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1-D Bin Packing: Algorithms and Optimization Techniques

Understand bin packing problem, optimal and approximation algorithms, and practical implementations for efficient packing. Learn from professor C. L. Liu's insightful slides on packing algorithms and strategies.

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1-D Bin Packing: Algorithms and Optimization Techniques

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  1. These slides on 1-D bin packing are adapted from slides from Professor C. L. Liu former President (1998-2002) Tsing Hua University, Hsinchu, Taiwan). Bin Packing (1-D)

  2. .1 .2 .5 .5 .7 .5 .6 .2 .4 Bin Packing (1-D) The bins; (capacity 1) Bin Packing Problem 1 …… .5 .7 .5 .2 .4 .2 .5 .1 .6 Items to be packed

  3. .1 .2 .5 .5 .7 .5 .6 .2 .4 Bin Packing (1-D) Bin Packing Problem 1 …… .5 .7 .5 .2 .4 .2 .5 .1 .6 Optimal Packing N0= 4

  4. .4 .2 .2 .5 .1 .2 .5 .7 .5 .5 .5 .4 .7 .5 .6 .6 .2 Next Fit Packing Algorithm Bin Packing Problem .5 .7 .5 .2 .4 .2 .5 .1 .6 .1 N0= 4 Next Fit Packing Algorithm N= 6

  5. Approximation Algorithms: Not optimal solution, but with some performance guarantee (eg, no worst than twice the optimal) Even though we don’t know what the optimal solution is!!! Bin Packing (1-D)

  6. aj-1 ak-1 ai-1 am-1 … . . . . . . . . . . . . . . . . . . aj ak am a1 ai al Next Fit Packing Algorithm Let a1+ a2 + …….. =  2   N – 1 N0    ≈ a1+……..+ ai > 1 ai+……..+ aj > 1 aj+……..+ ak > 1 al+……..+ am > 1 . . . . .

  7. aj-1 ak-1 ai-1 am-1 … . . . . . . . . . . . . . . . . . . aj ak am a1 ai al Next Fit Packing Algorithm (simpler proof) s(B1)+s(B2)> 1 s(B2)+s(B3)> 1 … … … … … s(BN-1)+s(BN)> 1 Let a1+ a2+ … =  2 > N – 1 2N0  2 N – 1 . . . . . 2( s(B1)+s(B2)+…+ s(BN) ) > N – 1

  8. .4 .2 .2 .1 .1 .2 .2 .7 .7 .5 .5 .5 .5 .5 .5 .4 .6 .6 First Fit PackingAlgorithm .5 .7 .5 .2 .4 .2 .5 .1 .6 Next Fit Packing Algorithm First Fit Packing Algorithm N= 5 (Proof omitted)

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