Online Bin Packing with Advice Joan Boyar 1 , Shahin Kamali 2 , Kim - - PowerPoint PPT Presentation

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Online Bin Packing with Advice Joan Boyar 1 , Shahin Kamali 2 , Kim - - PowerPoint PPT Presentation

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Online Bin Packing with Advice Joan Boyar 1 , Shahin Kamali 2 , Kim S. Larsen 1 , Alejandro L opez-Ortiz 2 1 University of Southern Denmark, Denmark 2 University of Waterloo, Canada July 7, 2014 Boyar Kamali Larsen L opez-Ortiz (1)

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SLIDE 1

Online Bin Packing with Advice

Joan Boyar1, Shahin Kamali2, Kim S. Larsen1, Alejandro L´

  • pez-Ortiz2

1 University of Southern Denmark, Denmark 2 University of Waterloo, Canada

July 7, 2014

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 1 / 28

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SLIDE 2

Overview

1

The bin packing problem: offline and online

2

Advice complexity results for bin packing

3

Open problems

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 2 / 28

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SLIDE 3

Section 1 The bin packing problem: offline and online

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 3 / 28

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SLIDE 4

Bin Packing Problem

Input: items of various sizes ∈ (0, 1] Output: packing of all items into unit size bins Goal: use minimum number of bins

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 4 / 28

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SLIDE 5

Bin Packing Problem

Input: items of various sizes ∈ (0, 1] Output: packing of all items into unit size bins Goal: use minimum number of bins Applications: storage, cutting stock...

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 4 / 28

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SLIDE 6

Offline Bin Packing Problem

The problem is NP-hard; Reduce from 2-PARTITION. First-Fit-Decreasing has an approximation ratio of 11/9 ≈ 1.22 [Johnson,Demers,Ullman,Garey,Graham, 1974] There is an asymptotic PTAS for the problem [de la Vega,Lueker, 1981]

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 5 / 28

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SLIDE 7

Online Bin Packing Problem

Request sequence is revealed in a sequential, online manner. Examples: Next-Fit First-Fit Best-Fit Harmonic, Harmonic++

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 6 / 28

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SLIDE 8

First-Fit vs. Next-Fit — Online

First-Fit Find the first open bin with enough space, and place the item there If such a bin does not exist, open a new bin

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 7 / 28

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SLIDE 9

First-Fit vs. Next-Fit — Online

First-Fit Find the first open bin with enough space, and place the item there If such a bin does not exist, open a new bin Next-Fit Put item in current open bit, if it fits Otherwise, close that bin and open a new current bin

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 7 / 28

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SLIDE 10

First-Fit vs. Next-Fit — Online

First-Fit Next-Fit

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 8 / 28

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SLIDE 11

First-Fit vs. Next-Fit — Online

First-Fit Next-Fit

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 8 / 28

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SLIDE 12

First-Fit vs. Next-Fit — Online

First-Fit Next-Fit

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 8 / 28

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SLIDE 13

First-Fit vs. Next-Fit — Online

First-Fit Next-Fit

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 8 / 28

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SLIDE 14

First-Fit vs. Next-Fit — Online

First-Fit Next-Fit

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 8 / 28

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SLIDE 15

First-Fit vs. Next-Fit — Online

First-Fit Next-Fit

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 8 / 28

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SLIDE 16

First-Fit vs. Next-Fit — Online

First-Fit Next-Fit

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 8 / 28

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SLIDE 17

First-Fit vs. Next-Fit — Online

First-Fit Next-Fit

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 8 / 28

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SLIDE 18

First-Fit vs. Next-Fit — Online

First-Fit Next-Fit Result: 4 Result: 6

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 8 / 28

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SLIDE 19

Competitive Analysis

Compare the performance of an online algorithm, Alg, with an optimal

  • ffline algorithm, Opt:

Opt knows the whole sequence in the beginning. Competitive ratio of Alg is the maximum ratio between the cost of Alg and Opt for serving the same sequence.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 9 / 28

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SLIDE 20

Competitive Analysis

Next-Fit has competitive ratio 2 [Johnson, 1974] Best-Fit and First-Fit have competitive ratio 1.7 [Johnson,Demers,Ullman,Garey,Graham, 1974] Best known online algorithm (Harmonic++) has competitive ratio 1.58889 [Seiden, 2002]

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 10 / 28

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SLIDE 21

Competitive Analysis

Next-Fit has competitive ratio 2 [Johnson, 1974] Best-Fit and First-Fit have competitive ratio 1.7 [Johnson,Demers,Ullman,Garey,Graham, 1974] Best known online algorithm (Harmonic++) has competitive ratio 1.58889 [Seiden, 2002] No online algorithm has a competitive ratio less than 1.54037 [Balogh,B´ ek´ esi,Galambos, 2012]

