A center curve under the Fr´echet distance

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A center curve under the Fr´echet

distance Jan Hitzschke

Introduction Preparations The Problem as an Inequality System A Center Curve Algorithm on the Line

A better Way to guess Visiting Orders End

A center curve under the Fr´ echet distance

Jan Hitzschke

30.03.2020

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Introduction

Preparations

The Problem as an Inequality System

A Center Curve Algorithm on the Line

A better Way to guess Visiting Orders

End

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Introduction

Preparations The Problem as an Inequality System A Center Curve Algorithm on the Line

The center curve problem

Definition

Acurve is a continuous mapp : [0,1]→R^m. LetCm be the set of all curves in R^m.

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Introduction

The center curve problem

Definition

I Assume all curves to be piece-wise linear.

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Introduction

The center curve problem

Definition

I Assume all curves to be piece-wise linear.

Problem

Given curves p¹, ...,pⁿ∈Cm,r∈Rand a distance measure d :Cm² →R, find a curveq ∈Cm such thatd(pⁱ,q)≤r or decide that none exists.

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Introduction

Example Center Curve

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Introduction

Example Center Curve

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Introduction

The Fr´ echet distance

Definition

Call a map f : [0,1]→[0,1] a paceif it is a continuous and increasing bijection. Let P be the set of all paces.

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Introduction

The Fr´ echet distance

Definition

Define theFr´echet distanceas the mapd_F :C_m² →Rwith d_F(p,q) := inf

f∈P max

x∈[0,1] kp(x)−q(f(x))k

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Introduction

The Fr´ echet distance

Definition

Define theFr´echet distanceas the mapd_F :Cm² →Rwith d_F(p,q) := inf

f∈P max

x∈[0,1] kp(x)−q(f(x))k Lemma

The Fr´echet distance is symmetric and satisfies the triangle inequality. It becomes a metric by setting

p ∼q ⇐⇒ d_F(p,q) = 0.

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Introduction

Open questions

I The Fr´echet center curve problem is NP-hard. Is it in NP?

I How can the set of solutions to the problem be represented?

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Example: Pace

d_F(p,q) := inf

f∈P max

x∈[0,1] kp(x)−q(f(x))k

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Example: Pace

d_F(p,q) := inf

f∈P max

x∈[0,1] kp(x)−q(f(x))k

(0,0)

(1,1)

1 x 1

y

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Example: Pace

d_F(p,q) := inf

f∈P max

x∈[0,1] kp(x)−q(f(x))k

q

(0,0) p

(1,1)

1 x 1

y

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Example: Pace

d_F(p,q) := inf

f∈P max

x∈[0,1] kp(x)−q(f(x))k

q

(0,0) p

(1,1)

1 x 1

y

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Redefining d

F

with Monotone Paths

Definition

Call the image T of a curve pT : [0,1]→R² amonotone pathif

1. p_T(0) = (0,0) andp_T(1) = (1,1)

2. x ≤y =⇒ pT(x)1 ≤pT(y)1 andpT(x)2 ≤pT(y)2. Denote the set of all monotone paths by T.

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Redefining d

F

with Monotone Paths

Definition

Call the image T of a curve pT : [0,1]→R² amonotone pathif

1. pT(0) = (0,0) andpT(1) = (1,1)

2. x ≤y =⇒ pT(x)1 ≤pT(y)1 andpT(x)2 ≤pT(y)2. Denote the set of all monotone paths by T.

Lemma

For any curves p,q ∈Cm their Fr´echet distance can be expressed as

d_F(p,q) = min

T∈T max

t∈T kp(t₁)−q(t₂)k

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Example: Monotone Path

d_F(p,q) = min

T∈T max

q

(0,0) p 1 x

1 y

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Example: Monotone Path

d_F(p,q) = min

T∈T max

q

(0,0) p 1 x

1 y

(1,1)

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Vertex Associations

Definition

Let p,q be piece-wise linear curves. A set A⊂T ∈T is called a vertex association, if Acontains a those points corresponding to vertices of either curve. Denote the set of all vertex associations byA.

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Vertex Associations

Definition

Let p,q be piece-wise linear curves. A set A⊂T ∈T is called a vertex association, if Acontains a those points corresponding to vertices of either curve. Denote the set of all vertex associations byA.

