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

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

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

4. Next Fit Packing Algorithm
Bin Packing Problem .5 .1 .2 .5 .6
N0 = 4 .7 .5 .4 .2
Next Fit Packing Algorithm .7 .2 .2 .1 .6 .5 .5 .5 .4
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 don’t know what the optimal solution is!!!

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

7. Next Fit Packing Algorithm (simpler proof)… a1
ai-1 . ai
aj-1
ak-1 aj ak
am-1 al am
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. First Fit Packing Algorithm
Next Fit Packing Algorithm .7 .2 .2 .1 .6 .5 .5 .5 .4
First Fit Packing Algorithm .5 .1 .2 .7 .2 .6 .5 .5 .4
N = 5
(Proof omitted)