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 10 / 28

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SLIDE 22

Competitive Analysis

Next-Fit has competitive ratio 2 [Johnson, 1974] Best-Fit and First-Fit have competitive ratio 1.7 [Johnson,Demers,Ullman,Garey,Graham, 1974] Best known online algorithm (Harmonic++) has competitive ratio 1.58889 [Seiden, 2002] No online algorithm has a competitive ratio less than 1.54037 [Balogh,B´ ek´ esi,Galambos, 2012] Recall that offline First-Fit-Decreasing has approximation ratio ≈ 1.22. A big gap between quality of online and offline solutions. What about an “almost online” algorithm?

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 10 / 28

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SLIDE 23

Section 2 Advice complexity results for bin packing

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 11 / 28

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SLIDE 24

Advice Model for Online Bin Packing Problem

Relax “absolutely no knowledge” assumption:

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 12 / 28

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SLIDE 25

Advice Model for Online Bin Packing Problem

Relax “absolutely no knowledge” assumption: Same advice model as previous talk [B¨

  • ckenhauer,Komm,Kr´

aloviˇ c,Kr´ aloviˇ c,M¨

  • mke, 2009]

Algorithms get b(n) bits of advice for sequences of length n:

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 12 / 28

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SLIDE 26

Advice Model for Online Bin Packing Problem

Relax “absolutely no knowledge” assumption: Same advice model as previous talk [B¨

  • ckenhauer,Komm,Kr´

aloviˇ c,Kr´ aloviˇ c,M¨

  • mke, 2009]

Algorithms get b(n) bits of advice for sequences of length n: The advice is generated by an offline oracle.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 12 / 28

slide-27
SLIDE 27

Advice Model for Online Bin Packing Problem

Relax “absolutely no knowledge” assumption: Same advice model as previous talk [B¨

  • ckenhauer,Komm,Kr´

aloviˇ c,Kr´ aloviˇ c,M¨

  • mke, 2009]

Algorithms get b(n) bits of advice for sequences of length n: The advice is generated by an offline oracle. The advice is written on a tape and can be accessed by the online algorithm at any time.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 12 / 28

slide-28
SLIDE 28

Advice Model for Online Bin Packing Problem

Relax “absolutely no knowledge” assumption: Same advice model as previous talk [B¨

  • ckenhauer,Komm,Kr´

aloviˇ c,Kr´ aloviˇ c,M¨

  • mke, 2009]

Algorithms get b(n) bits of advice for sequences of length n: The advice is generated by an offline oracle. The advice is written on a tape and can be accessed by the online algorithm at any time. There are other advice models for bin packing

Original: [Dobrev, Kr´ aloviˇ c, Markou, 2009] Advice with request: [Fraigniaud,Korman,Ros´ en, 2011]

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 12 / 28

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SLIDE 29

Relevant Questions

For a sequence of fixed length How many bits of advice are required (sufficient) to achieve an

  • ptimal solution?

How many bits of advice are sufficient to outperform all online algorithms? How good can the competitive ratio be with advice of linear/sublinear size?

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 13 / 28

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SLIDE 30

Relevant Questions

For a sequence of fixed length How many bits of advice are required (sufficient) to achieve an

  • ptimal solution?

How many bits of advice are sufficient to outperform all online algorithms? How good can the competitive ratio be with advice of linear/sublinear size? Is there useful advice one could reasonably get (without knowing Opt)?

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 13 / 28

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SLIDE 31

Optimal Solution with Advice

How many bits of advice are sufficient to achieve an optimal solution? Advice for each item: index of target bin in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient 1 2 3 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 14 / 28

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SLIDE 32

Optimal Solution with Advice

How many bits of advice are sufficient to achieve an optimal solution? Advice for each item: index of target bin in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient 1 2 3 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 14 / 28

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SLIDE 33

Optimal Solution with Advice

How many bits of advice are sufficient to achieve an optimal solution? Advice for each item: index of target bin in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient 1 2 3 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 14 / 28

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SLIDE 34

Optimal Solution with Advice

How many bits of advice are sufficient to achieve an optimal solution? Advice for each item: index of target bin in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient 2 3 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 14 / 28

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SLIDE 35

Optimal Solution with Advice

How many bits of advice are sufficient to achieve an optimal solution? Advice for each item: index of target bin in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient 3 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 14 / 28

