SLIDE 1
On the complexity of the middle curve problem
Maike Buchin 1 Nicole Funk 2 Amer Krivošija 2
1Ruhr-Universität Bochum, Germany 2TU Dortmund, Germany
- 13. März 2020
1 / 11
Introduction 2 / 11
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 - - PowerPoint PPT Presentation
On the complexity of the middle curve problem Maike Buchin 1 Nicole Funk 2 Amer Krivoija 2 1 Ruhr-Universitt Bochum, Germany 2 TU Dortmund, Germany 13. Mrz 2020 1 / 11 Introduction 2 / 11 Introduction 2 / 11 Introduction 2 / 11
SLIDE 2
SLIDE 3
Introduction 2 / 11
SLIDE 4
Introduction 2 / 11
SLIDE 5
Introduction 2 / 11
SLIDE 6
Introduction 2 / 11
SLIDE 7
Introduction 2 / 11
SLIDE 8
Introduction Preliminaries 3 / 11
SLIDE 9
Introduction Preliminaries
Definition polygonal curve P sequence of vertices p1, . . . , pℓ in Rd, connected by line segments ℓ : complexity of P
3 / 11
SLIDE 10
Introduction Preliminaries 3 / 11
SLIDE 11
Introduction Preliminaries
continuous Fréchet distance dcF
3 / 11
SLIDE 12
Introduction Preliminaries
continuous Fréchet distance dcF
3 / 11
SLIDE 13
Introduction Preliminaries
continuous Fréchet distance dcF
3 / 11
SLIDE 14
Introduction Preliminaries
continuous Fréchet distance dcF
3 / 11
SLIDE 15
Introduction Preliminaries
discrete Fréchet distance ddF
3 / 11
SLIDE 16
Introduction Preliminaries
discrete Fréchet distance ddF
3 / 11
SLIDE 17
Introduction Preliminaries
discrete Fréchet distance ddF
3 / 11
SLIDE 18
Introduction Preliminaries
discrete Fréchet distance ddF
3 / 11
SLIDE 19
Introduction Preliminaries
P = {P1, . . . , Pn} set of polygonal curves δ ≥ 0 dF ∈ {ddF, dcF}
3 / 11
SLIDE 20
Introduction Preliminaries
P = {P1, . . . , Pn} set of polygonal curves δ ≥ 0 dF ∈ {ddF, dcF} Definition middle curve with distance δ M = m1, . . . , mℓ mi ∈
Pj∈P Pj
s.t. max{dF(M, Pj)|Pj ∈ P} ≤ δ
3 / 11
SLIDE 21
Introduction Middle curve problem
Definition middle curve with distance δ M = m1, . . . , mℓ mi ∈
Pj∈P Pj
s.t. max{dF(M, Pj)|Pj ∈ P} ≤ δ
4 / 11
SLIDE 22
Introduction Middle curve problem
Problem MIDDLE CURVE [Ahn et al. ’16] Given P = p1, . . . , pn with complexity ≤ m and δ > 0 unordered MIDDLE CURVE middle curve with distance δ?
- rdered MIDDLE CURVE
unordered MIDDLE CURVE + vertices respect the order in their original curves? restricted MIDDLE CURVE
- rdered MIDDLE CURVE
+ vertices get matched to themselves in their original curve? Definition middle curve with distance δ M = m1, . . . , mℓ mi ∈
Pj∈P Pj
s.t. max{dF(M, Pj)|Pj ∈ P} ≤ δ
4 / 11
SLIDE 23
Introduction Middle curve problem
Problem MIDDLE CURVE [Ahn et al. ’16] Given P = p1, . . . , pn with complexity ≤ m and δ > 0 unordered MIDDLE CURVE middle curve with distance δ?
- rdered MIDDLE CURVE
unordered MIDDLE CURVE + vertices respect the order in their original curves? restricted MIDDLE CURVE
- rdered MIDDLE CURVE
+ vertices get matched to themselves in their original curve? running time of algorithm [Ahn et al. ’16]:
- rdered case: O(m2n)
unordered case: O(mn log m) restricted case: O(mn logn m)
4 / 11
SLIDE 24
Introduction Middle curve problem
Problem MIDDLE CURVE [Ahn et al. ’16] Given P = p1, . . . , pn with complexity ≤ m and δ > 0 unordered MIDDLE CURVE middle curve with distance δ?
