšŽπš‘-6 𝚜𝚞𝚌𝚌-1,

Β Β Β Β Β Β Β Β Β Β Β Β πš’πš—πšπšŽπš‘-7 𝚜𝚞𝚌𝚌-7, πš’πš—πšπšŽπš‘-8 𝚜𝚞𝚌𝚌-5βŒͺ),

Β Β Β πšπš˜πš–πšŠπš’πš—(〈D 1 ,D 2 ,D 3 ,D 4 ,D 5 ,D 6 ,D 7 ,D 8 βŒͺ,0,8),

Β Β Β DS 1 ∈[0,8],Β Β πšŽπš•πšŽπš–πšŽπš—πš(1,〈0,D 2 ,D 3 ,D 4 ,D 5 ,D 6 ,D 7 ,D 8 βŒͺ,DS 1 ),Β Β D 1 -0=1,

Β Β Β DS 2 ∈[0,8],Β Β πšŽπš•πšŽπš–πšŽπš—πš(5,〈1,0,D 3 ,D 4 ,D 5 ,D 6 ,D 7 ,D 8 βŒͺ,DS 2 ),Β Β D 2 -D 5 =1,

Β Β Β DS 3 ∈[0,8],Β Β πšŽπš•πšŽπš–πšŽπš—πš(5,〈1,D 2 ,0,D 4 ,D 5 ,D 6 ,D 7 ,D 8 βŒͺ,DS 3 ),Β Β D 3 -D 5 =1,

Β Β Β DS 4 ∈[0,8],Β Β πšŽπš•πšŽπš–πšŽπš—πš(7,〈1,D 2 ,D 3 ,0,D 5 ,D 6 ,D 7 ,D 8 βŒͺ,DS 4 ),Β Β D 4 -D 7 =1,

Β Β Β DS 5 ∈[0,8],Β Β πšŽπš•πšŽπš–πšŽπš—πš(1,〈1,D 2 ,D 3 ,D 4 ,0,D 6 ,D 7 ,D 8 βŒͺ,DS 5 ),Β Β D 5 -1=1,

Β Β Β DS 6 ∈[0,8],Β Β πšŽπš•πšŽπš–πšŽπš—πš(1,〈1,3,3,D 4 ,2,0,D 7 ,D 8 βŒͺ,DS 6 ),Β Β D 6 -1=1,

Β Β Β DS 7 ∈[0,8],Β Β πšŽπš•πšŽπš–πšŽπš—πš(7,〈1,3,3,D 4 ,2,2,0,D 8 βŒͺ,DS 7 ),Β Β D 7 -0=1,

Β Β Β DS 8 ∈[0,8],Β Β πšŽπš•πšŽπš–πšŽπš—πš(5,〈1,3,3,2,2,2,1,0βŒͺ,DS 8 ),Β Β D 8 -2=1,

Β Β Β πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’(〈1,5,5,7,1,1,7,5βŒͺ, βŒ©πšŸπšŠπš•-1 πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-3,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš•-2 πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-0,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš•-3 πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-0,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš•-4 πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-0,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš•-5 πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-3,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš•-6 πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-0,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš•-7 πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-2,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš•-8 πš—πš˜πšŒπšŒπšžπš›πš›πšŽπš—πšŒπšŽ-0βŒͺ),

Β Β Β 1⇔3>0,0⇔0>0,0⇔0>0,0⇔0>0,

Β Β Β 1⇔3>0,0⇔0>0,1⇔2>0,0⇔0>0,

Β Β Β πš˜πš™πšŽπš—_πš–πš’πš—πš’πš–πšžπš–(𝙼𝙸𝙽,βŒ©πšŸπšŠπš›-3 πš‹πš˜πš˜πš•-1,πšŸπšŠπš›-0 πš‹πš˜πš˜πš•-0,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš›-0 πš‹πš˜πš˜πš•-0,πšŸπšŠπš›-0 πš‹πš˜πš˜πš•-0,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš›-3 πš‹πš˜πš˜πš•-1,πšŸπšŠπš›-0 πš‹πš˜πš˜πš•-0,

Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β πšŸπšŠπš›-2 πš‹πš˜πš˜πš•-1,πšŸπšŠπš›-0 πš‹πš˜πš˜πš•-0βŒͺ),

Β Β Β πš–πšŠπš‘πš’πš–πšžπš–(π™Όπ™°πš‡,〈1,3,3,2,2,2,1,3βŒͺ),

Β Β Β π™Όπ™°πš‡-𝙼𝙸𝙽=𝚁=1.

See also

related: πš‹πšŠπš•πšŠπš—πšŒπšŽΒ (balanced tree versus balanced assignment).

root concept: πšπš›πšŽπšŽ.

used in reformulation: πšπš˜πš–πšŠπš’πš—, πšŽπš•πšŽπš–πšŽπš—πš, πšπš•πš˜πš‹πšŠπš•_πšŒπšŠπš›πšπš’πš—πšŠπš•πš’πšπš’, πš–πšŠπš‘πš’πš–πšžπš–, πš˜πš™πšŽπš—_πš–πš’πš—πš’πš–πšžπš–, πšπš›πšŽπšŽ.

Keywords

constraint type: graph constraint, graph partitioning constraint.

final graph structure: connected component, tree.

modelling: balanced tree.

Arc input(s)

π™½π™Ύπ™³π™΄πš‚

Arc generator
πΆπΏπΌπ‘„π‘ˆπΈβ†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš—πš˜πšπšŽπšœ1,πš—πš˜πšπšŽπšœ2)

Arc arity
Arc constraint(s)
πš—πš˜πšπšŽπšœ1.𝚜𝚞𝚌𝚌=πš—πš˜πšπšŽπšœ2.πš’πš—πšπšŽπš‘
Graph property(ies)
β€’ πŒπ€π—_𝐍𝐒𝐂𝐂≀1
β€’ 𝐍𝐂𝐂=π™½πšƒπšπ™΄π™΄πš‚
β€’ 𝐑𝐀𝐍𝐆𝐄_𝐃𝐑𝐆=𝚁

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.338.2 respectively show the initial and final graph associated with the Example slot. Since we use the πšπ™°π™½π™Άπ™΄_π™³πšπ™Ά graph property, we respectively display the longest and shortest paths of the final graph with a bold and a dash line.

Figure 5.338.2. Initial and final graph of the πšπš›πšŽπšŽ_πš›πšŠπš—πšπšŽ constraint
ctrs/tree_rangeActrs/tree_rangeB
(a) (b)