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SLIDE 36

Optimal Solution with Advice

How many bits of advice are sufficient to achieve an optimal solution? Advice for each item: index of target bin in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient 3 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 14 / 28

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SLIDE 37

Optimal Solution with Advice

How many bits of advice are sufficient to achieve an optimal solution? Advice for each item: index of target bin in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 14 / 28

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SLIDE 38

Optimal Solution with Advice

How many bits of advice are sufficient to achieve an optimal solution? Advice for each item: index of target bin in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 14 / 28

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SLIDE 39

Optimal Solution with Advice

In fact, (n − 2 Opt(σ)) log Opt(σ) bits of advice are required to achieve an

  • ptimal packing.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 15 / 28

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SLIDE 40

Optimal Solution with Advice

In fact, (n − 2 Opt(σ)) log Opt(σ) bits of advice are required to achieve an

  • ptimal packing.

Comparison: n⌈log Opt(σ)⌉ bits of advice are sufficient for optimality. (n − 2 Opt(σ)) log Opt(σ) bits of advice are required to guarantee

  • ptimality.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 15 / 28

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SLIDE 41

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

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SLIDE 42

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

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SLIDE 43

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3].

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

slide-44
SLIDE 44

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3]. Algorithm: Reserve a space of size 2/3 for each of them Apply First-Fit for the other items. Advice: 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

slide-45
SLIDE 45

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3]. Algorithm: Reserve a space of size 2/3 for each of them Apply First-Fit for the other items. Advice: 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

slide-46
SLIDE 46

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3]. Algorithm: Reserve a space of size 2/3 for each of them Apply First-Fit for the other items. Advice: 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

slide-47
SLIDE 47

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3]. Algorithm: Reserve a space of size 2/3 for each of them Apply First-Fit for the other items. Advice: 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

slide-48
SLIDE 48

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3]. Algorithm: Reserve a space of size 2/3 for each of them Apply First-Fit for the other items. Advice: 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

slide-49
SLIDE 49

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3]. Algorithm: Reserve a space of size 2/3 for each of them Apply First-Fit for the other items. Advice: 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

slide-50
SLIDE 50

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3]. Algorithm: Reserve a space of size 2/3 for each of them Apply First-Fit for the other items. Advice: 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

slide-51
SLIDE 51

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3]. Algorithm: Reserve a space of size 2/3 for each of them Apply First-Fit for the other items. Advice: 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

slide-52
SLIDE 52

Breaking the Lower Bound — Effectively

Recall: All online algorithms have a competitive ratio of at least 1.54. Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5. Advice: The number of items in range (1/2, 2/3]. Algorithm: Reserve a space of size 2/3 for each of them Apply First-Fit for the other items. Advice: 1

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 16 / 28

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SLIDE 53

Advice of Linear Size

An online algorithm which receives 2 bits of advice per request (plus an additive lower order term).

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 17 / 28

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SLIDE 54

Advice of Linear Size

An online algorithm which receives 2 bits of advice per request (plus an additive lower order term). Achieves a competitive ratio of 4/3 + ε, for any positive value of ε.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 17 / 28

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SLIDE 55

Advice of Linear Size

An online algorithm which receives 2 bits of advice per request (plus an additive lower order term). Achieves a competitive ratio of 4/3 + ε, for any positive value of ε. A variety of bin packing techniques are used in the proof.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 17 / 28

slide-56
SLIDE 56

Advice of Linear Size

An online algorithm which receives 2 bits of advice per request (plus an additive lower order term). Achieves a competitive ratio of 4/3 + ε, for any positive value of ε. A variety of bin packing techniques are used in the proof. Advice depends on Opt’s packing.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 17 / 28

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SLIDE 57

A Lower Bound

A linear amount of advice is required to achieve a competitive ratio better than 9/8. Get a trade-off — better ratio requires more advice

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 18 / 28

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SLIDE 58

A Lower Bound

A linear amount of advice is required to achieve a competitive ratio better than 9/8. Get a trade-off — better ratio requires more advice Reduction order: Binary string guessing problem − → Binary separation problem Binary separation problem − → Bin packing problem

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 18 / 28

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SLIDE 59

Binary String Guessing Problem

Binary string guessing problem (with known history): 2-SGKH [Emek,Fraigniaud,Korman,Ros´ en, 2011] [B¨

  • ckenhauer,Hromkovic,Komm,Krug,Smula,Sprock, 2013]

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 19 / 28

slide-60
SLIDE 60

Binary String Guessing Problem

Binary string guessing problem (with known history): 2-SGKH [Emek,Fraigniaud,Korman,Ros´ en, 2011] [B¨

  • ckenhauer,Hromkovic,Komm,Krug,Smula,Sprock, 2013]

Guess the next bit in a bit string revealed in an online manner

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 19 / 28

slide-61
SLIDE 61

Binary String Guessing Problem

Binary string guessing problem (with known history): 2-SGKH [Emek,Fraigniaud,Korman,Ros´ en, 2011] [B¨

  • ckenhauer,Hromkovic,Komm,Krug,Smula,Sprock, 2013]

Guess the next bit in a bit string revealed in an online manner 0, 1, 0, ?

