On the complexity of the middle curve problem Maike Buchin 1 Nicole - PowerPoint PPT Presentation

On the complexity of the middle curve problem Maike Buchin 1 Nicole Funk 2 Amer Krivošija 2 1 Ruhr-Universität Bochum, Germany 2 TU Dortmund, Germany 13. März 2020 1 / 11

Introduction 2 / 11

Introduction 2 / 11

Introduction 2 / 11

Introduction 2 / 11

Introduction 2 / 11

Introduction 2 / 11

Introduction Preliminaries 3 / 11

Introduction Preliminaries Definition polygonal curve P sequence of vertices � p 1 , . . . , p ℓ � in R d , connected by line segments ℓ : complexity of P 3 / 11

Introduction Preliminaries 3 / 11

Introduction Preliminaries continuous Fréchet distance d cF 3 / 11

Introduction Preliminaries continuous Fréchet distance d cF 3 / 11

Introduction Preliminaries continuous Fréchet distance d cF 3 / 11

Introduction Preliminaries continuous Fréchet distance d cF 3 / 11

Introduction Preliminaries discrete Fréchet distance d dF 3 / 11

Introduction Preliminaries discrete Fréchet distance d dF 3 / 11

Introduction Preliminaries discrete Fréchet distance d dF 3 / 11

Introduction Preliminaries discrete Fréchet distance d dF 3 / 11

Introduction Preliminaries P = { P 1 , . . . , P n } set of polygonal curves δ ≥ 0 d F ∈ { d dF , d cF } 3 / 11

Introduction Preliminaries P = { P 1 , . . . , P n } set of polygonal curves δ ≥ 0 d F ∈ { d dF , d cF } Definition middle curve with distance δ M = � m 1 , . . . , m ℓ � m i ∈ � P j ∈P P j s.t. max { d F ( M , P j ) | P j ∈ P} ≤ δ 3 / 11

Introduction Middle curve problem Definition middle curve with distance δ M = � m 1 , . . . , m ℓ � m i ∈ � P j ∈P P j s.t. max { d F ( M , P j ) | P j ∈ P} ≤ δ 4 / 11

Introduction Middle curve problem Problem M IDDLE C URVE [Ahn et al. ’16] Given P = � p 1 , . . . , p n � with complexity ≤ m and δ > 0 unordered M IDDLE C URVE middle curve with distance δ ? ordered M IDDLE C URVE unordered M IDDLE C URVE + vertices respect the order in their original curves? restricted M IDDLE C URVE ordered M IDDLE C URVE + vertices get matched to themselves in their original curve? Definition middle curve with distance δ M = � m 1 , . . . , m ℓ � m i ∈ � P j ∈P P j s.t. max { d F ( M , P j ) | P j ∈ P} ≤ δ 4 / 11

Introduction Middle curve problem Problem M IDDLE C URVE [Ahn et al. ’16] Given P = � p 1 , . . . , p n � with complexity ≤ m and δ > 0 unordered M IDDLE C URVE middle curve with distance δ ? ordered M IDDLE C URVE unordered M IDDLE C URVE + vertices respect the order in their original curves? restricted M IDDLE C URVE ordered M IDDLE C URVE + vertices get matched to themselves in their original curve? running time of algorithm [Ahn et al. ’16]: ordered case : O ( m 2 n ) unordered case : O ( m n log m ) restricted case : O ( m n log n m ) 4 / 11

Introduction Middle curve problem Problem M IDDLE C URVE [Ahn et al. ’16] Given P = � p 1 , . . . , p n � with complexity ≤ m and δ > 0 unordered M IDDLE C URVE middle curve with distance δ ? ordered M IDDLE C URVE unordered M IDDLE C URVE + vertices respect the order in their original curves? restricted M IDDLE C URVE ordered M IDDLE C URVE + vertices get matched to themselves in their original curve? Theorem M IDDLE C URVE is NP-complete. 4 / 11

NP-Completeness Shortest Common Supersequence Theorem M IDDLE C URVE is NP-complete. 5 / 11

NP-Completeness Shortest Common Supersequence Theorem M IDDLE C URVE is NP-complete. Proof based on [Buchin, Driemel and Struijs 19] and [Buchin et al. 17] reduction from Shortest Common Supersequence 5 / 11

NP-Completeness Shortest Common Supersequence Theorem M IDDLE C URVE is NP-complete. Proof based on [Buchin, Driemel and Struijs 19] and [Buchin et al. 17] reduction from Shortest Common Supersequence Problem S HORTEST C OMMON S UPERSEQUENCE (SCS) Given a set of sequences S over binary alphabet Σ = { A , B } positive integer t Exists a sequence s ∗ of length at most t , that is a supersequence of all sequences in S ? 5 / 11

NP-Completeness Shortest Common Supersequence Theorem M IDDLE C URVE is NP-complete. Proof based on [Buchin, Driemel and Struijs 19] and [Buchin et al. 17] reduction from Shortest Common Supersequence Problem S HORTEST C OMMON S UPERSEQUENCE (SCS) Given a set of sequences S over binary alphabet Σ = { A , B } positive integer t Exists a sequence s ∗ of length at most t , that is a supersequence of all sequences in S ? example: S = { AB , BB } t = 3 s ∗ = ABB 5 / 11

NP-Completeness Reduction Given SCS -instance ( S , t ) , construct M IDDLE C URVE instance for i + j = t ( G ∪ { A i , B j } , 1 ) 6 / 11

