When Lipschitz Walks Your Dog: Algorithm Engineering of the - - PowerPoint PPT Presentation

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Jan 03, 2024 275 likes •1.15k views

When Lipschitz Walks Your Dog: Algorithm Engineering of the Discrete Fr echet Distance under Translation Karl Bringmann, Marvin K unnemann, and Andr e Nusser Woof! Woof! Lipschitz Wau! Wau! translated dog Teaser Karl Bringmann,

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SLIDE 1

When Lipschitz Walks Your Dog: Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser

Lipschitz

Wau! Wau! Woof! Woof!

translated dog

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SLIDE 2

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Teaser

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SLIDE 3

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Teaser

Fr´ echet Distance Under Translation:

human dog

Trajectory Similarity: Fr´ echet Distance:

Pigeon GPS Trajectories Handwritten Characters

  • traversal based
  • fast in practice
  • traversal based
  • only impractical

algorithms (before)

  • translation invariant
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SLIDE 4

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Teaser

Best algorithm: O(n4.66) Conditional lower bound: n4−o(1) All algorithms build O(n4) arrangement! Lipschitz Meets Fr´ echet: Fr´ echet under translation is 1-Lipschitz in τ! τ1 τ2 Use continuous

  • ptimization:
  • branch & bound!

τ1 τ2

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SLIDE 5

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Teaser

Take-Home Message:

arrangement-based geometric algorithm methods from continuous optimization

fast practical algorithm :)

exact approximation exact expensive

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SLIDE 6

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

End of Teaser

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SLIDE 7

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Why Trajectory Similarity?

Pigeons’ GPS Trajectories: Handwritten Character Trajectories:

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SLIDE 8

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance

Intuition

human dog

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SLIDE 9

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance

Intuition

human dog

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SLIDE 10

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance

Intuition

human dog

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SLIDE 11

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance

Intuition

human dog

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SLIDE 12

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance

Intuition

human dog

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SLIDE 13

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance

Intuition

human dog

What is the traversal that achieves the shortest leash length? Question:

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SLIDE 14

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance

Formal Definition

δF (π, σ) := minf,g∈T maxt∈[0,1]

  • πf(t) − σg(t)
  • π, σ = polygonal curves of length n

T = set of monotone and surjective functions from [0, 1] to {1, . . . , n}

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SLIDE 15

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation

δT (π, σ) := minτ∈R2δF (π, σ+τ)

Definition

Intuition: Allow arbitrary translations τ ∈ R2 of curve σ.

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SLIDE 16

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation

δT (π, σ) := minτ∈R2δF (π, σ+τ)

Definition

Intuition: Allow arbitrary translations τ ∈ R2 of curve σ.

σ π

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SLIDE 17

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation

δT (π, σ) := minτ∈R2δF (π, σ+τ)

Definition

Intuition: Allow arbitrary translations τ ∈ R2 of curve σ.

σ π σ + τ

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SLIDE 18

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation

δT (π, σ) := minτ∈R2δF (π, σ+τ)

Definition

Intuition: Allow arbitrary translations τ ∈ R2 of curve σ.

σ π σ + τ Decision Problem:

  • Given π, σ, δ
  • δT (π, σ) ≤ δ?
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SLIDE 19

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation

δT (π, σ) := minτ∈R2δF (π, σ+τ)

Definition

Intuition: Allow arbitrary translations τ ∈ R2 of curve σ.

σ π σ + τ Decision Problem:

  • Given π, σ, δ
  • δT (π, σ) ≤ δ?

Focus on this in the talk!

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SLIDE 20

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Goal:

Performant implementation computing the discrete Fr´ echet distance under translation

  • n practical inputs.
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SLIDE 21

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Related Work

Theory:

  • Discrete Fr´

echet distance under translation in ˜ O(n5) [Agarwal, Ben Avraham, Kaplan, Sharir arXiv’15]

  • Discrete Fr´

echet distance under translation in ˜ O(n4.66) [Bringmann, K¨ unnemann, N. SODA’19]

  • SETH based lower bound of n4−o(1) for discrete Fr´

echet distance under translation [Bringmann, K¨ unnemann, N. SODA’19]

curve length

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SLIDE 22

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Related Work

Theory: Practice:

  • GIS Cup on (fixed-translation) Fr´

echet distance near neighbors search [Werner, Oliver; Baldus et al.; Buchin et al.; D¨ utsch et al. SIGSPATIAL’17]

  • State of the art (fixed-translation) Fr´

echet distance implementation [Bringmann, K¨ unnemann, N. SoCG’19]

  • Discrete Fr´

echet distance under translation in ˜ O(n5) [Agarwal, Ben Avraham, Kaplan, Sharir arXiv’15]

  • Discrete Fr´

echet distance under translation in ˜ O(n4.66) [Bringmann, K¨ unnemann, N. SODA’19]

  • SETH based lower bound of n4−o(1) for discrete Fr´

echet distance under translation [Bringmann, K¨ unnemann, N. SODA’19]

curve length

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SLIDE 23

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2

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SLIDE 24

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2

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SLIDE 25

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2

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SLIDE 26

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2

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SLIDE 27

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2

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SLIDE 28

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2

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SLIDE 29

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2

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SLIDE 30

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2

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SLIDE 31

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2 Observation: All translations in a cell of the arrangement have the same closeness relation.

