tangles faster computing height-optimal · 2019. 9. 18. · tangles faster oksanafirman...

Computing Height-OptimalTangles Faster

Oksana FirmanPhilipp Kindermann

Alexander WolffJohannes Zink

Julius-Maximilians-Universitat Wurzburg,

Germany

Alexander RavskyPidstryhach Institute for Applied Problems

of Mechanics and Mathematics,National Academy of Sciences of Ukraine,

Lviv, Ukraine

PK AW JZ OF Wu AR Lviv

Introduction

Given a set of ny -monotone wires

Introduction

Given a set of ny -monotone wires

1 ≤ i < j ≤ n

swap i j

i j

Introduction

Given a set of ny -monotone wires

1 ≤ i < j ≤ n

swap i j

disjoint swaps

Introduction

Given a set of ny -monotone wires

1 ≤ i < j ≤ n

swap i j

disjoint swaps

adjacentpermutations

Introduction

Given a set of ny -monotone wires

1 ≤ i < j ≤ n

swap i j

disjoint swaps

multiple swaps

adjacentpermutations

Introduction

Given a set of ny -monotone wires

1 ≤ i < j ≤ n

swap i j

disjoint swaps

multiple swaps

tangle T ofheight h(T )

π1

π2

π3

π6

adjacentpermutations

π4

π5

Introduction

Given a set of ny -monotone wires

1 ≤ i < j ≤ n

1 2 · · · n

swap i j

disjoint swaps

multiple swaps

tangle T ofheight h(T )

π1

π2

π3

π6

π′1

π′2

π′3

π′5

adjacentpermutations

π4

π5

π′4

1 2 · · · n

Introduction

Given a set of ny -monotone wires

. . . and given a list ofswaps L

1 ≤ i < j ≤ n

1 2 · · · n

swap i j

disjoint swaps

multiple swaps

tangle T ofheight h(T )

π1

π2

π3

π6

adjacentpermutations

π4

π5

Introduction

Given a set of ny -monotone wires

as a multiset (`i j)

. . . and given a list ofswaps L

1 ≤ i < j ≤ n

1 2 · · · n

swap i j

disjoint swaps

multiple swaps

tangle T ofheight h(T )

π1

π2

π3

π6

1

3112

adjacentpermutations

π4

π5

Introduction

Given a set of ny -monotone wires

as a multiset (`i j)

. . . and given a list ofswaps L

1 ≤ i < j ≤ n

1 2 · · · n

swap i j

disjoint swaps

multiple swaps

tangle T ofheight h(T )

π1

π2

π3

π6

1

3112

Tangle T (L) realizes list L.

adjacentpermutations

π4

π5

Introduction

Given a set of ny -monotone wires

as a multiset (`i j)

. . . and given a list ofswaps L

1 ≤ i < j ≤ n

1 2 · · · n

swap i j

disjoint swaps

multiple swaps

tangle T ofheight h(T )

π1

π2

π3

π6

1

3112

Tangle T (L) realizes list L.

adjacentpermutations

π4

π5

Introduction

Given a set of ny -monotone wires

as a multiset (`i j)

. . . and given a list ofswaps L

1 ≤ i < j ≤ n

1 2 · · · n

swap i j

disjoint swaps

multiple swaps

tangle T ofheight h(T )

π1

π2

π3

π6

1

3112

Tangle T (L) realizes list L.

adjacentpermutations

not feasible

π4

π5

Introduction

Given a set of ny -monotone wires

as a multiset (`i j)

. . . and given a list ofswaps L

1 ≤ i < j ≤ n

1 2 · · · n

swap i j

disjoint swaps

multiple swaps

tangle T ofheight h(T )

π1

π2

π3

π6

A tangle T (L) is height-optimal if it has the minimum heightamong all tangles realizing the list L.

1

3112

Tangle T (L) realizes list L.

adjacentpermutations

π4

π5

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.GD 2018

mlist

Related Work• Olszewski et al. Visualizing the template

of a chaotic attractor.GD 2018

mlist

Algorithm for findingoptimal tangles

Related Work• Olszewski et al. Visualizing the template

of a chaotic attractor.GD 2018

mlist

Algorithm for findingoptimal tangles

Complexity ??

