powerpoint presentation15780/slides/780s18-2.pdf · powerpoint presentation author: ariel procaccia...
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o 𝑑 4𝑑
o 4𝑑𝑑
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• 𝑔 𝑥 =
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𝑎
𝑏𝑐
𝑑
𝑒
1 3 2
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1
𝑔 = 0 𝑔 = 1
𝑎
𝑔 = 2
𝑏
𝑔 = 4
𝑑
𝑔 = 6
𝑒
𝑔 = 3
𝑐
𝑔 = 6
1
𝑔 = 7
𝑑
#1 #2 #3 #4#5 #6 #7
𝑠 𝑡
𝑠 𝑡
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• ℎ 𝑥 =𝑥
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𝑎
𝑏𝑐
𝑑
𝑒
1 3 2
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1
ℎ = 6 ℎ = 5
𝑎
ℎ = 6
𝑏
ℎ = 2
𝑑
ℎ = 1
𝑒
ℎ = 2
𝑑
ℎ = 0
1
#1 #2 #3 #4 #5
ℎ = 6 ℎ = 5 ℎ = 2 ℎ = 0
ℎ = 1ℎ = 6ℎ = 7
𝑠 𝑡
𝑠 𝑡
• 𝑓 𝑥 = ℎ 𝑥 + 𝑔 𝑥
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𝑎
𝑏𝑐
𝑑
𝑒
1 3 2
51
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ℎ = 6 ℎ = 5 ℎ = 2 ℎ = 0
ℎ = 1ℎ = 6ℎ = 7
𝑑
𝑒
𝑐
𝑠 𝑡
𝑡
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𝑎
𝑏
2
2
ℎ = 3
ℎ = 2
ℎ = 0
ℎ = 1
2
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𝑠 𝑡
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𝑎
5
1
ℎ = 7 ℎ = 0
ℎ = 6
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𝑠 𝑡
• ℎ 𝑥ℎ 𝑥 ≤ ℎ∗ 𝑥 ,
ℎ∗
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• ℎ ≡ 0
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o
𝑡𝑡∗
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𝑥
𝑡∗
𝑡
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o 𝑥𝑡∗
o 𝑓 𝑥 = 𝑔 𝑥 + ℎ 𝑥≤ 𝑔 𝑥 + ℎ∗ 𝑥= 𝑔 𝑡∗ < 𝑔 𝑡 = 𝑓(𝑡)
o 𝑥𝑡 ∎
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𝑥
𝑡∗
𝑡
• ℎ1• ℎ2
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ℎ1ℎ2ℎ1 ℎ2
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5
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6 1
87
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2
54 6
1
87
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• ℎ1• ℎ2
• ℎ ℎ′ ∀𝑥, ℎ 𝑥 ≥ ℎ′(𝑥)
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ℎ1 ℎ21. ℎ1 ℎ22. ℎ2 ℎ13. ℎ1 ℎ2
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5
4
6 1
87
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54 6
1
87
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𝑨∗(𝒉𝟏) 𝑨∗(𝒉𝟐)
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𝑎
𝑏
1
1
ℎ = 2
ℎ = 4
ℎ = 1
ℎ = 1
1
2
𝑐 3ℎ = 0
𝑠 𝑡
• 𝑐 𝑥, 𝑦 = 𝑥 𝑦
• ℎ 𝑥, 𝑦,ℎ 𝑥 ≤ 𝑐(𝑥, 𝑦) + ℎ(𝑦)
• ℎ 𝑡 = 0 𝑡
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⇒
⇒
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𝑥
𝑦
𝑡
• ℎ1• ℎ2
•
ℎ1ℎ2ℎ1 ℎ2
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5
4
6 1
87
3
2
54 6
1
87
32
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•o ℎ 𝑥 ≤ 𝑐(𝑥, 𝑦) + ℎ(𝑦)
o 𝑓(𝑥) 𝑦 𝑥𝑓 𝑥 = 𝑔 𝑥 + ℎ 𝑥 ≤ 𝑔 𝑥 + 𝑐 𝑥, 𝑦 + ℎ 𝑦 = 𝑔(𝑦) + ℎ(𝑦) = 𝑓(𝑦)
o 𝑥 𝑥𝑦 𝑥
o 𝑓 𝑥
o ∎
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o ℎ
o ℎ
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