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SLIDE 32

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2 Observation: All translations in a cell of the arrangement have the same closeness relation. for each cell, pick some τ and check dF (π, σ + τ)

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SLIDE 33

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

Arrangement

  • Idea: Partition the plane into equivalent regions.

π σ δ τ1 τ2 Observation: All translations in a cell of the arrangement have the same closeness relation. O(n4) complexity for each cell, pick some τ and check dF (π, σ + τ)

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SLIDE 34

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

  • All known algorithms build an O(n4) arrangement.
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SLIDE 35

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

  • All known algorithms build an O(n4) arrangement.
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SLIDE 36

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

  • All known algorithms build an O(n4) arrangement.
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SLIDE 37

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach I: Discrete Algorithms

  • All known algorithms build an O(n4) arrangement.

Ewwww.....

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SLIDE 38

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

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SLIDE 39

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

τ1 τ2

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SLIDE 40

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization |dF(π, σ + τ) − dF(π, σ + τ ′)| ≤ τ − τ ′

Observation: Fr´ echet under Translation is 1-Lipschitz, i.e.,

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SLIDE 41

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

Lipschitz Optimization Approach:

  • branch & bound

For each box:

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SLIDE 42

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

Lipschitz Optimization Approach:

  • branch & bound

For each box:

  • if dF (π, σ+

) ≤ δ – return LESS

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SLIDE 43

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

Lipschitz Optimization Approach:

  • branch & bound

For each box:

  • if dF (π, σ+

) ≤ δ – return LESS

  • if dF (π, σ+

) > δ+ – skip box

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SLIDE 44

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

Lipschitz Optimization Approach:

  • branch & bound

For each box:

  • if dF (π, σ+

) ≤ δ – return LESS

  • if dF (π, σ+

) > δ+ – skip box

  • if both fail: split
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SLIDE 45

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

Issues

  • In general, only approximate decisions possible.
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SLIDE 46

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

Issues

  • Locally highly non-convex:
  • In general, only approximate decisions possible.
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SLIDE 47

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

Issues

  • Locally highly non-convex:
  • In general, only approximate decisions possible.
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SLIDE 48

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Approach II: Continuous Optimization

Issues

  • Locally highly non-convex:
  • In general, only approximate decisions possible.
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SLIDE 49

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Core Idea

Combine Both Approaches!

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SLIDE 50

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Core Idea

Combine Both Approaches!

1) Use Lipschitz optimization to identify important regions

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SLIDE 51

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Core Idea

Combine Both Approaches!

1) Use Lipschitz optimization to identify important regions 2) Use arrangement algorithm inside these regions

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SLIDE 52

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

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SLIDE 53

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Approach:

  • augment branch & bound approach
  • for each box:

– estimate arrangement size

  • if it is small: build arrangement

exact decision!

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SLIDE 54

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Approach:

  • augment branch & bound approach
  • for each box:

– estimate arrangement size

  • if it is small: build arrangement

exact decision!

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SLIDE 55

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Approach:

  • augment branch & bound approach
  • for each box:

– estimate arrangement size

  • if it is small: build arrangement

exact decision!

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SLIDE 56

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Approach:

  • augment branch & bound approach
  • for each box:

– estimate arrangement size

  • if it is small: build arrangement

exact decision!

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SLIDE 57

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Approach:

  • augment branch & bound approach
  • for each box:

– estimate arrangement size

  • if it is small: build arrangement

exact decision!

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SLIDE 58

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Approach:

  • augment branch & bound approach
  • for each box:

– estimate arrangement size

  • if it is small: build arrangement

exact decision!

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SLIDE 59

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Approach:

  • augment branch & bound approach
  • for each box:

– estimate arrangement size

  • if it is small: build arrangement

exact decision! Issue: When to build the arrangement? Main Ingredients:

  • 1. Arrangement size estimation
  • 2. Threshold parameter
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SLIDE 60

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Approach:

  • augment branch & bound approach
  • for each box:

– estimate arrangement size

  • if it is small: build arrangement

exact decision! Issue: When to build the arrangement? Main Ingredients:

  • 1. Arrangement size estimation
  • 2. Threshold parameter

modified kd-tree

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SLIDE 61

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Approach:

  • augment branch & bound approach
  • for each box:

– estimate arrangement size

  • if it is small: build arrangement

exact decision! Issue: When to build the arrangement? Main Ingredients:

  • 1. Arrangement size estimation
  • 2. Threshold parameter

empirically choose modified kd-tree

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SLIDE 62

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution I: Exact Decider

Implementation Details

  • Adaption of (fixed-translation) [SoCG’19] implementation to discrete case
  • Lazy translation
  • Parameter choice for arrangement size estimation
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SLIDE 63

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

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SLIDE 64

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Approaches Epsilon-approximate Set:

O(ǫ) O(ǫ)

