Computing Height-Optimal Tangles Faster Oksana Firman W u PK AW - PowerPoint PPT Presentation

Computing Height-Optimal Tangles Faster Oksana Firman W¨ u PK AW JZ OF AR Lviv Philipp Kindermann Alexander Wolff Johannes Zink Julius-Maximilians-Universit¨ at W¨ urzburg, Germany Alexander Ravsky Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

Introduction Given a set of n y -monotone wires

Introduction 1 ≤ i < j ≤ n Given a set of n swap ij y -monotone wires i j

Introduction 1 ≤ i < j ≤ n Given a set of n swap ij y -monotone wires disjoint swaps

Introduction 1 ≤ i < j ≤ n Given a set of n swap ij y -monotone wires disjoint swaps adjacent permutations

Introduction 1 ≤ i < j ≤ n Given a set of n swap ij y -monotone wires disjoint swaps adjacent permutations multiple swaps

Introduction 1 ≤ i < j ≤ n Given a set of n swap ij y -monotone wires disjoint swaps π 1 adjacent π 2 permutations π 3 multiple swaps π 4 tangle T of height h ( T ) π 5 π 6

Introduction 1 ≤ i < j ≤ n Given a set of n swap ij y -monotone wires · · · disjoint swaps · · · 1 2 1 2 n n π ′ π 1 1 adjacent π ′ π 2 permutations 2 π ′ π 3 multiple swaps 3 π ′ π 4 tangle T of 4 height h ( T ) π ′ π 5 5 π 6

Introduction 1 ≤ i < j ≤ n Given a set of n . . . and given a list of swap ij y -monotone wires swaps L · · · disjoint swaps 1 2 n π 1 adjacent π 2 permutations π 3 multiple swaps π 4 tangle T of height h ( T ) π 5 π 6

Introduction 1 ≤ i < j ≤ n Given a set of n . . . and given a list of swap ij y -monotone wires swaps L · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 π 4 1 tangle T of 2 height h ( T ) π 5 π 6

Introduction 1 ≤ i < j ≤ n Given a set of n . . . and given a list of swap ij y -monotone wires swaps L · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 π 4 1 tangle T of 2 height h ( T ) π 5 π 6 Tangle T ( L ) realizes list L .

Introduction 1 ≤ i < j ≤ n Given a set of n . . . and given a list of swap ij y -monotone wires swaps L · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 π 4 1 tangle T of 2 height h ( T ) π 5 π 6 Tangle T ( L ) realizes list L .

Introduction 1 ≤ i < j ≤ n Given a set of n . . . and given a list of swap ij y -monotone wires swaps L · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 π 4 1 tangle T of 2 height h ( T ) π 5 π 6 not feasible Tangle T ( L ) realizes list L .

Introduction 1 ≤ i < j ≤ n Given a set of n . . . and given a list of swap ij y -monotone wires swaps L · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 π 4 1 tangle T of 2 height h ( T ) π 5 π 6 Tangle T ( L ) realizes list L . A tangle T ( L ) is height-optimal if it has the minimum height among all tangles realizing the list L .

Related Work • Olszewski et al. Visualizing the template of a chaotic attractor. GD 2018

Related Work • Olszewski et al. Visualizing the template � of a chaotic attractor. list GD 2018

Related Work • Olszewski et al. Visualizing the template � of a chaotic attractor. list GD 2018 Algorithm for finding optimal tangles

Related Work • Olszewski et al. Visualizing the template � of a chaotic attractor. list GD 2018 Algorithm for finding ? Complexity ? optimal tangles

Related Work • Olszewski et al. Visualizing the template � of a chaotic attractor. list GD 2018 Algorithm for finding ? Complexity ? optimal tangles • Wang. Novel routing schemes for IC layout part I: Two-layer channel routing. DAC 1991 Given: initial and final permutations

Related Work • Olszewski et al. Visualizing the template � of a chaotic attractor. list GD 2018 Algorithm for finding ? Complexity ? optimal tangles • Wang. Novel routing schemes for IC layout part I: Two-layer channel routing. DAC 1991 Given: initial and final permutations • Bereg et al. Drawing Permutations with Few Corners. GD 2013 Objective: minimize the number of bends

Overview • Complexity: NP-hardness by reduction from 3-Partition . • New algorithm: using dynamic programming; asymptotically faster than [Olszewski et al., GD’18]. � n 2 �� 2 | L | � � ϕ 2 | L | � 2 ϕ n n O 5 | L | / n n n 2 + 1 O • Experiments: comparison with [Olszewski et al., GD’18]

Complexity Theorem. Tangle-Height Minimization is NP-hard.

