Approximation Algorithms Performance Guaranteed

1
Approximation Algorithms(Performance Guaranteed)

• This section on Approximation Algorithms
• Is for your information and enjoyment only.
• Not included in the Final Exam of UIT2201

2
Bin Packing (1-D)

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

3
Approximation Algorithms(Performance Guaranteed)
Bin Packing Problem

1
.5 .7 .5 .2 .4 .2 .5 .1 .6
Optimal Packing
N0 4
4
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
Bin Packing (1-D)

• Approximation Algorithms Not optimal solution,
• but with some performance guarantee
• (eg, no worst than twice the optimal)
• Even though we dont know what the
  optimal solution is!!!

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Next Fit Packing Algorithm
Let a1 a2 .. ? 2 ? ? N 1
N0 ? ? ? ?
a1.. ai gt 1 ai.. aj gt 1 aj.. ak gt
1 al.. am gt 1
. . . . .
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Next Fit Packing Algorithm (simpler proof)
s(B1)s(B2) gt 1 s(B2)s(B3) gt 1
s(BN-1)s(BN) gt 1
Let a1 a2 ? 2 ? gt N 1 2N0 ? 2?
? N 1
. . . . .
2(s(B1)s(B2) s(BN)) gt N 1
8
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)