- rdered MIDDLE CURVE
unordered MIDDLE CURVE + vertices respect the order in their original curves? restricted MIDDLE CURVE
- rdered MIDDLE CURVE
+ vertices get matched to themselves in their original curve? Theorem
MIDDLE CURVE is NP-complete.
4 / 11
SLIDE 25
NP-Completeness Shortest Common Supersequence
Theorem
MIDDLE CURVE is NP-complete.
5 / 11
SLIDE 26
NP-Completeness Shortest Common Supersequence
Theorem
MIDDLE CURVE is NP-complete.
Proof based on [Buchin, Driemel and Struijs 19] and [Buchin et al. 17] reduction from Shortest Common Supersequence
5 / 11
SLIDE 27
NP-Completeness Shortest Common Supersequence
Theorem
MIDDLE CURVE is NP-complete.
Proof based on [Buchin, Driemel and Struijs 19] and [Buchin et al. 17] reduction from Shortest Common Supersequence Problem SHORTEST COMMON SUPERSEQUENCE (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
SLIDE 28
NP-Completeness Shortest Common Supersequence
Theorem
MIDDLE CURVE is NP-complete.
Proof based on [Buchin, Driemel and Struijs 19] and [Buchin et al. 17] reduction from Shortest Common Supersequence Problem SHORTEST COMMON SUPERSEQUENCE (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
SLIDE 29
NP-Completeness Reduction
Given SCS-instance (S, t), construct MIDDLE CURVE instance for i + j = t
(G ∪ {Ai, Bj}, 1)
6 / 11
SLIDE 30
NP-Completeness Reduction
Given SCS-instance (S, t), construct MIDDLE CURVE instance for i + j = t
(G ∪ {Ai, Bj}, 1)
A →
3 2 1 −1 −2 −3 ×t ×t
B →
3 2 1 −1 −2 −3 ×t ×t
6 / 11
SLIDE 31
NP-Completeness Reduction
Given SCS-instance (S, t), construct MIDDLE CURVE instance for i + j = t
(G ∪ {Ai, Bj}, 1)
A →
3 2 1 −1 −2 −3 ×t ×t
B →
3 2 1 −1 −2 −3 ×t ×t
3 2 1 −1 −2 −3
γ(AB)
3 2 1 −1 −2 −3
γ(BB)
6 / 11
SLIDE 32
NP-Completeness Reduction
Given SCS-instance (S, t), construct MIDDLE CURVE instance for i + j = t
(G ∪ {Ai, Bj}, 1)
Ai =
3 2 1 −1 −2 −3 ×i
Bj =
3 2 1 −1 −2 −3 ×j
6 / 11
SLIDE 33
NP-Completeness Reduction
Given SCS-instance (S, t), construct MIDDLE CURVE instance for i + j = t
(G ∪ {Ai, Bj}, 1)
Ai =
3 2 1 −1 −2 −3 ×i
Bj =
3 2 1 −1 −2 −3 ×j
3 2 1 −1 −2 −3
A1
3 2 1 −1 −2 −3
B2
6 / 11
SLIDE 34
NP-Completeness Reduction
Given SCS-instance (S, t), construct MIDDLE CURVE instance for i + j = t
(G ∪ {Ai, Bj}, 1)
6 / 11
SLIDE 35
NP-Completeness Reduction
SCS ⇐ unordered MIDDLE CURVE SCS ⇒ restricted MIDDLE CURVE
7 / 11
SLIDE 36
NP-Completeness Reduction
SCS ⇐ unordered MIDDLE CURVE SCS ⇒ restricted MIDDLE CURVE Theorem
MIDDLE CURVE for the discrete Fréchet distance is NP-hard.
holds for every variant
7 / 11
SLIDE 37
NP-Completeness Reduction
SCS ⇐ unordered MIDDLE CURVE SCS ⇒ restricted MIDDLE CURVE Theorem
MIDDLE CURVE for the continuous Fréchet distance is NP-hard.
holds for every variant
7 / 11
SLIDE 38
NP-Completeness Reduction
SCS ⇐ unordered MIDDLE CURVE SCS ⇒ restricted MIDDLE CURVE Theorem
MIDDLE CURVE for the continuous Fréchet distance is NP-hard.
holds for every variant test Fréchet in O(mℓ log(mℓ)) [Alt and Godau ’95] Theorem
MIDDLE CURVE is NP-complete for the discrete and continuous Frechet
distance.