Complexity Theorem. Tangle-Height Minimization is NP-hard. Proof. Reduction from 3-Partition

Complexity Theorem. Tangle-Height Minimization is NP-hard. Proof. Reduction from 3-Partition 3-Partition Given: Multiset A of 3 m positive integers. a 3 m − 2 a 3 m − 1 a 1 a 2 a 3 · · · a 3 m

Complexity Theorem. Tangle-Height Minimization is NP-hard. Proof. Reduction from 3-Partition 3-Partition Given: Multiset A of 3 m positive integers. Question: Can A be partitioned into m groups of three elements s.t. each group sums up to the same value B ? a 3 m − 2 a 3 m − 1 a 1 a 2 a 3 · · · a 3 m · · · � � � 1 = B 2 = B m = B

Complexity Theorem. Tangle-Height Minimization is NP-hard. B 4 < a i < B Proof. 2 Reduction from 3-Partition 3-Partition B is poly in m Given: Multiset A of 3 m positive integers. Question: Can A be partitioned into m groups of three elements s.t. each group sums up to the same value B ? a 3 m − 2 a 3 m − 1 a 1 a 2 a 3 · · · a 3 m · · · � � � 1 = B 2 = B m = B

Complexity Theorem. Tangle-Height Minimization is NP-hard. B 4 < a i < B Proof. 2 Reduction from 3-Partition B is poly in m Given: Multiset A of 3 m positive integers. Question: Can A be partitioned into m groups of three elements s.t. each group sums up to the same value B ? a 3 m − 2 a 3 m − 1 a 1 a 2 a 3 · · · a 3 m · · · � � � 1 = B 2 = B m = B Given: Instance A of 3-Partition .

Complexity Theorem. Tangle-Height Minimization is NP-hard. B 4 < a i < B Proof. 2 Reduction from 3-Partition B is poly in m Given: Multiset A of 3 m positive integers. Question: Can A be partitioned into m groups of three elements s.t. each group sums up to the same value B ? a 3 m − 2 a 3 m − 1 a 1 a 2 a 3 · · · a 3 m · · · � � � 1 = B 2 = B m = B Given: Instance A of 3-Partition . Task: Construct L s.t. there is T realizing L with height at most H = 2 m 3 ( � A )+7 m 2 iff A is a yes-instance.

Complexity Theorem. Tangle-Height Minimization is NP-hard. Proof. Reduction from 3-Partition a 3 m − 2 a 3 m − 1 a 1 a 2 a 3 · · · a 3 m · · · � � � 1 = B 2 = B m = B � A Given: Instance A of 3-Partition . Task: Construct L s.t. there is T realizing L with height at most H = 2 m 3 ( � A )+7 m 2 iff A is a yes-instance.

Complexity Theorem. Tangle-Height Minimization is NP-hard. Proof. Reduction from 3-Partition a 3 m − 2 a 3 m − 1 a 1 a 2 a 3 · · · a 3 m · · · � � � 1 = B 2 = B m = B +1 � A Given: Instance A of 3-Partition . Task: Construct L s.t. there is T realizing L with height at most H = 2 m 3 ( � A )+7 m 2 iff A is a yes-instance.

Complexity Theorem. Tangle-Height Minimization is NP-hard. Proof. Reduction from 3-Partition a 3 m − 2 a 3 m − 1 a 1 a 2 a 3 · · · a 3 m · · · � � � 1 = B 2 = B m = B +1 � A +1 Given: Instance A of 3-Partition . Task: Construct L s.t. there is T realizing L with height at most H = 2 m 3 ( � A )+7 m 2 iff A is a yes-instance.

Complexity Theorem. Tangle-Height Minimization is NP-hard. Proof. Reduction from 3-Partition a 3 m − 2 a 3 m − 1 a 1 a 2 a 3 · · · a 3 m · · · � � � 1 = B 2 = B m = B +1 � A +1 Given: Instance A of 3-Partition . Task: construct L s.t. there is T realizing L with height at most H = 2 m 3 ( � A )+7 m 2 iff A is a yes-instance +1

Transforming the Instance A into a List L ω ′ ω ω ′ ω

Transforming the Instance A into a List L ω ′ ω 2 m swaps ω ′ ω

Transforming the Instance A into a List L α ′ ω ′ α 1 ω 1 ω ′ α ′ ω α 1 1

Transforming the Instance A into a List L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 ω ′ α ′ ω α 1 1

Transforming the Instance A into a List L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 ω ′ α ′ ω α 1 1

Transforming the Instance A into a List L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 ω ′ α ′ ω α 1 1

Transforming the Instance A into a List L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 ω ′ α ′ ω α 1 1

Transforming the Instance A into a List L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 What is not possible? split ω ′ α ′ ω α 1 1

Transforming the Instance A into a List L α ′ α ′ ω ′ α 2 α 1 ω 1 2 M = 2 m 3 Ma 1 Ma 2 ω ′ α ′ α ′ ω α 2 α 1 1 2

Transforming the Instance A into a List L α ′ α ′ ω ′ α 2 α 1 ω 1 2 M = 2 m 3 Ma 1 What is not possible? put it on the same level with other α - α ′ swaps Ma 2 ω ′ α ′ α ′ ω α 2 α 1 1 2