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 19 / 28

slide-62
SLIDE 62

Binary String Guessing Problem

Binary string guessing problem (with known history): 2-SGKH [Emek,Fraigniaud,Korman,Ros´ en, 2011] [B¨

  • ckenhauer,Hromkovic,Komm,Krug,Smula,Sprock, 2013]

Guess the next bit in a bit string revealed in an online manner 0, 1, 0, ? A linear amount advice is required to correctly guess more than half

  • f the bits.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 19 / 28

slide-63
SLIDE 63

Binary String Guessing Problem

Binary string guessing problem (with known history): 2-SGKH [Emek,Fraigniaud,Korman,Ros´ en, 2011] [B¨

  • ckenhauer,Hromkovic,Komm,Krug,Smula,Sprock, 2013]

Guess the next bit in a bit string revealed in an online manner 0, 1, 0, ? A linear amount advice is required to correctly guess more than half

  • f the bits.

Theorem

On inputs of length n, any deterministic algorithm for 2-SGKH that is guaranteed to guess correctly on more than αn bits, for 1/2 ≤ α < 1, needs to read at least (1 + (1 − α) log(1 − α) + α log(α))n bits of advice. Note: If we assume the number, n0, of 0s is given, we need at least (1 + (1 − α) log(1 − α) + α log(α))n − e(n0) bits of advice, where e(n0) = ⌈log(n0 + 1)⌉ + 2⌈log(⌈log(n0 + 1)⌉ + 1)⌉ + 1 (self-delimiting code).

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 19 / 28

slide-64
SLIDE 64

Binary Separation Problem

Binary separation problem: For a sequence of n1 + n2 items decide whether an item belongs to the n1 smaller items or n2 larger items.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 20 / 28

slide-65
SLIDE 65

Binary Separation Problem

Binary separation problem: For a sequence of n1 + n2 items decide whether an item belongs to the n1 smaller items or n2 larger items. 1

2 (s), 3 4 (l), 5 8 (s), 11 16 (?)

  • Boyar

Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 20 / 28

slide-66
SLIDE 66

Binary Separation Problem

Binary separation problem: For a sequence of n1 + n2 items decide whether an item belongs to the n1 smaller items or n2 larger items. 1

2 (s), 3 4 (l), 5 8 (s), 11 16 (?)

  • Don’t have to choose in [0, 1].

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 20 / 28

slide-67
SLIDE 67

Binary Separation Problem

Binary separation problem: For a sequence of n1 + n2 items decide whether an item belongs to the n1 smaller items or n2 larger items. 1

2 (s), 3 4 (l), 5 8 (s), 11 16 (?)

  • Don’t have to choose in [0, 1].

Don’t have to choose the exact middle value.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 20 / 28

slide-68
SLIDE 68

Reduction from Binary separation to Bin packing

Idea: Create small and large items, so Alg has to decide which is which.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 21 / 28

slide-69
SLIDE 69

Reduction from Binary separation to Bin packing

Idea: Create small and large items, so Alg has to decide which is which. Give n2 items of size 1

2 + ǫ — begin items, B.

Boyar Kamali Larsen L´

  • pez-Ortiz

(1) Trends in Online Algorithms 2014 July 7, 2014 21 / 28

slide-70
SLIDE 70

Reduction from Binary separation to Bin packing

Idea: Create small and large items, so Alg has to decide which is which. Give n2 items of size 1

2 + ǫ — begin items, B.

Alg (and Opt) must put them in separate bins.

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slide-71
SLIDE 71

Reduction from Binary separation to Bin packing

Idea: Create small and large items, so Alg has to decide which is which. Give n2 items of size 1

2 + ǫ — begin items, B.

Alg (and Opt) must put them in separate bins. Give large items, L and small items, S: Opt places large items with begin items. Opt places small items, one per bin. Alg much choose.

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slide-72
SLIDE 72

Reduction from Binary separation to Bin packing

Idea: Create small and large items, so Alg has to decide which is which. Give n2 items of size 1

2 + ǫ — begin items, B.