Lemma

Let p,q be two piece-wise linear curves. Then dF = min

A∈A max

(x,y)∈Akp(x)−q(y)k

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Example: Vertex Association

d_F = min

A∈A max

q

(0,0) p

(1,1)

1 x 1

y

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Example: Vertex Association

d_F = min

A∈A max

q

(0,0) p

(1,1)

1 x 1

y

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Example: Vertex Association

d_F = min

A∈A max

q

(0,0) p

(1,1)

1 x 1

y

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Example: Vertex Association

d_F = min

A∈A max

q

(0,0) p

(1,1)

1 x 1

y

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Example: Vertex Association

d_F = min

A∈A max

q

(0,0) p

(1,1)

1 x 1

y

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Visiting Orders

Lemma

Given piece-wise linear curvesp¹, ...,pⁿ each of lengthl and a center curveq of radius r, there is a curveq⁰ with

I q⁰ is piece-wise linear withl⁰≤nl vertices.

I d(pⁱ,q⁰)≤r for all i = 1, ...,n

I |Aⁱ|=l⁰, i.e. each vertex of pⁱ associates to a vertex of q⁰.

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Visiting Orders

Lemma

I d(pⁱ,q⁰)≤r for all i = 1, ...,n.

This gives an enumeration of the input vertices by the order of their associated vertices in q⁰.

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Visiting Orders

Lemma

I d(pⁱ,q⁰)≤r for all i = 1, ...,n

This gives an enumeration of the input vertices by the order of their associated vertices in q⁰.

Definition

Given n piece-wise linear curves with l vertices each, let N =n(l−2). Then avisiting orderΓ is a sequence Γ⁰,Γ¹, ...,Γ^N ∈ {1, ...,l}ⁿ where

I Γ⁰= (1, ...,1) I Γ^k = Γ^k⁻¹+e_j_k.

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Example: Visiting Order

p¹ p² p³

1 1 1

2 2

3 3

2 3 4

4 4

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Example: Visiting Order

p¹ p² p³

2 2

3 3

2 3 4

4 4 1

1 1

Γ⁰



 1 1 1





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Example: Visiting Order

p¹ p² p³

2

3 3

2 3 4

4 4 1

1 1

2

Γ⁰ Γ¹



 1 1 1







 2 1 1





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Example: Visiting Order

p¹ p² p³

3 3

2 3 4

4 4 1

1 1

2 2

Γ⁰ Γ¹ Γ²



 1 1 1







 2 1 1







 2 2 1





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Example: Visiting Order

p¹ p² p³

3

2 3 4

4 4 1

1 1

2 2

3

Γ⁰ Γ¹ Γ² Γ³



 1 1 1







 2 1 1







 2 2 1







 3 2 1





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Example: Visiting Order

p¹ p²

p³ 2 3 4

4 4 1

1 1

2 2

3 3

Γ⁰ Γ¹ Γ² Γ³ Γ⁴



 1 1 1







 2 1 1







 2 2 1







 3 2 1







 3 3 1





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Example: Visiting Order

p¹ p²

p³ 3 4

4 4 1

1 1

2 2

3 3 2

Γ⁰ Γ¹ Γ² Γ³ Γ⁴ Γ⁵



 1 1 1







 2 1 1







 2 2 1







 3 2 1







 3 3 1







 3 3 2





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Example: Visiting Order

p¹ p²

p³ 4

4 4 1

1 1

2 2

3 3

2 3

Γ⁰ Γ¹ Γ² Γ³ Γ⁴ Γ⁵ Γ⁶



 1 1 1







 2 1 1







 2 2 1







 3 2 1







 3 3 1







 3 3 2







 3 3 3





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Example: Visiting Order

p¹ p² p³

1 1 1

2 2

3 3

2 3 4

4 4

Γ⁰ Γ¹ Γ² Γ³ Γ⁴ Γ⁵ Γ⁶



 1 1 1







 2 1 1







 2 2 1







 3 2 1







 3 3 1







 3 3 2







 3 3 3





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Preparing the variables

I Assume to have guessed the correct visiting order Γ.

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Preparing the variables

I Denote the vertices of pⁱ as p₁ⁱ,p₂ⁱ, ...,pⁱ_l.