Related Work• Olszewski et al. Visualizing the template

of a chaotic attractor.GD 2018

mlist

• Wang. Novel routing schemes for IClayout part I: Two-layer channel routing.DAC 1991

Given:initial andfinal permutations

Algorithm for findingoptimal tangles

Complexity ??

Related Work• Olszewski et al. Visualizing the template

of a chaotic attractor.GD 2018

mlist

• Wang. Novel routing schemes for IClayout part I: Two-layer channel routing.DAC 1991

Given:initial andfinal permutations

Objective: minimizethe number of bends

• Bereg et al. Drawing Permutations with Few Corners.GD 2013

Algorithm for findingoptimal tangles

Complexity ??

Overview

• Complexity:NP-hardness byreduction from3-Partition.

• New algorithm: using dynamic programming;asymptotically faster than [Olszewski et al., GD’18].

O(ϕ2|L|

5|L|/nn)

O(( 2|L|

n2 + 1) n2

2 ϕnn)

• Experiments: comparison with [Olszewski et al., GD’18]

Complexity

Tangle-Height Minimization is NP-hard.Theorem.

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof.

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof.3-Partition

Given: Multiset A of 3m positive integers.

a1 a2 a3 a3m−2 a3m· · · a3m−1

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof.3-Partition

∑1 = B · · · ∑

m = B

Given: Multiset A of 3m positive integers.Can A be partitioned into m groups ofthree elements s.t. each group sums up tothe same value B?

Question:

a1 a2 a3 a3m−2 a3m· · ·∑2 = B

a3m−1

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof.3-Partition

B4 < ai <

B2

B is poly in m

∑1 = B · · · ∑

m = B

Given: Multiset A of 3m positive integers.Can A be partitioned into m groups ofthree elements s.t. each group sums up tothe same value B?

Question:

a1 a2 a3 a3m−2 a3m· · ·∑2 = B

a3m−1

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof. B4 < ai <

B2

B is poly in m

∑1 = B · · · ∑

m = B

Given: Multiset A of 3m positive integers.Can A be partitioned into m groups ofthree elements s.t. each group sums up tothe same value B?

Given: Instance A of 3-Partition.

Question:

a1 a2 a3 a3m−2 a3m· · ·∑2 = B

a3m−1

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof. B4 < ai <

B2

B is poly in m

∑1 = B · · · ∑

m = B

Given: Multiset A of 3m positive integers.Can A be partitioned into m groups ofthree elements s.t. each group sums up tothe same value B?

Given: Instance A of 3-Partition.Task: Construct L s.t. there is T realizing L with height at

most H = 2m3(∑

A)+7m2 iff A is a yes-instance.

Question:

a1 a2 a3 a3m−2 a3m· · ·∑2 = B

a3m−1

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof.

Given: Instance A of 3-Partition.Task: Construct L s.t. there is T realizing L with height at

most H = 2m3(∑

A)+7m2 iff A is a yes-instance.

∑A

∑1 = B · · · ∑

m = B

a1 a2 a3 a3m−2 a3m−1 a3m· · ·∑2 = B

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof.

Given: Instance A of 3-Partition.Task: Construct L s.t. there is T realizing L with height at

most H = 2m3(∑

A)+7m2 iff A is a yes-instance.

∑A

∑1 = B · · · ∑

m = B

a1 a2 a3 a3m−2 a3m−1 a3m· · ·∑2 = B +1

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof.

Given: Instance A of 3-Partition.Task: Construct L s.t. there is T realizing L with height at

most H = 2m3(∑

A)+7m2 iff A is a yes-instance.

∑A +1

∑1 = B · · · ∑

m = B

a1 a2 a3 a3m−2 a3m−1 a3m· · ·∑2 = B +1

Complexity

Reduction from 3-Partition

Tangle-Height Minimization is NP-hard.Theorem.

