Aimene Belfodil1,2, Sergei O. Kuznetsov3,

Céline Robardet1, Mehdi Kaytoue1

𝑙1

𝑙2

𝑙3

𝓖 𝒊𝟏 𝒊𝟐 𝒊𝟑 class

𝒂 ✗ ✗ +

𝒃 ✗ ✗ +

𝒄 ✗ ✗ −

𝒅 ✗ ✗ ✗ +

𝒆 ✗ ✗ −

𝒇 ✗ −

𝓖 𝒊𝟏 𝒊𝟐 𝒊𝟑 class

𝒂 ✗ ✗ +

𝒃 ✗ ✗ +

𝒄 ✗ ✗ −

𝒅 ✗ ✗ ✗ +

𝒆 ✗ ✗ −

𝒇 ✗ −

𝑖2 AND 𝑖3

𝓖 𝒊𝟏 𝒊𝟐 𝒊𝟑 class

𝒂 ✗ ✗ +

𝒃 ✗ ✗ +

𝒄 ✗ ✗ −

𝒅 ✗ ✗ ✗ +

𝒆 ✗ ✗ −

𝒇 ✗ −

𝑖2 AND 𝑖3

{𝑏, 𝑐, 𝑑} = 𝟑

𝓖 𝒊𝟏 𝒊𝟐 𝒊𝟑 class

𝒂 ✗ ✗ +

𝒃 ✗ ✗ +

𝒄 ✗ ✗ −

𝒅 ✗ ✗ ✗ +

𝒆 ✗ ✗ −

𝒇 ✗ −

𝑖2 AND 𝑖3

{𝑏, 𝑐, 𝑑} = 𝟑

=𝟐

𝟑−𝟑

𝟔=𝟏

𝟔

𝓖 𝒊𝟏 𝒊𝟐 𝒊𝟑 class

𝒂 ✗ ✗ +

𝒃 ✗ ✗ +

𝒄 ✗ ✗ −

𝒅 ✗ ✗ ✗ +

𝒆 ✗ ✗ −

𝒇 ✗ −

𝑖2 AND 𝑖3

{𝑏, 𝑐, 𝑑} = 𝟑

=𝟐

𝟑−𝟑

𝟔=𝟏

𝟔

𝑖2 𝑖3+

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

1 2 4 5

1

2

3

4

3 𝑥

𝑦

𝓖 𝒙 𝒚

𝒂 1 4

𝒃 2 2

𝒄 2 3

𝒅 3 4

𝒆 4 3

𝒇 5 1

𝒙 𝒚

1 2 4 5

1

2

3

4

3 𝑥

𝑦

𝓖 𝒙 𝒚

𝒂 1 4

𝒃 2 2

𝒄 2 3

𝒅 3 4

𝒆 4 3

𝒇 5 1

•

1 ≤ 𝑥 ≤ 42 ≤ 𝑦 ≤ 4

𝑨𝑵𝑫.

𝐈𝐧𝐭𝐞𝐧𝐭.

{𝑎, 𝑏, 𝑐, 𝑑, 𝑒}𝐄𝐱𝐭𝐞𝐧𝐭.𝒙 𝒚

1 2 4 5

1

2

3

4

3 𝑥

𝑦

𝓖 𝒙 𝒚

𝒂 1 4

𝒃 2 2

𝒄 2 3

𝒅 3 4

𝒆 4 3

𝒇 5 1

•

•

1 ≤ 𝑥 ≤ 42 ≤ 𝑦 ≤ 4

𝑨𝑵𝑫.

𝐈𝐧𝐭𝐞𝐧𝐭.

{𝑎, 𝑏, 𝑐, 𝑑, 𝑒}𝐄𝐱𝐭𝐞𝐧𝐭.𝒙 𝒚

1 2 4 5

1

2

3

4

3 𝑥

𝑦

𝓖 𝒙 𝒚

𝒂 1 4

𝒃 2 2

𝒄 2 3

𝒅 3 4

𝒆 4 3

𝒇 5 1

•

•

•

1 ≤ 𝑥 ≤ 42 ≤ 𝑦 ≤ 4

𝑨𝑵𝑫.