7 / 11
SLIDE 39
Parameterized middle curve Parameterization
Problem MIDDLE CURVE Given P = p1, . . . , pn and δ > 0 unordered MIDDLE CURVE middle curve with distance δ?
- rdered MIDDLE CURVE
unordered MIDDLE CURVE + vertices respect the order in their original curves? restricted MIDDLE CURVE
- rdered MIDDLE CURVE
+ vertices get matched to themselves in their original curve?
8 / 11
SLIDE 40
Parameterized middle curve Parameterization
Problem PARAM MIDDLE CURVE Given P = p1, . . . , pn and δ > 0 and a parameter ℓ > 0 unordered PARAM MIDDLE CURVE middle curve with distance δ and complexity ℓ?
- rdered PARAM MIDDLE CURVE
unordered PARAM MIDDLE CURVE + vertices respect the order in their original curves? restricted PARAM MIDDLE CURVE
- rdered PARAM MIDDLE CURVE
+ vertices get matched to themselves in their original curve?
8 / 11
SLIDE 41
Parameterized middle curve Parameterization
Problem PARAM MIDDLE CURVE Given P = p1, . . . , pn and δ > 0 and a parameter ℓ > 0 unordered PARAM MIDDLE CURVE middle curve with distance δ and complexity ℓ?
- rdered PARAM MIDDLE CURVE
unordered PARAM MIDDLE CURVE + vertices respect the order in their original curves? restricted PARAM MIDDLE CURVE
- rdered PARAM MIDDLE CURVE
+ vertices get matched to themselves in their original curve? Theorem Every variant of the PARAM MIDDLE CURVE Instance can be decided in O((mn)ℓnmℓ log(mℓ)) time via brute force.
8 / 11
SLIDE 42
Parameterized middle curve Approximation
Problem (k, ℓ)-CENTER Given polygonal curves G = {G1, . . . , Gn} and distance measure d Find set of curves C = {C1, . . . , Ck}, each of complexity of at most ℓ, that minimizes maxG∈G mink
i=1 d(Ci, G)
9 / 11
SLIDE 43
Parameterized middle curve Approximation 10 / 11
SLIDE 44
Parameterized middle curve Approximation 10 / 11
SLIDE 45
Parameterized middle curve Approximation
2δ
10 / 11
SLIDE 46
Parameterized middle curve Approximation 10 / 11
SLIDE 47
Parameterized middle curve Approximation 10 / 11
SLIDE 48
Parameterized middle curve Approximation 10 / 11
SLIDE 49
Parameterized middle curve Approximation 10 / 11
SLIDE 50
Parameterized middle curve Approximation
n curves m input complexity δ > 0 ℓ middle curve complexity Theorem Given α-approximation algorithm for (1, ℓ)-CENTER with running time T We can compute a (2α)-approximation of the unordered PARAM MIDDLE
CURVE in O(ℓmn + T) time.
10 / 11
SLIDE 51
Parameterized middle curve Approximation
n curves m input complexity δ > 0 ℓ middle curve complexity Corollary using [Buchin, Driemel and Struijs 19] We can compute a (2 + ǫ)-approximation of the unordered PARAM MIDDLE
CURVE in O(((( 4
√ d ǫ
+ 1)dℓ)ℓ + log(ℓ + n))ℓmn) time.
10 / 11
SLIDE 52
Open question
Open question
complexity of PARAM MIDDLE CURVE
11 / 11
SLIDE 53
Open question
Open question
complexity of PARAM MIDDLE CURVE Guesses:
11 / 11
SLIDE 54
Open question
Open question
complexity of PARAM MIDDLE CURVE Guesses: XP ✓ hard ✗
11 / 11
SLIDE 55
Open question
Open question
complexity of PARAM MIDDLE CURVE Guesses: XP ✓ hard ✗ paraNP ✓ hard ✗
11 / 11
SLIDE 56
Open question
Open question
complexity of PARAM MIDDLE CURVE Guesses: XP ✓ hard ✗ paraNP ✓ hard ✗ W[P] ✓ hard ?
11 / 11
SLIDE 57
Open question
Open question
complexity of PARAM MIDDLE CURVE Guesses: XP ✓ hard ✗ paraNP ✓ hard ✗ W[P] ✓ hard ? W[i] ✗
11 / 11