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Bin Packing (1-D)

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

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Bin Packing (1-D)

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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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