Alg (and Opt) must put them in separate bins. Give large items, L and small items, S: Opt places large items with begin items. Opt places small items, one per bin. Alg much choose. For each small item of size 1

2 − ǫi,

give an item of size 1

2 + ǫi — matching items, M.

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slide-73
SLIDE 73

Reduction from Binary separation to Bin packing

Idea: Create small and large items, so Alg has to decide which is which. Give n2 items of size 1

2 + ǫ — begin items, B.

Alg (and Opt) must put them in separate bins. Give large items, L and small items, S: Opt places large items with begin items. Opt places small items, one per bin. Alg much choose. For each small item of size 1

2 − ǫi,

give an item of size 1

2 + ǫi — matching items, M.

Opt packs matching items with small items, using n1 + n2 bins.

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slide-74
SLIDE 74

Reduction from Binary separation to Bin packing

Large item + matching item > 1.

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slide-75
SLIDE 75

Reduction from Binary separation to Bin packing

Large item + matching item > 1. A large item may not be with a begin item because bad guess for that item bad guess for small item — no space

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slide-76
SLIDE 76

Reduction from Binary separation to Bin packing

Large item + matching item > 1. A large item may not be with a begin item because bad guess for that item bad guess for small item — no space Let x = max{number bad guesses for small, number bad guesses for large}

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slide-77
SLIDE 77

Reduction from Binary separation to Bin packing

Large item + matching item > 1. A large item may not be with a begin item because bad guess for that item bad guess for small item — no space Let x = max{number bad guesses for small, number bad guesses for large} x large items not paired with begin items.

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slide-78
SLIDE 78

Reduction from Binary separation to Bin packing

Large item + matching item > 1. A large item may not be with a begin item because bad guess for that item bad guess for small item — no space Let x = max{number bad guesses for small, number bad guesses for large} x large items not paired with begin items. At most 2 fit in a bin together.

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slide-79
SLIDE 79

Reduction from Binary separation to Bin packing

Large item + matching item > 1. A large item may not be with a begin item because bad guess for that item bad guess for small item — no space Let x = max{number bad guesses for small, number bad guesses for large} x large items not paired with begin items. At most 2 fit in a bin together. 4 errors in binary separation ⇒ ≥ 1 more bin

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slide-80
SLIDE 80

Reduction from Binary separation to Bin packing

B B B B S S S S L L L L M M M M B B B B L S S M M L S L S L M M Opt Alg

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slide-81
SLIDE 81

Reduction from Binary separation to Bin packing

B B B B S S S S L L L L M M M M B B B B L S S M M L S L S L M M Opt Alg Result: 8 Result: 9

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slide-82
SLIDE 82

Lower bound result for bin packing

Theorem

On inputs of length n, to achieve a competitive ratio of c (1 < c < 9/8), an online algorithm must get at least (1 + (4c − 4) log(4c − 4) + (5 − 4c) log(5 − 4c))n − e(n) bits of advice. Recall that e(n) = ⌈log(n + 1)⌉ + 2⌈log(⌈log(n + 1)⌉ + 1)⌉ + 1. [Renault, Ros´ en, van Stee, 2013] For a fixed competitive ratio, there exists an online algorithm which only needs linear advice: They present an algorithm for online bin packing which is (1 + 3δ)-competitive (or asymptotically (1 + 2δ)-competitive), using s = 1

δ log 2 δ2 + log 2 δ2 + 3 bits of advice per request.

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slide-83
SLIDE 83

Section 3 Open problems

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slide-84
SLIDE 84

Open Problems

Linear advice is needed to be c-competitive, c < 9/8. Linear advice is sufficient for any fixed c. There is a huge gap, though. (2 + o(1))n advice is sufficient to be 4/3 + ǫ-competitive. Can one get a better ratio with so few bits? O(log n) advice is sufficient to be 3/2-competitive. How many bits are required to break the 1.54 lower bound?

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SLIDE 85

Thank you for your attention.

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slide-86
SLIDE 86

Reduction from 2-SGKH to Binary separation

1: small = 0; large = 1 2: repeat 3:

mid = (large − small) / 2

4:

class guess = SeparationAlgorithm.ClassifyThis(mid)

5:

if class guess = “large” then

6:

bit guess = 0

7:

else

8:

bit guess = 1

9:

actual bit = Guess(bit guess) {The actual value is received after guessing (2-SGKH).}

10:

if actual bit = 0 then

11:

large = mid {We let “large” be the correct decision.}

12:

else

13:

small = mid {We let “small” be the correct decision.}

14: until end of sequence

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