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SLIDE 65

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Approaches Epsilon-approximate Set: Binary Search via Decision Problem:

O(ǫ) O(ǫ)

  • Binary search over δ using decider
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SLIDE 66

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Approaches Epsilon-approximate Set: Binary Search via Decision Problem: Lipschitz-only Optimization:

O(ǫ) O(ǫ)

  • Binary search over δ using decider
  • Use plain Lipschitz optimization
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SLIDE 67

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Approaches Epsilon-approximate Set: Binary Search via Decision Problem: Lipschitz-only Optimization:

O(ǫ) O(ǫ)

  • Binary search over δ using decider
  • Use plain Lipschitz optimization

Lipschitz-meets-Fr´ echet: next slide

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SLIDE 68

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Lipschitz-meets-Fr´ echet For each box: Approach:

  • 1. Maintain local lower bound
  • 2. Maintain global upper bound
  • 3. Arrangement size estimation

adapt!

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SLIDE 69

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Lipschitz-meets-Fr´ echet For each box:

  • update global upper bound:

– min{ub, dF (π, σ+ )} Approach:

  • 1. Maintain local lower bound
  • 2. Maintain global upper bound
  • 3. Arrangement size estimation

adapt!

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SLIDE 70

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Lipschitz-meets-Fr´ echet For each box:

  • update global upper bound:

– min{ub, dF (π, σ+ )}

  • update local lower bound:

– max{lb, dF (π, σ+ )− } Approach:

  • 1. Maintain local lower bound
  • 2. Maintain global upper bound
  • 3. Arrangement size estimation

adapt!

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SLIDE 71

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Lipschitz-meets-Fr´ echet For each box:

  • update global upper bound:

– min{ub, dF (π, σ+ )}

  • update local lower bound:

– max{lb, dF (π, σ+ )− }

  • if ub > lb + ǫ: split

Approach:

  • 1. Maintain local lower bound
  • 2. Maintain global upper bound
  • 3. Arrangement size estimation

adapt!

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SLIDE 72

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Lipschitz-meets-Fr´ echet For each box:

  • update global upper bound:

– min{ub, dF (π, σ+ )}

  • update local lower bound:

– max{lb, dF (π, σ+ )− }

  • if ub > lb + ǫ: split

Approach:

  • 1. Maintain local lower bound
  • 2. Maintain global upper bound
  • 3. Arrangement size estimation

adapt!

b i n a r y s e a r c h

  • v

e r a r r a n g e m e n t !

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SLIDE 73

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Contribution II: From Decider to Value Computation

Lipschitz-meets-Fr´ echet: Implementation Details

  • Initial estimates
  • Arrangement size estimation
  • Priority queue on lower bound → no regret strategy!
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SLIDE 74

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Data Sets

Data set Type #Curves Mean #vertices Sigspatial synthetic GPS-like 20199 247.8 Characters 20 handwritten chars 2858 120.9 (142.9 per character)

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SLIDE 75

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Running Times

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SLIDE 76

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Running Times

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SLIDE 77

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Running Times

  • Hard instances are distances

slightly less than actual distance

  • Running times of hard instances

in the order of 100ms

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SLIDE 78

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Black box calls vs. arrangement size

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SLIDE 79

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Black box calls vs. arrangement size

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SLIDE 80

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Black box calls vs. arrangement size

  • several orders of

magnitudes less calls to black-box decider

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SLIDE 81

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Approach Time Black-Box Calls LMF 148,032 ms 13,323,232 (141.0 ms per instance) (12,688.8 per instance) Binary Search 536,853 ms 45,909,628 (511.3 ms per instance) (43,723.5 per instance) Lipschitz-only 4,204,521 ms 820,468,224 (4,004.3 ms per instance) (781,398.3 per instance) Value Computation Times

slide-82
SLIDE 82

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Binary Search vs. LMF

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SLIDE 83

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Experiments

Binary Search vs. LMF

  • LMF better on

hard instances

slide-84
SLIDE 84

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Summary

arrangement-based geometric algorithm methods from continuous optimization

e x a c t a p p r

  • x

i m a t i

  • n

expensive

slide-85
SLIDE 85

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Summary

arrangement-based geometric algorithm methods from continuous optimization

fast practical algorithm :)

e x a c t a p p r

  • x

i m a t i

  • n

exact expensive

slide-86
SLIDE 86

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Summary

arrangement-based geometric algorithm methods from continuous optimization

fast practical algorithm :)

e x a c t a p p r

  • x

i m a t i

  • n

exact expensive

Future Directions:

  • Apply approach to other

problems

  • Find optimal point of

building the arrangement

slide-87
SLIDE 87

Karl Bringmann, Marvin K¨ unnemann, and Andr´ e Nusser Algorithm Engineering of the Discrete Fr´ echet Distance under Translation

Summary

arrangement-based geometric algorithm methods from continuous optimization

fast practical algorithm :)

e x a c t a p p r

  • x

i m a t i

  • n

exact expensive

Future Directions:

  • Apply approach to other

problems

  • Find optimal point of

building the arrangement

Code: https://gitlab.com/anusser/frechet distance under translation

Thanks!