Proof.

Given: Instance A of 3-Partition.

∑A +1

Task: construct L s.t. there is T realizing L with height atmost H = 2m3(

∑A )+7m2 iff A is a yes-instance+1

∑1 = B · · · ∑

m = B

a1 a2 a3 a3m−2 a3m−1 a3m· · ·∑2 = B +1

Transforming the Instance A into a List Lω ω′

ω ω′

Transforming the Instance A into a List Lω ω′

ω ω′

2m swaps

Transforming the Instance A into a List L

α1

ω ω′

ω ω′

α1 α′1

α′1

Transforming the Instance A into a List L

α1

ω ω′

ω ω′

α1 α′1

α′1

Ma1M = 2m3

Transforming the Instance A into a List L

α1

ω ω′

ω ω′

α1 α′1

α′1

Ma1M = 2m3

Transforming the Instance A into a List L

α1

ω ω′

ω ω′

α1 α′1

α′1

Ma1M = 2m3

Transforming the Instance A into a List L

α1

ω ω′

ω ω′

α1 α′1

α′1

Ma1M = 2m3

Transforming the Instance A into a List L

α1

ω ω′

ω ω′

α1 α′1

α′1

Ma1M = 2m3

What is not possible?

split

Transforming the Instance A into a List L

α1

ω ω′

ω ω′

α1 α′1

α′1

α2 α′2

α2 α′2

Ma1M = 2m3

Ma2

Transforming the Instance A into a List L

α1

ω ω′

ω ω′

α1 α′1

α′1

α2 α′2

α2 α′2

Ma1M = 2m3

Ma2

What is not possible?