𝐈𝐧𝐭𝐞𝐧𝐭.

{𝑎, 𝑏, 𝑐, 𝑑, 𝑒}𝐄𝐱𝐭𝐞𝐧𝐭.𝒙 𝒚

1 2 4 5

1

2

3

4

3 𝑥

𝑦

𝓖 𝒙 𝒚

𝒂 1 4

𝒃 2 2

𝒄 2 3

𝒅 3 4

𝒆 4 3

𝒇 5 1

(𝒙, 𝒚)

1 2 4 5

1

2

3

4

3 𝑥

𝑦

𝓖 𝒙 𝒚

𝒂 1 4

𝒃 2 2

𝒄 2 3

𝒅 3 4

𝒆 4 3

𝒇 5 1

(𝒙, 𝒚)

1 2 4 5

1

2

3

4

3 𝑥

𝑦

𝑦 ≤ 4𝑥 + 𝑦 ≤ 7

𝑥 − 2𝑦 ≤ −22𝑥 + 𝑦 ≥ 6

𝑨𝑵𝑫𝑨𝑵𝑫𝑨𝑵𝑫.

𝐈𝐧𝐭𝐞𝐧𝐭.

•

𝓖 𝒙 𝒚

𝒂 1 4

𝒃 2 2

𝒄 2 3

𝒅 3 4

𝒆 4 3

𝒇 5 1

(𝒙, 𝒚)

1 2 4 5

1

2

3

4

3 𝑥

𝑦

𝑦 ≤ 4𝑥 + 𝑦 ≤ 7

𝑥 − 2𝑦 ≤ −22𝑥 + 𝑦 ≥ 6

𝑨𝑵𝑫𝑨𝑵𝑫𝑨𝑵𝑫.

𝐈𝐧𝐭𝐞𝐧𝐭.

•

•

𝓖 𝒙 𝒚

𝒂 1 4

𝒃 2 2

𝒄 2 3

𝒅 3 4

𝒆 4 3

𝒇 5 1

(𝒙, 𝒚)

1 2 4 5

1

2

3

4

3 𝑥

𝑦

𝑦 ≤ 4𝑥 + 𝑦 ≤ 7

𝑥 − 2𝑦 ≤ −22𝑥 + 𝑦 ≥ 6

𝑨𝑵𝑫𝑨𝑵𝑫𝑨𝑵𝑫.

𝐈𝐧𝐭𝐞𝐧𝐭.

•

•

•

[𝒂, 𝒄, 𝒆, 𝒅]

𝓖 𝒙 𝒚

𝒂 1 4

𝒃 2 2

𝒄 2 3

𝒅 3 4

𝒆 4 3

𝒇 5 1

(𝒙, 𝒚)

𝒢

𝒢

𝒢

𝒢

1

2

3

1

2

3

𝐵

𝐶

𝐸

𝐴

𝐷

𝐶

𝐸

𝐴

𝐷 𝐵

𝐸

𝐴

𝐷𝐵

𝐶

𝐸

𝐷 𝐵

𝐶

𝐸

𝐴

𝐵

𝐶

𝐷 𝐵

𝐶

𝐸

𝐶

𝐸

𝐷𝐵

𝐸

𝐷 𝐵

𝐶

𝐴

𝐵

𝐸

𝐴

𝐸

𝐴

𝐷

𝐶

𝐸

𝐴

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

⬆

𝑨 < 𝑩 …

[𝐴, 𝐵, 𝐶, 𝐷, 𝐸]

⬆

𝑨 < 𝑩 …

[𝐴, 𝐵, 𝐶, 𝐷, 𝐸]

⬆

𝑨 < 𝑩 …

[𝐴, 𝐵, 𝐶, 𝐷, 𝐸]