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Preparing the variables

I Denote vertices of the center curveq asq₀,q₁, ...,q_N+1.

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Preparing the variables

I Denote vertices of the center curveq asq₀,q₁, ...,q_N+1. I Each q_k is associated to a point between p_Γⁱk

i

andp_Γⁱk i+1.

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Preparing the variables

i

andp_Γⁱk i+1. I The exact position can be represented by a value

λⁱ_k ∈[0,1].

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Preparing the variables

i

andp_Γⁱk i+1. I The exact position can be represented by a value

λⁱ_k ∈[0,1].

I λⁱ_k = 0 if Γ^k_i −Γ^k−1_i = 1 (say i =j_k).

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The Inequality System

The set of solutions to the center curve problem is represented as

Q_Γ(p,r) =











(q, λ)∈R^N×m+n×N :

kq_k−s_kⁱk ≤ r (1−λⁱ_k)p_Γⁱk

i

−λⁱ_kpⁱ_Γk

i+1 = s_kⁱ

kq₀−pⁱ₁k ≤ r kq_N+1−p_lⁱk ≤ r λ^j_k^k = 0

∀i 6=j_k λⁱ_k−1 ≤ λⁱ_k 0≤λⁱ_k ≤ 1











wherek = 1, ...,N and i = 1, ...,n in each line.

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The Inequality System

Q_Γ(p,r) =











(q, λ)∈R^N×m+n×N :

kq_k−s_kⁱk ≤ r (1−λⁱ_k)p_Γⁱk

i

−λⁱ_kpⁱ_Γk

i+1 = s_kⁱ

kq₀−pⁱ₁k ≤ r kq_N+1−p_lⁱk ≤ r λ^j_k^k = 0

∀i 6=j_k λⁱ_k−1 ≤ λⁱ_k 0≤λⁱ_k ≤ 1











How to solve this?

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The Inequality System

QΓ(p,r) =











(q, λ)∈R^N×m+n×N :

kq_k−s_kⁱk ≤ r (1−λⁱ_k)pⁱ

Γ^k_i −λⁱ_kpⁱ

Γ^k_i+1 = s_kⁱ kq₀−pⁱ₁k ≤ r kq_N+1−p_lⁱk ≤ r λ^j_k^k = 0

∀i 6=jk λⁱ_k−1 ≤ λⁱ_k 0≤λⁱ_k ≤ 1











How to solve this?

Q_Γ(p,r) is convex. =⇒ Use or adapt convex optimization techniques!

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Relaxations

I If theλⁱ_k are known, this reduces to solvingN+ 2 center problems of points.

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Relaxations

I If theqk are known, this reduces to computingn vertex associations for a given visiting order. This can be done efficiently by a greedy algorithm.

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Relaxations

I If theqk are known, this reduces to computingn vertex associations for a given visiting order. This can be done efficiently by a greedy algorithm.

I If the constraints λⁱ_k−1 ≤λⁱ_k are dropped, allq_k are independent from each other.

=⇒ Each is the solution to a center problem of points and lines.

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The Ellipsoid Method

Separation

Each inequality constraining Q(p,r) can be made into a polynomial time separation oracle.

(But not if r was a variable!)

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The Ellipsoid Method

Separation

(But not if r was a variable!) Proof.

Given a vector y the constrainty^Ty ≤r² can be checked efficiently.

Ifkyk>r, for anyx ∈QΓ(p,r), by Cauchy-Schwarz:

y^Tx ≤ kykkxk ≤ kykr <kyk²=y^Ty

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The Ellipsoid Method

Separation

Ifkyk>r, for anyx ∈Q_Γ(p,r), by Cauchy-Schwarz:

y^Tx ≤ kykkxk ≤ kykr<kyk²

We can use the ellipsoid method

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The Ellipsoid Method

Separation

Ifkyk>r, for anyx ∈Q_Γ(p,r), by Cauchy-Schwarz:

y^Tx ≤ kykkxk ≤ kykr <kyk²=y^Ty

We can use the ellipsoid method

to get a point x∈Q_Γ(p,r) or decide thatvol(Q_Γ(p,r))<C for anyC >0.

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The Ellipsoid Method

I LetQ_Γ(p,r) result from QΓ(p,r) by increasing all right hand sides by.

