On Minimal-Perimeter Latice Animals Gill Barequet, Gil Ben-Shachar - PowerPoint PPT Presentation

On Minimal-Perimeter Latice Animals Gill Barequet, Gil Ben-Shachar Dept. of Computer Science, Technion, Haifa EuroCG 2020, Würzburg, Germany

What is a Lattice Animal? Polyominoes Polyhexes Polyiamonds Polycubes

Definitions Term Definition Notation Lattice Animal A set of connected cells on some lattice 𝑅

Definitions Term Definition Notation Lattice Animal A set of connected cells on some lattice 𝑅 Area The number of cells |𝑅| Area = 8 Area = 19

Definitions Term Definition Notation Lattice Animal A set of connected cells on some lattice 𝑅 Area The number of cells |𝑅| Perimeter Empty adjacent cells 𝒬(𝑅) Perimeter size = 19 Perimeter size = 13 Area = 8 Area = 19

Definitions Term Definition Notation Lattice Animal A set of connected cells on some lattice 𝑅 Area The number of cells |𝑅| Perimeter Empty adjacent cells 𝒬(𝑅) Border Lattice animal cells with empty adjacent cells ℬ(𝑅) Perimeter size = 19 Perimeter size = 13 Area = 8 Area = 19

Minimal-Perimeter Lattice Animals • Definition: a minimal-perimeter lattice animal (MPA) is a lattice animal which have the minimum possible perimeter from within all lattice animals of the same size. • Examples:

Motivation [Asinowski, Barequet and Zheng. 2017]

Minimal-Perimeter Lattice Animals 𝑁 7 = 𝑁 17 = 𝑁 31 =

Inflation of Polyominoes Inflation of a polyomino 𝑅 , 𝐽 𝑅 , is 𝐽 𝑅 = 𝑅 ∪ 𝒬 𝑅 𝐽 𝑅 The deflated polyomino D 𝑅 is 𝐸 𝑅 = 𝑅\ℬ(𝑅) 𝑅 𝐸 𝑅

Inflation of Polyominoes • Theorem: For 𝑜 ≥ 3 and any 𝑙 ∈ 𝑂, |𝑁 𝑜 | = |𝑁 𝑜+𝑙𝜗 𝑜 +2𝑙 𝑙−1 | 𝑁 7 = 𝑁 17 = 𝑁 31 = |𝑁 2477537 | = 4

Minimal-Perimeter Lattice Animals • Definition: a minimal-perimeter lattice animal (MPA) is a lattice animal which have the minimum possible perimeter from within all lattice animals of the same size. • Examples:

Genralization to Lattice Animals • Does inflation induce a bijection in other lattices? • The following set of conditions are sufficient: 1) The minimal perimeter size is monotonically increasing (w.r.t the area) Heaviest 2) |𝒬 𝑅 | = |ℬ 𝑅 | + 𝑑 for some 𝑑 requirements 3) Deflation of a MPA creates a valid lattice animal

Proof structure • The idea is to show a bijection between sets of MPAs. • First direction: Inflation of an MPA creates a new (unique) MPA. • Second direction: If one MPA of area 𝑜 is created by an inflation, then any MPA of area 𝑜 can be deflated to a smaller MPA.

Proof: First direction • Theorem: For a minimal-perimeter a nimal 𝑅, 𝐽(𝑅) is a minimal- perimeter animal as well. • Proof idea  Assume 𝐽(𝑅) is not minimal-perimeter animal.  ∃𝑅 ′ s.t. 𝑅 ′ = 𝐽 𝑅 and 𝒬 𝑅 ′ < 𝒬(𝐽 𝑅 )  After some calculations (using condition #2 ) – 𝐸 𝑅 ′ > 𝑅 , and 𝒬 𝑅 ′ < 𝒬 𝑅  Contradicts condition #1.  ⇒ 𝑅 is not a minimal-perimeter animal.

First direction: Corollary • Inflation of MPAs creates an infinite chain of new MPAs.

Second direction • Lemma: If 𝑅 ∈ 𝑁 𝑜+𝜗 𝑜 then 𝐸 𝑅 ∈ 𝑁 𝑜 • Proof: • Let Q ∈ 𝑁 𝑜+𝜗 𝑜 • ℬ 𝑅 = 𝜗 𝑜 , thus 𝐸 𝑅 = 𝑜 . ℬ 𝑅 = 𝜗(𝑜) ℬ 𝑅 = 𝜗(𝑜)

Second direction • Lemma: If 𝑅 ∈ 𝑁 𝑜+𝜗 𝑜 then 𝐸 𝑅 ∈ 𝑁 𝑜 • Proof: • Let Q ∈ 𝑁 𝑜+𝜗 𝑜 • ℬ 𝑅 = 𝜗 𝑜 , thus 𝐸 𝑅 = 𝑜 . • | 𝒬 𝐸 𝑅 | ≥ 𝜗 𝑜 and 𝒬 𝐸 𝑅 ⊆ ℬ 𝑅 • 𝒬 D Q = ℬ 𝑅 ⇒ 𝐽 𝐸 𝑅 = 𝑅 . • ⇒ 𝑁 𝑜 ≥ |𝑁 𝑜+𝜗 𝑜 | 𝒬 𝑅 = 𝜗(𝑜)

Genralization to Lattice Animals • Does inflation induce a this bijection in other lattices? • The following set of conditions are sufficient: 1) The minimal perimeter size is monotonically increasing Heaviest 2) |𝒬 𝑅 | = |ℬ 𝑅 | + 𝑑 for some 𝑑 requirements 3) Deflation of a MPA creates a valid lattice animal

Proof for polyhexes 1) The minimal perimeter size is monotonically increasing Known [Vainsencher and Bruckstein, 2008] 2) |𝒬 𝑅 | = |ℬ 𝑅 | + 𝑑 for some 𝑑 3) Deflation of a minimal-perimeter polyhex creates a valid polyhex Easy to see …

Proof for polyhexes • How to prove that 𝒬 𝑅 = ℬ 𝑅 + 𝑑? • Classify each cell or perimeter cell to one of the following patterns: Regular cells: Perimeter cells: • Show that: 𝒬 Q = ℬ 𝑅 + 3 ⋅ # + 2 ⋅ # + # − 3 ⋅ # − 2 ⋅ # − #

Proof for polyhexes • How to prove that 𝒬 𝑅 = ℬ 𝑅 + 𝑑? • Show that: 𝒬 Q = ℬ 𝑅 + 3 ⋅ # + 2 ⋅ # + # − 3 ⋅ # − 2 ⋅ # − # • Use some calculations to get: 𝒬 𝑅 = ℬ 𝑅 + 6 • (For polyominoes it is 𝒬 𝑅 = ℬ 𝑅 + 4)

Proof for polyhexes

Polyiamonds

Polyiamonds vs.

Polyiamonds

Polycubes

Counting Minimal - Perimeter Polyhexes

Questions • Is there a bijection between sets of minimal-perimeter polycubes? • Are all the conditions are necessary?

Thank you!