put it on the same levelwith other α-α′ swaps

Transforming the Instance A into a List L

α1

ω ω′

ω ω′

α6 α1· · · α′1 α′6

α6· · · α′1 α′6· · ·

· · ·

Ma1

Ma4

Ma5

Ma2

Ma3

Ma6

M = 2m3

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′δ2β1δ1 β2

β2δ2 β1δ1

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′δ2β1δ1 β2

β2δ2 β1δ1

β2δ2 β1δ1

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′δ2β1δ1 β2

β2δ2 β1δ1

β2δ2 β1δ1

β2 δ2β1 δ1

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′δ2β1δ1 β2

β2δ2 β1δ1

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′δ2β1δ1 β2

β2δ2 β1δ1

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′δ2β1δ1 β2

β2δ2 β1δ1

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′δ2β1δ1 β2

β2δ2 β1δ1

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′δ2β1δ1 β2

β2δ2 β1δ1

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′δ2β1δ1 β2

β2δ2 β1δ1

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

2MB

MB

M = 2m3

Making Sure That the “Pockets” Can’t Be Squeezedω ω′

ω ω′

γ′1 γ′2δ′1 β′2δ′2β′1

β′1 γ′1γ′2δ′1β′2δ

′2

2MB

MB

γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

2MB

MB

M = 2m3

Making Sure That the “Pockets” Can’t Be Squeezed

α1

ω ω′

ω ω′

γ′1 γ′2δ′1 β′2δ′2β′1

β′1 γ′1γ′2δ′1β′2δ

′2

α6 α1· · · α′1 α′6

α6· · · α′1 α′6· · ·

2MB

MB

· · ·

γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

Ma1

Ma4

Ma5

Ma2

Ma3

Ma6

2MB

MB

M = 2m3

Proof of Correctness

α1

ω ω′

ω ω′

γ′1 γ′2δ′1 β′2δ′2β′1

β′1 γ′1γ′2δ′1β′2δ

′2

α6 α1· · · α′1 α′6

α6· · · α′1 α′6· · ·

2MB

MB

· · ·

γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

Ma1

Ma4

Ma5

Ma2

Ma3

Ma6

2MB

MB

M = 2m3

A is a yes-instance

H = 2m3(∑

A) + 7m2

is the maximum allowedheight for the reduction

Proof of Correctness

α1

ω ω′

ω ω′

γ′1 γ′2δ′1 β′2δ′2β′1

β′1 γ′1γ′2δ′1β′2δ

′2

α6 α1· · · α′1 α′6

α6· · · α′1 α′6· · ·

2MB

MB

· · ·

γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

Ma1

Ma4

Ma5

Ma2

Ma3

Ma6

2MB

MB

M = 2m3

A is a yes-instance

by construction

H = 2m3(∑

A) + 7m2

is the maximum allowedheight for the reduction

Proof of Correctness

α1

ω ω′

ω ω′

γ′1 γ′2δ′1 β′2δ′2β′1

β′1 γ′1γ′2δ′1β′2δ

′2

α6 α1· · · α′1 α′6

α6· · · α′1 α′6· · ·

2MB

MB

· · ·

γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

Ma1

Ma4

Ma5

Ma2

Ma3

Ma6

2MB

MB

M = 2m3

A is a yes-instance

by construction

height ≤ H

H = 2m3(∑

A) + 7m2

is the maximum allowedheight for the reduction

Proof of Correctness

α1

ω ω′

ω ω′

γ′1 γ′2δ′1 β′2δ′2β′1

β′1 γ′1γ′2δ′1β′2δ

′2

α6 α1· · · α′1 α′6

α6· · · α′1 α′6· · ·

2MB

MB

· · ·

γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

Ma1

Ma4

Ma5

Ma2

Ma3

Ma6

2MB

MB

M = 2m3

A is a yes-instanceno

H = 2m3(∑

A) + 7m2

is the maximum allowedheight for the reduction

Proof of Correctness

α1

ω ω′

ω ω′

γ′1 γ′2δ′1 β′2δ′2β′1

β′1 γ′1γ′2δ′1β′2δ

′2

α6 α1· · · α′1 α′6

α6· · · α′1 α′6· · ·

2MB

MB

· · ·

γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

Ma1

Ma4

Ma5

Ma2

Ma3

Ma6

2MB

MB

M = 2m3

A is a yes-instanceno

H = 2m3(∑

A) + 7m2

is the maximum allowedheight for the reduction

minimum height2m3(

∑A + 1)

Proof of Correctness

α1

ω ω′

ω ω′

γ′1 γ′2δ′1 β′2δ′2β′1

β′1 γ′1γ′2δ′1β′2δ

′2

α6 α1· · · α′1 α′6

α6· · · α′1 α′6· · ·

2MB

MB

· · ·

γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

Ma1

Ma4

Ma5

Ma2

Ma3

Ma6

2MB

MB

M = 2m3

A is a yes-instanceno

height > H

H = 2m3(∑

A) + 7m2

is the maximum allowedheight for the reduction

minimum height2m3(

∑A + 1)

Proof of Correctness

α1

ω ω′

ω ω′

γ′1 γ′2δ′1 β′2δ′2β′1

β′1 γ′1γ′2δ′1β′2δ

′2

α6 α1· · · α′1 α′6

α6· · · α′1 α′6· · ·

2MB

MB

· · ·

γ2γ1 δ2β1δ1 β2

β2γ2γ1 δ2 β1δ1

Ma1

Ma4

Ma5

Ma2

Ma3

Ma6

2MB

MB

M = 2m3

A is a yes-instanceno

height > H

H = 2m3(∑

A) + 7m2

is the maximum allowedheight for the reduction

minimum height2m3(

∑A + 1)

Tangle-Height Minimization is NP-hard.Theorem.

X

Overview

• Complexity:NP-hardness byreduction from3-Partition.

• New algorithm: using dynamic programming;asymptotically faster than [Olszewski et al., GD’18].

O(ϕ2|L|

5|L|/nn)

O(( 2|L|

n2 + 1) n2

2 ϕnn)

• Experiments: comparison with [Olszewski et al., GD’18]

Improving Exact Algorithms

Simple lists

Tangle-Height Minimization can be solved in . . .

General lists

Improving Exact Algorithms

Simple lists

Tangle-Height Minimization can be solved in . . .

