Online Bin Packing with Advice Joan Boyar 1 , Shahin Kamali 2 , Kim S. Larsen 1 , Alejandro L´ opez-Ortiz 2 March 8, 2014 1 University of Southern Denmark, Denmark 2 University of Waterloo, Canada Online Bin Packing with Advice 1 / 19 �
Problem Statement Bin Packing Problem The input is a set of items of various sizes. Item sizes are in range (0 , 1] E.g., < 9 , 3 , 8 , 5 , 1 , 1 , 3 , 2 , 4 , 2 , 4 , 5 , 5 , 8 , 6 , 4 , 5 > . The goal is to pack these items into a minimum number of bins of uniform capacity. � � � � � � � � � � � � � � � � � Online Bin Packing with Advice 2 / 19 �
Problem Statement Bin Packing Problem In the offline version of the problem, you have access to the whole set at the beginning. E.g., you can sort items (E.g., First Fit Decreasing). Online Bin Packing with Advice 3 / 19 �
Problem Statement Bin Packing Problem In the offline version of the problem, you have access to the whole set at the beginning. E.g., you can sort items (E.g., First Fit Decreasing). The problem is NP-hard [Garey and Johnson, 1979]. First Fit Decreasing has an approximation ratio of 11 / 9 ≈ 1 . 22 [Yue, 1991]. There is an asymptotic PTAS for the problem [Fernandez de la Vega and Lueker, 1981]. Online Bin Packing with Advice 3 / 19 �
Problem Statement Online Bin Packing Problem The input is a sequence of items of various sizes which are revealed in a sequential, online manner. E.g., < 9 , 3 , 8 , 5 , 1 , 1 , 3 , 2 , 4 , 2 , 4 , 5 , 5 , 8 , 6 , 4 , . . . , 5 > . Almost Any Fit algorithms E.g., First Fit and Best Fit algorithms (widely applied). Harmonic class of algorithms Harmonic-based on placing items of almost equal sizes together. Online Bin Packing with Advice 4 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 9 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 9 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 9 8 8 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 5 9 8 8 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 5 5 9 8 8 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 1 1 5 5 9 8 8 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 1 1 5 5 9 8 8 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 1 1 5 5 9 8 8 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 2 1 1 5 5 4 4 4 9 8 8 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 2 1 1 2 2 5 5 4 4 4 9 8 8 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 2 1 1 2 2 5 5 4 4 4 9 8 8 4 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 2 1 1 2 2 5 5 5 5 4 4 4 9 8 8 4 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 2 1 1 2 2 5 5 5 5 4 4 4 9 8 8 5 5 4 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 2 1 1 2 2 5 5 5 5 4 4 4 9 8 8 8 8 5 5 4 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 2 1 1 2 2 5 5 5 5 4 4 4 9 8 8 8 8 6 5 5 4 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 2 1 1 2 2 4 5 5 5 5 4 4 4 9 8 8 8 8 6 5 5 4 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Algorithms First Fit Algorithm Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist. < 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 > 1 1 2 2 1 1 2 2 4 5 5 5 5 4 4 4 9 8 8 8 8 6 5 5 5 5 4 3 3 3 3 Online Bin Packing with Advice 5 / 19 �
Worst Case Analysis Competitive Analysis Compare the performance of an online algorithm with an optimal offline algorithm Opt : Opt knows the whole sequence in the beginning. Competitive ratio of an algorithm A is the maximum ratio between the cost of A and Opt for serving the same sequence. Online Bin Packing with Advice 6 / 19 �
Worst Case Analysis Competitive Analysis Competitive ratio of Best Fit and First Fit are both 1.7 [Johnson et al., 1974]. The best existing online algorithm (Harmonic++) has a competitive ratio of 1.582 [Seiden, 2002]. No algorithm has a competitive ratio less than 1.54 [Vliet, 1992]. Online Bin Packing with Advice 7 / 19 �
Worst Case Analysis Competitive Analysis Competitive ratio of Best Fit and First Fit are both 1.7 [Johnson et al., 1974]. The best existing online algorithm (Harmonic++) has a competitive ratio of 1.582 [Seiden, 2002]. No algorithm has a competitive ratio less than 1.54 [Vliet, 1992]. Recall that offline First Fit Decreasing has an approximation ratio of 1.22. A big gap between quality of online and offline solutions. What about an ‘almost online’ algorithm? Online Bin Packing with Advice 7 / 19 �
Advice Complexity Advice Model for Online Bin Packing Problem Relax ‘absolutely no knowledge’ assumption behind Look-ahead? Online Bin Packing with Advice 8 / 19 �
Advice Complexity Advice Model for Online Bin Packing Problem Relax ‘absolutely no knowledge’ assumption behind Look-ahead? Closed bin packing? (length of sequence is known to the algorithm) Online Bin Packing with Advice 8 / 19 �
Advice Complexity Advice Model for Online Bin Packing Problem Relax ‘absolutely no knowledge’ assumption behind Look-ahead? Closed bin packing? (length of sequence is known to the algorithm) Under the advice model, the online algorithms are provided with b bits of advice: 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 [Renault et al., 2013]. Online Bin Packing with Advice 8 / 19 �
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