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CMU 15-251
Online Algorithms
Teachers:
Anil Ada
Ariel Procaccia (this time)
Ski rental
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𝐵
•
• 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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•
•
•
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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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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•
•
•
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List update
• 𝛼𝛼
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Summary
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o
o
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o
•
o
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