General lists

[Olszewski et al., GD’18]

2O(n2)

n – number of wires

Improving Exact Algorithms

Simple lists

Tangle-Height Minimization can be solved in . . .

General lists

[Olszewski et al., GD’18]

2O(n2) 2O(n log n)

our runtime

n – number of wires

Improving Exact Algorithms

Simple lists

Tangle-Height Minimization can be solved in . . .

General lists

[Olszewski et al., GD’18]

2O(n2) 2O(n log n)

our runtime

[Olszewski et al., GD’18]

O(ϕ2|L|

5|L|/nn)

n – number of wires– length of the list L (=

∑`i j )

– golden ratio (≈ 1.618)|L|ϕ

Improving Exact Algorithms

Simple lists

Tangle-Height Minimization can be solved in . . .

General lists

[Olszewski et al., GD’18]

2O(n2) 2O(n log n)

our runtime

[Olszewski et al., GD’18]

O(ϕ2|L|

5|L|/nn)

O(( 2|L|

n2 + 1) n2

2 ϕnn)our runtime

n – number of wires– length of the list L (=

∑`i j )

– golden ratio (≈ 1.618)|L|ϕ

Improving Exact Algorithms

Simple lists

Tangle-Height Minimization can be solved in . . .

General lists

[Olszewski et al., GD’18]

2O(n2) 2O(n log n)

our runtime

[Olszewski et al., GD’18]

O(ϕ2|L|

5|L|/nn)

O(( 2|L|

n2 + 1) n2

2 ϕnn)our runtime

polynomial in |L|

n – number of wires– length of the list L (=

∑`i j )

– golden ratio (≈ 1.618)|L|ϕ

polynomial in |L|for fixed n

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency.

L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency.i

L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency.i

for each wire i :// find a position where it is after applying L′

L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency.i

for each wire i :// find a position where it is after applying L′

i 7→ i + |{j : j > i and `′i j is odd}| − |{j : j < i and `′i j is odd}|

L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency.i

for each wire i :// find a position where it is after applying L′

i 7→ i + |{j : j > i and `′i j is odd}| − |{j : j < i and `′i j is odd}|

L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency.i

for each wire i :// find a position where it is after applying L′

i 7→ i + |{j : j > i and `′i j is odd}| − |{j : j < i and `′i j is odd}|

L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency.i

for each wire i :// find a position where it is after applying L′

i 7→ i + |{j : j > i and `′i j is odd}| − |{j : j < i and `′i j is odd}|

L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency.i

for each wire i :// find a position where it is after applying L′

i 7→ i + |{j : j > i and `′i j is odd}| − |{j : j < i and `′i j is odd}|check whether the map is indeed a permutation

L′ is a sublist of L if`′i j ≤ `i j

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency.

idn L′

L′ is a sublist of L if`′i j ≤ `i j

Compute the final permutation idn L′.

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency. π1π2...

πhidn L′

Choose the shortest tangle T (L′′).

L′ is a sublist of L if`′i j ≤ `i j

Compute the final permutation idn L′.

πh and idn L′ are adjacentL′′∪ add. swaps = L′

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency. π1π2...

πhidn L′

Choose the shortest tangle T (L′′).

L′ is a sublist of L if`′i j ≤ `i j

Compute the final permutation idn L′.

πh and idn L′ are adjacentL′′∪ add. swaps = L′

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency. π1π2...

πhAdd the final permutation to its end.idn L′

Choose the shortest tangle T (L′′).

L′ is a sublist of L if`′i j ≤ `i j

Compute the final permutation idn L′.

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency. π1π2...

πhAdd the final permutation to its end.idn L′

Running time

O(λ · (Fn+1 − 1) · n) ≤

Choose the shortest tangle T (L′′).

L′ is a sublist of L if`′i j ≤ `i j

Compute the final permutation idn L′.

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency. π1π2...

πhAdd the final permutation to its end.idn L′

Running time

O(λ · (Fn+1 − 1) · n) ≤

Choose the shortest tangle T (L′′).

