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

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

Teaser Trajectory Similarity: Fr´ echet Distance: human • traversal based • fast in practice dog Pigeon GPS Trajectories Fr´ echet Distance Under Translation: • traversal based • only impractical algorithms (before) • translation invariant Handwritten Characters Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Teaser Best algorithm: O ( n 4 . 66 ) Conditional lower bound: n 4 − o (1) All algorithms build O ( n 4 ) arrangement! Fr´ echet under translation is 1-Lipschitz in τ ! Lipschitz Meets Fr´ echet: Use continuous optimization: • branch & bound! τ 2 τ 2 τ 1 τ 1 Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Teaser Take-Home Message: expensive arrangement-based geometric algorithm exact methods from continuous optimization approximation fast practical algorithm :) exact Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

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

Why Trajectory Similarity? Pigeons’ GPS Trajectories: Handwritten Character Trajectories: Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

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

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

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

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

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

Discrete Fr´ echet Distance Intuition human dog What is the traversal that achieves the shortest leash length? Question: Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Discrete Fr´ echet Distance Formal Definition � � δ F ( π, σ ) := min f,g ∈T max t ∈ [0 , 1] � π f ( t ) − σ g ( t ) � π, σ = polygonal curves of length n T = set of monotone and surjective functions from [0 , 1] to { 1 , . . . , n } Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation Definition Intuition: Allow arbitrary translations τ ∈ R 2 of curve σ . δ T ( π, σ ) := min τ ∈ R 2 δ F ( π, σ + τ ) Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation Definition Intuition: Allow arbitrary translations τ ∈ R 2 of curve σ . δ T ( π, σ ) := min τ ∈ R 2 δ F ( π, σ + τ ) σ π Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation Definition Intuition: Allow arbitrary translations τ ∈ R 2 of curve σ . δ T ( π, σ ) := min τ ∈ R 2 δ F ( π, σ + τ ) σ π σ + τ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation Definition Intuition: Allow arbitrary translations τ ∈ R 2 of curve σ . δ T ( π, σ ) := min τ ∈ R 2 δ F ( π, σ + τ ) σ Decision Problem: • Given π, σ, δ • δ T ( π, σ ) ≤ δ ? π σ + τ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Discrete Fr´ echet Distance under Translation Definition Intuition: Allow arbitrary translations τ ∈ R 2 of curve σ . δ T ( π, σ ) := min τ ∈ R 2 δ F ( π, σ + τ ) σ Decision Problem: • Given π, σ, δ • δ T ( π, σ ) ≤ δ ? π σ + τ Focus on this in the talk! Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Goal: Performant implementation computing the discrete Fr´ echet distance under translation on practical inputs. Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Related Work Theory: curve length echet distance under translation in ˜ O ( n 5 ) • Discrete Fr´ [Agarwal, Ben Avraham, Kaplan, Sharir arXiv’15] echet distance under translation in ˜ O ( n 4 . 66 ) • Discrete Fr´ [Bringmann, K¨ unnemann, N. SODA’19] • SETH based lower bound of n 4 − o (1) for discrete Fr´ echet distance under translation [Bringmann, K¨ unnemann, N. SODA’19] Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Related Work Theory: curve length echet distance under translation in ˜ O ( n 5 ) • Discrete Fr´ [Agarwal, Ben Avraham, Kaplan, Sharir arXiv’15] echet distance under translation in ˜ O ( n 4 . 66 ) • Discrete Fr´ [Bringmann, K¨ unnemann, N. SODA’19] • SETH based lower bound of n 4 − o (1) for discrete Fr´ echet distance under translation [Bringmann, K¨ unnemann, N. SODA’19] 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] Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Approach I: Discrete Algorithms Arrangement • Idea: Partition the plane into equivalent regions. τ 2 σ π τ 1 δ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Approach I: Discrete Algorithms Arrangement • Idea: Partition the plane into equivalent regions. τ 2 σ π τ 1 δ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Approach I: Discrete Algorithms Arrangement • Idea: Partition the plane into equivalent regions. τ 2 σ π τ 1 δ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Approach I: Discrete Algorithms Arrangement • Idea: Partition the plane into equivalent regions. τ 2 σ π τ 1 δ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Approach I: Discrete Algorithms Arrangement • Idea: Partition the plane into equivalent regions. τ 2 σ π τ 1 δ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Approach I: Discrete Algorithms Arrangement • Idea: Partition the plane into equivalent regions. τ 2 σ π τ 1 δ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Approach I: Discrete Algorithms Arrangement • Idea: Partition the plane into equivalent regions. τ 2 σ π τ 1 δ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Approach I: Discrete Algorithms Arrangement • Idea: Partition the plane into equivalent regions. τ 2 σ π τ 1 δ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation

Approach I: Discrete Algorithms Arrangement • Idea: Partition the plane into equivalent regions. τ 2 σ π Observation: All translations in a cell of the arrangement have the same closeness relation. τ 1 δ Karl Bringmann, Marvin K¨ unnemann, Algorithm Engineering of the Discrete and Andr´ e Nusser Fr´ echet Distance under Translation