Ο 𝑛 ⋅ 𝑙𝑜𝑔 𝑛

⬆*𝑛

𝑨 < 𝑩 …

, 𝐾 > 𝐻

[𝐴, 𝐵, 𝐶, 𝐷, 𝐸]

Ο 𝑛 ⋅ 𝑙𝑜𝑔 𝑛

⬆*𝑛

𝑨 < 𝑩 …

, 𝐾 > 𝐻

[𝐴, 𝐵, 𝐶, 𝐷, 𝐸]

Ο 𝑛 ⋅ 𝑙𝑜𝑔 𝑛

⟹

⬆*𝑛

𝑨 < 𝑩 …

, 𝐾 > 𝐻

[𝐴, 𝐵, 𝐶, 𝐷, 𝐸]

Ο 𝑛 ⋅ 𝑙𝑜𝑔 𝑛

⟹

⬆*𝑛

𝑨 < 𝑩 …

, 𝐾 > 𝐻

[𝐴, 𝐵, 𝐶, 𝐷, 𝐸]

Ο 𝑛 ⋅ 𝑙𝑜𝑔 𝑛

𝐽 > 𝐻

⟹

⬆*𝑛

𝑨 < 𝑩 …

⬆

𝐴, 𝑩, 𝐶

⬆

𝐴, 𝑩, 𝐶

⬆

𝐴, 𝑩, 𝐶

𝐁

⬆

𝐴, 𝑭,𝑫, 𝐶

Ο 𝑘 ⋅ 𝑙𝑜𝑔 𝑘

*𝑘

⬆

𝐴, 𝑭,𝑫, 𝐶

Ο 𝑘 ⋅ 𝑙𝑜𝑔 𝑘

*𝑘

⬆

𝐴, 𝑭,𝑫, 𝐶

Ο 𝑘 ⋅ 𝑙𝑜𝑔 𝑘

*𝑘

⟹

𝐸 ⊂ ℝ2 𝑒 ∈ ℝ2 𝒄𝒉(𝑬)

𝐸

𝒄𝒉(𝑬 ∪ {𝒆}) ≤ 𝒄𝒉(𝑬) + 1

𝐸 ⊂ ℝ2 𝑒 ∈ ℝ2 𝒄𝒉(𝑬)

𝐸

𝒄𝒉(𝑬 ∪ {𝒆}) ≤ 𝒄𝒉(𝑬) + 1

𝐸 ⊂ ℝ2 𝑒 ∈ ℝ2 𝒄𝒉(𝑬)

𝐸

𝒄𝒉(𝑬 ∪ {𝒆}) ≤ 𝒄𝒉(𝑬) + 1

𝑑 𝑑′

𝑑′ = 𝑑 + 1

𝐸 ⊂ ℝ2 𝑒 ∈ ℝ2 𝒄𝒉(𝑬)

𝐸

𝒄𝒉(𝑬 ∪ {𝒆}) ≤ 𝒄𝒉(𝑬) + 1

𝑑 𝑑′

𝑑′ = 𝑑 + 1

≤ 𝜏

𝐴𝐵

𝐶 𝐷

𝐸

𝐴𝐵

𝐶 𝐷

𝐸𝐹

𝐴𝐵

𝐶 𝐷

𝐸𝐹

𝐴𝐵

𝐶 𝐷

𝑑 = 𝐴, 𝐵, 𝐶, 𝐷𝐴 𝑑′ = [𝐹, 𝐵, 𝐶, 𝐷]

𝐸𝐹

𝐴𝐵

𝐶 𝐷

𝑑 = 𝐴, 𝐵, 𝐶, 𝐷𝐴 𝑑′ = [𝐹, 𝐵, 𝐶, 𝐷]

𝒅′ = 𝒅 + 𝟏.

