PowerPoint Presentationaada/courses/15251f15/www/slides/lec17.pdfSki rental ≥8. 1. 𝐺𝐼= t⋅ 𝑇𝐼 2. 𝐺𝐼= u⋅ 𝑇(𝐼) 3. 𝐺𝐼= 𝐵 2 ⋅ 𝑇(𝐼) 4. 𝐺𝐼=

CMU 15-251

Online Algorithms

Teachers:

Anil Ada

Ariel Procaccia (this time)

Ski rental

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𝐵

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• 𝐵 = 5

2

Ski rental

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• 𝐵

3

⇒ 10

⇒ 4

Ski rental

𝐵 ≥ 8.𝐵

1. 𝐴𝐿𝐺 𝐼 = 2 ⋅ 𝑂𝑃𝑇 𝐼

2. 𝐴𝐿𝐺 𝐼 = 3 ⋅ 𝑂𝑃𝑇(𝐼)

3. 𝐴𝐿𝐺 𝐼 =𝐵

2⋅ 𝑂𝑃𝑇(𝐼)

4. 𝐴𝐿𝐺 𝐼 = 𝐵 ⋅ 𝑂𝑃𝑇(𝐼)

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Competitive ratio

• c > 1 𝐴𝐿𝐺𝑐 𝐼𝐴𝐿𝐺 𝐼 ≤ 𝑐 ⋅ 𝑂𝑃𝑇(𝐼)

• 𝑐 < 1 𝐴𝐿𝐺𝑐 𝐼𝐴𝐿𝐺 𝐼 ≥ 𝑐 ⋅ 𝑂𝑃𝑇(𝐼)

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𝐴𝐿𝐺 𝑂𝑃𝑇(𝐼)

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Ski rental, revisited

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• 𝐵 − 12𝐵−1

𝐵

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Ski rental, revisited

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𝛼 𝛼 <2𝐵−1

𝐵

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o 𝐾𝐾 + 1

o 𝐾 + 2

o 𝐾 ≥ 𝐵:𝑂𝑃𝑇 𝐼 = 𝐵, 𝐴𝐿𝐺 𝐼 = 𝐾 + 𝐵 ≥ 2𝐵

o 𝐾 ≤ 𝐵 − 2: 𝑂𝑃𝑇 𝐼 = 𝐾 + 1,𝐴𝐿𝐺 𝐼 = 𝐾 + 𝐵 ≥ 2𝐾 + 2 ∎

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Pancakes, revisited

8

≈

𝐵2𝐵−1

𝐵

Ski rental, revisited

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Paging

• 𝑁 𝑘

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•

•

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10

Paging

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Paging

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Paging

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1,… , 𝑘)

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•

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•

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Example: LIFO

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Paging

• 𝛼𝛼

1. 𝛼 = 2

2. 𝛼 = 𝑘

3. 𝛼 = 𝑁

4. 𝛼 = ∞

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Paging

• 𝛼𝛼

1. 𝛼 = 2

2. 𝛼 = 𝑘

3. 𝛼 = 𝑁

4. 𝛼 = ∞

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Paging

• 𝑘

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o

𝑘

o 𝑘 = 3:

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Paging

• 𝑘

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o 𝑚 = 𝑝𝑗𝑖 𝑗

𝑖

o 𝑝1𝑖 , … , 𝑝𝑘

𝑖 , 𝑝1𝑖+1

o 𝑝2𝑖 , … , 𝑝𝑘

𝑖

𝑝1𝑖+1

⇒𝑂𝑃𝑇 ≥ 𝑚

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Paging

• 𝑘

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o

o 𝐴𝐿𝐺 ≤ 𝑘𝑚 ∎

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Paging

• 𝑘

• ∎

•𝛼 𝛼 < 𝑘

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Paging

•

o

{1, … , 𝑘 + 1} ⇒

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Paging

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o 𝑘∎

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List update

• 𝑛

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List update

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List update

• 𝛼𝛼

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Summary

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o

o

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o

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o

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