NP-Completeness Reduction Given SCS -instance ( S , t ) , construct M IDDLE C URVE instance for i + j = t ( G ∪ { A i , B j } , 1 ) 3 3 2 2 1 1 A → B → 0 0 − 1 − 1 × t × t × t × t − 2 − 2 − 3 − 3 6 / 11

NP-Completeness Reduction Given SCS -instance ( S , t ) , construct M IDDLE C URVE instance for i + j = t ( G ∪ { A i , B j } , 1 ) 3 3 2 2 1 1 A → B → 0 0 − 1 − 1 × t × t × t × t − 2 − 2 − 3 − 3 3 3 2 2 1 1 0 0 − 1 − 1 − 2 − 2 − 3 − 3 γ ( AB ) γ ( BB ) 6 / 11

NP-Completeness Reduction Given SCS -instance ( S , t ) , construct M IDDLE C URVE instance for i + j = t ( G ∪ { A i , B j } , 1 ) 3 3 2 2 1 1 A i = B j = 0 0 × i × j − 1 − 1 − 2 − 2 − 3 − 3 6 / 11

NP-Completeness Reduction Given SCS -instance ( S , t ) , construct M IDDLE C URVE instance for i + j = t ( G ∪ { A i , B j } , 1 ) 3 3 2 2 1 1 A i = B j = 0 0 × i × j − 1 − 1 − 2 − 2 − 3 − 3 3 3 2 2 1 1 0 0 − 1 − 1 − 2 − 2 − 3 − 3 A 1 B 2 6 / 11

NP-Completeness Reduction Given SCS -instance ( S , t ) , construct M IDDLE C URVE instance for i + j = t ( G ∪ { A i , B j } , 1 ) 6 / 11

NP-Completeness Reduction SCS ⇐ unordered M IDDLE C URVE SCS ⇒ restricted M IDDLE C URVE 7 / 11

NP-Completeness Reduction SCS ⇐ unordered M IDDLE C URVE SCS ⇒ restricted M IDDLE C URVE Theorem M IDDLE C URVE for the discrete Fréchet distance is NP-hard. holds for every variant 7 / 11

NP-Completeness Reduction SCS ⇐ unordered M IDDLE C URVE SCS ⇒ restricted M IDDLE C URVE Theorem M IDDLE C URVE for the continuous Fréchet distance is NP-hard. holds for every variant 7 / 11

NP-Completeness Reduction SCS ⇐ unordered M IDDLE C URVE SCS ⇒ restricted M IDDLE C URVE Theorem M IDDLE C URVE for the continuous Fréchet distance is NP-hard. holds for every variant test Fréchet in O ( m ℓ log( m ℓ )) [Alt and Godau ’95] Theorem M IDDLE C URVE is NP-complete for the discrete and continuous Frechet distance. 7 / 11

Parameterized middle curve Parameterization Problem M IDDLE C URVE Given P = � p 1 , . . . , p n � and δ > 0 unordered M IDDLE C URVE middle curve with distance δ ? ordered M IDDLE C URVE unordered M IDDLE C URVE + vertices respect the order in their original curves? restricted M IDDLE C URVE ordered M IDDLE C URVE + vertices get matched to themselves in their original curve? 8 / 11

Parameterized middle curve Parameterization Problem P ARAM M IDDLE C URVE Given P = � p 1 , . . . , p n � and δ > 0 and a parameter ℓ > 0 unordered P ARAM M IDDLE C URVE middle curve with distance δ and complexity ℓ ? ordered P ARAM M IDDLE C URVE unordered P ARAM M IDDLE C URVE + vertices respect the order in their original curves? restricted P ARAM M IDDLE C URVE ordered P ARAM M IDDLE C URVE + vertices get matched to themselves in their original curve? 8 / 11

Parameterized middle curve Parameterization Problem P ARAM M IDDLE C URVE Given P = � p 1 , . . . , p n � and δ > 0 and a parameter ℓ > 0 unordered P ARAM M IDDLE C URVE middle curve with distance δ and complexity ℓ ? ordered P ARAM M IDDLE C URVE unordered P ARAM M IDDLE C URVE + vertices respect the order in their original curves? restricted P ARAM M IDDLE C URVE ordered P ARAM M IDDLE C URVE + vertices get matched to themselves in their original curve? Theorem Every variant of the P ARAM M IDDLE C URVE Instance can be decided in O (( mn ) ℓ nm ℓ log( m ℓ )) time via brute force. 8 / 11

Parameterized middle curve Approximation Problem ( k , ℓ ) - CENTER Given polygonal curves G = { G 1 , . . . , G n } and distance measure d Find set of curves C = { C 1 , . . . , C k } , each of complexity of at most ℓ , that minimizes max G ∈G min k i = 1 d ( C i , G ) 9 / 11

Parameterized middle curve Approximation 10 / 11

Parameterized middle curve Approximation 10 / 11

Parameterized middle curve Approximation 2 δ 10 / 11

Parameterized middle curve Approximation 10 / 11

Parameterized middle curve Approximation 10 / 11

Parameterized middle curve Approximation 10 / 11

Parameterized middle curve Approximation 10 / 11

Parameterized middle curve Approximation n curves m input complexity δ > 0 middle ℓ curve complexity Theorem Given α -approximation algorithm for ( 1 , ℓ ) - CENTER with running time T We can compute a ( 2 α ) -approximation of the unordered P ARAM M IDDLE C URVE in O ( ℓ mn + T ) time. 10 / 11