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Approximation Algorithms （ Performance Guaranteed ）

This section on Approximation Algorithms Is for your information and enjoyment only. Not included in the Final Exam of UIT2201. Approximation Algorithms （ Performance Guaranteed ）. These slides on 1-D bin packing are adapted from slides from Professor C. L. Liu

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Approximation Algorithms （ Performance Guaranteed ）

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  1. This section on Approximation Algorithms Is for your information and enjoyment only. Not included in the Final Exam of UIT2201 Approximation Algorithms（Performance Guaranteed）

  2. These slides on 1-D bin packing are adapted from slides from Professor C. L. Liu (then of Tsing Hua University, Taiwan). Bin Packing (1-D)

  3. .1 .2 .5 .5 .7 .5 .6 .2 .4 Approximation Algorithms（Performance Guaranteed） 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 Approximation Algorithms（Performance Guaranteed） 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 Approximation Algorithms（Performance Guaranteed） .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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