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The Ellipsoid Method

I LetQ_Γ(p,r) result from Q_Γ(p,r) by increasing all right hand sides by.

I IfQ_Γ(p,r)6=∅, then vol(Q_Γ(p,r))≥C(m, ).

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The Ellipsoid Method

I IfQ_Γ(p,r)6=∅, then vol(Q_Γ(p,r))≥C(m, ).

I Ifx ∈Q_Γ(p,r) is found, thenx ∈QΓ(p,r+) with small changes.

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The Ellipsoid Method

I IfQΓ(p,r)6=∅, then vol(Q_Γ(p,r))≥C(m, ).

I Ifx ∈Q_Γ(p,r) is found, thenx ∈QΓ(p,r+) with small changes.

Theorem

For inputs p,r, fixed visiting orderσ and >0 there is a polynomial time algorithm that

I gives a center curve with radius ≤r+or I decides there is no center curve of radiusr.

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Adapting the Simplex Method

Lemma

Each linear closed constraint of a closed convex set P is attained with equality by some point x∈P unless it is redundant.

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Adapting the Simplex Method

Lemma

Proposition

By guessing which are attained as equalities, all linear inequalities constraining QΓ(p,r) can be transformed or removed.

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Adapting the Simplex Method

Lemma

Proposition

By guessing which are attained as equalities, all linear inequalities constraining Q_Γ(p,r) can be transformed or removed.

Question

Can a similar thing be done for nonlinear inequalities?

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Assumptions on nonlinear constraints

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Assumptions on nonlinear constraints

x y

z

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Assumptions on nonlinear constraints

x y

z y’

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Assumptions on nonlinear constraints

x y

z y’

y’

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Assumptions on nonlinear constraints

x y

z y’

y’

x

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Assumptions on nonlinear constraints

x y

z y’

y’

x y’

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Observations on the line

Lemma

A center curveq can be chosen such that for allk = 1, ...N, one of the conditions qk =qk−1 or kq_k−p_Γ^σk^k

σk

k=r is satisfied.

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Observations on the line

Lemma

A center curveq can be chosen such that for allk = 1, ...N, one of the conditions q_k =q_k₋₁ or kq_k−p_Γ^σ_k^k

σk

k=r is satisfied.

Fact

For a point y ∈R, there are exactly two points y1 <y2∈R with |y_i −y|=r for anyr >0.

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Observations on the line

Lemma

A center curveq can be chosen such that for allk = 1, ...N, one of the conditions q_k =q_k−1 or kq_k−p_Γ^σ_k^k

σk

k=r is satisfied.

Fact

For a point y ∈R, there are exactly two points y1 <y2∈R with |y_i −y|=r for anyr >0.

Idea

Build the center curve iteratively by either staying at the current vertex or moving to the closer one of the two candidates.

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A center curve Algorithm on the Line

Theorem

Given curves p¹, ...,pⁿ in Randr ≥0, a curveq can be computed in time O(N) such thatq is a center curve with radiusr if one exists.

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A center curve Algorithm on the Line

Theorem

Given curves p¹, ...,pⁿ in Randr ≥0, a curveq can be computed in time O(N) such thatq is a center curve with radiusr if one exists.

Lemma

Whether d_F(pⁱ,q)≤r can be checked in polynomial time.

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Counting Visiting Orders

Lemma

Given n curves with l vertices each, the number|V|of possible visiting orders is|V|= _(l−2)!^N! n.

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Counting Visiting Orders

Lemma

For this number, there are bounds n

2 N−n

≤ |V| ≤n^N

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Counting Visiting Orders

Lemma

2 N−n

≤ |V| ≤n^N

I An algorithm trying all visiting orders can have exponential, but no polynomial runtime.

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Counting Visiting Orders

Lemma

2 N−n

≤ |V| ≤n^N

I An algorithm trying all visiting orders can have exponential, but no polynomial runtime.

I Is there a way to not test every single visiting order?

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n-wise Fr´ echet distance

Definition

Given curves p¹, ...,pⁿ their n-wise Fr´echet distance is defined as

d_F(p¹, ...,pⁿ) = inf

f1,...,fn∈P max

1≤i,j≤n max

x∈[0,1]kpⁱ(f_i(x))−p^j(f_j(x))k