L′ is a sublist of L if`′i j ≤ `i j

Compute the final permutation idn L′.

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency. π1π2...

πhAdd the final permutation to its end.idn L′

Running time

O(λ · (Fn+1 − 1) · n) ≤

Choose the shortest tangle T (L′′).

Fn is the n-th Fibonacci number

L′ is a sublist of L if`′i j ≤ `i j

Compute the final permutation idn L′.

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency. π1π2...

πhAdd the final permutation to its end.idn L′

Running time

O(λ · (Fn+1 − 1) · n) ≤

Choose the shortest tangle T (L′′).

L′ is a sublist of L if`′i j ≤ `i j

Compute the final permutation idn L′.

λ =∏i<j

(`i j + 1) ≤(

2|L|n2

+ 1)n2/2

Fn ∈ O(ϕn)

Dynamic Programming Algorithm

O(( 2|L|

n2 + 1)n2/2

ϕnn)

Let L = (`i j) be the given list of swaps.

λ = # of distinct sublists of L.

Consider them in order of increasing length.

Let L′ be the next list to consider.

Check its consistency. π1π2...

πhAdd the final permutation to its end.idn L′

Running time

O(λ · (Fn+1 − 1) · n) ≤

Choose the shortest tangle T (L′′).

L′ is a sublist of L if`′i j ≤ `i j

Compute the final permutation idn L′.

λ =∏i<j

(`i j + 1) ≤(

2|L|n2

+ 1)n2/2

Fn ∈ O(ϕn)

Overview

• Complexity:NP-hardness byreduction from3-Partition.

• New algorithm: using dynamic programming;asymptotically faster than [Olszewski et al., GD’18].

O(ϕ2|L|

5|L|/nn)

O(( 2|L|

n2 + 1) n2

2 ϕnn)

• Experiments: comparison with [Olszewski et al., GD’18]

[Olszewski et al., GD’18]

O(ϕ2|L|

5|L|/nn)

O(( 2|L|

n2 + 1) n2

2 ϕnn)Our algorithm

run

tim

e[s

ec]

length |L| of the list L

0 10 20 30 40 50

0.0001

0.01

1

100

3600

Experiments

0 10 20 30 40 0 10 20 30 40

Open Problems

Problem 1

Is it NP-hard to test the feasibility of a given (non-simple) list?

Open Problems

Problem 1

Is it NP-hard to test the feasibility of a given (non-simple) list?

Problem 2

Can we decide a feasibility of a list faster than finding itsoptimal realization?

Open Problems

Problem 1

Is it NP-hard to test the feasibility of a given (non-simple) list?

Problem 2

Can we decide a feasibility of a list faster than finding itsoptimal realization?

Problem 3A list (`i j) is non-separableif ∀i<k<j :

(`ik = `kj = 0 implies `i j = 0

).

i k j

Open Problems

Problem 1

Is it NP-hard to test the feasibility of a given (non-simple) list?

Problem 2

Can we decide a feasibility of a list faster than finding itsoptimal realization?

Problem 3A list (`i j) is non-separableif ∀i<k<j :

(`ik = `kj = 0 implies `i j = 0

).

i k j

necessary

Open Problems

Problem 1

Is it NP-hard to test the feasibility of a given (non-simple) list?

Problem 2

Can we decide a feasibility of a list faster than finding itsoptimal realization?

Problem 3

For lists where all entries are even, is this sufficient?

A list (`i j) is non-separableif ∀i<k<j :

(`ik = `kj = 0 implies `i j = 0

).

i k j

necessary

Open Problems

Problem 1

Is it NP-hard to test the feasibility of a given (non-simple) list?

Problem 2

Can we decide a feasibility of a list faster than finding itsoptimal realization?

Problem 3

For lists where all entries are even, is this sufficient?

A list (`i j) is non-separableif ∀i<k<j :

(`ik = `kj = 0 implies `i j = 0

).

i k j

necessary

Thank you!