Computing Optimal Tangles Faster PK AW JZ OF W u AR Lviv - PowerPoint PPT Presentation

Computing Optimal Tangles Faster PK AW JZ OF W¨ u AR Lviv Alexander Wolff Oksana Firman Philipp Kindermann 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 y -monotone wires

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

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

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

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

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

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

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

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

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

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

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

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

Related work • Olszewski et al. Visualizing the template � of a chaotic attractor. list GD 2018 Algorithm to find ? Complexity ? the optimal tangle

Related work • Olszewski et al. Visualizing the template � of a chaotic attractor. list GD 2018 Algorithm to find ? Complexity ? the optimal tangle • 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 to find ? Complexity ? the optimal tangle • 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 • Improved the algorithm of [Olszewski et al., GD’18] Using the Dynamic Program � n 2 �� 2 | L | � � ϕ 2 | L | � 2 ϕ n n O 5 | L | / n n n 2 + 1 O • Experiments

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: a multiset A of 3 m positive integers · · · a 1 a 2 a 3 a 3 m − 2 a 3 m − 1 a 3 m

Complexity Theorem Tangle-Height Minimization is NP-hard. Proof Reduction from 3-Partition 3-Partition Given: a multiset A of 3 m positive integers Objective: decide whether A can be partitioned into m groups of three elements each that all sum up to the same value B · · · a 1 a 2 a 3 a 3 m − 2 a 3 m − 1 a 3 m · · · � � � m = B 2 = B 1 = B

Complexity Theorem Tangle-Height Minimization is NP-hard. Proof B 4 < a i < B 2 Reduction from 3-Partition 3-Partition B is poly in m Given: a multiset A of 3 m positive integers Objective: decide whether A can be partitioned into m groups of three elements each that all sum up to the same value B · · · a 1 a 2 a 3 a 3 m − 2 a 3 m − 1 a 3 m · · · � � � m = B 2 = B 1 = B

Complexity Theorem Tangle-Height Minimization is NP-hard. Proof B 4 < a i < B 2 Reduction from 3-Partition B is poly in m Given: a multiset A of 3 m positive integers Objective: decide whether A can be partitioned into m groups of three elements each that all sum up to the same value B · · · a 1 a 2 a 3 a 3 m − 2 a 3 m − 1 a 3 m · · · � � � m = B 2 = B 1 = B Given: an instance A of 3-Partition

Complexity Theorem Tangle-Height Minimization is NP-hard. Proof B 4 < a i < B 2 Reduction from 3-Partition B is poly in m Given: a multiset A of 3 m positive integers Objective: decide whether A can be partitioned into m groups of three elements each that all sum up to the same value B · · · a 1 a 2 a 3 a 3 m − 2 a 3 m − 1 a 3 m · · · � � � m = B 2 = B 1 = B Given: an 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 1 a 2 a 3 a 3 m − 2 a 3 m − 1 a 3 m · · · � � m = B � 2 = B 1 = B � A Given: an 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 1 a 2 a 3 a 3 m − 2 a 3 m − 1 a 3 m · · · � � m = B � 2 = B 1 = B +1 � A Given: an 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 1 a 2 a 3 a 3 m − 2 a 3 m − 1 a 3 m · · · � � m = B � 2 = B 1 = B +1 � A +1 Given: an 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 1 a 2 a 3 a 3 m − 2 a 3 m − 1 a 3 m · · · � � m = B � 2 = B 1 = B +1 � A +1 Given: an 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

Constructing the list L ω ′ ω ω ′ ω

Constructing the list L ω ′ ω 2 m swaps ω ′ ω

Constructing the list L α ′ ω ′ α 1 ω 1 ω ′ α ′ ω α 1 1

Constructing the list L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 ω ′ α ′ ω α 1 1

Constructing the list L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 ω ′ α ′ ω α 1 1

Constructing the list L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 ω ′ α ′ ω α 1 1

Constructing the list L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 ω ′ α ′ ω α 1 1

Constructing the list L α ′ ω ′ α 1 ω 1 M = 2 m 3 Ma 1 What is not allowed? split ω ′ α ′ ω α 1 1

Constructing the 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

Constructing the list L α ′ α ′ ω ′ α 2 α 1 ω 1 2 M = 2 m 3 Ma 1 What is not allowed? put it on the same level with other α - α ′ swaps Ma 2 ω ′ α ′ α ′ ω α 2 α 1 1 2

Constructing the list L α ′ α ′ ω ′ · · · α 6 α 1 · · · ω 6 1 M = 2 m 3 Ma 1 Ma 4 Ma 5 Ma 2 Ma 3 Ma 6 ω ′ α ′ ω α ′ α 6 · · · α 1 · · · 1 6