⬆•

𝑨 < 𝑩 …

𝐴, 𝐵, 𝑪, 𝐷

⬆•

𝐴, 𝐵, 𝑪, 𝐷

𝑨 < 𝑩 …

⬆•

𝑨 < 𝑩 …

[𝐴, 𝐵, 𝐶,𝑯,𝐷]

⬆•

𝑨 < 𝑩 …

[𝐴, 𝐵, 𝐶,𝑯,𝐷]

(𝑱)

⬆•

𝑨 < 𝑩 …

[𝐴, 𝐵, 𝐶,𝑯,𝐷]

(𝑱)

⟹ 𝑛𝑜𝑡 𝐺, 𝑛𝑜𝑡 𝐽,𝑛𝑜𝑡 𝐾, 𝒃𝒖𝒕 𝑳

⬆•

𝑨 < 𝑩 …

[𝐴, 𝐵, 𝐶,𝑯,𝐷]

(𝑱)

⟹ 𝑛𝑜𝑡 𝐺, 𝑛𝑜𝑡 𝐽,𝑛𝑜𝑡 𝐾, 𝒃𝒖𝒕 𝑳

⟹

⬆ ⬇ ⬆

✗ ✓ ✗

✗ ✓ ✗

✓ ✗ ✓

✗ ✓ ✗

✓ ✗ ✓

✗ ✗ ✓

1

2

3

𝟑𝟒𝟔𝟒

𝟑𝟒𝟔𝟒

𝟑𝟒𝟔𝟒

𝟑𝟒𝟔𝟒

DMKD - minor revision

1

2

3

1 ≤ 𝑥 ≤ 42 ≤ 𝑦 ≤ 4

𝑨𝑵𝑫.

1 ≤ 𝑥 ≤ 42 ≤ 𝑦 ≤ 4

𝑨𝑵𝑫.

1 ≤ 𝑥 ≤ 4 𝑨𝑵𝑫2 ≤ 𝑦 ≤ 4 𝑨𝑵𝑫𝒙 ≤ 𝒚 .

1 ≤ 𝑥 ≤ 42 ≤ 𝑦 ≤ 4

𝑨𝑵𝑫.

1 ≤ 𝑥 ≤ 4 𝑨𝑵𝑫2 ≤ 𝑦 ≤ 4 𝑨𝑵𝑫𝒙 ≤ 𝒚 .

≤

1 ≤ 𝑥 ≤ 42 ≤ 𝑦 ≤ 4

𝑨𝑵𝑫.

…

1 ≤ 𝑥 ≤ 4 𝑨𝑵𝑫2 ≤ 𝑦 ≤ 4 𝑨𝑵𝑫𝒙 ≤ 𝒚 .

≤

ℝ2 𝑛

ℝ2 𝑛

• Ο 𝑛𝟑

ℝ2 𝑛

• Ο 𝑛𝟑

ℝ2 𝑛

• Ο 𝑛𝟑

• Ο 𝑛4

ℝ2 𝑛

• Ο 𝑛𝟑

• Ο 𝑛4

• 𝑂(𝑛5)

ℝ2 𝑛

• Ο 𝑛𝟑

• Ο 𝑛4

• 𝑂(𝑛5)• 𝑂(2𝑛)

ℝ2 𝑛

• Ο 𝑛𝟑

• Ο 𝑛4

• 𝑂(𝑛5)• 𝑂(2𝑛)•

ℝ2 𝑛

• Ο 𝑛𝟑

• Ο 𝑛4

• 𝑂(𝑛5)• 𝑂(2𝑛)•

•

ℝ2 𝑛

• Ο 𝑛𝟑

• Ο 𝑛4

• 𝑂(𝑛5)• 𝑂(2𝑛)•

•

•

ℝ2 𝑛

• Ο 𝑛𝟑

• Ο 𝑛4

• 𝑂(𝑛5)• 𝑂(2𝑛)•

mailto:aimene.Belfodil@insa-lyon.frhttps://github.com/BelfodilAimene/MiningConvexPolygonPatternshttps://www.ijcai.org/proceedings/2017/0197.pdf