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)
The bins;
(capacity 1)
Bin Packing Problem
……
1
.5
.7
.5
.5
.5
.2
.4
.2
.4
.2
.2
.7
Items to be packed
.5
.5
.1
.6
.1
.6
Bin Packing (1-D)
Bin Packing Problem
……
1
.5
.7
.5
.2
.4
.2
.5
.1
.6
Optimal Packing
.1
.5
.5
.2
.7
.5
.4
.6
.2
N0 = 4
Next Fit Packing
Algorithm
Bin Packing Problem
.5
.7
.5
.2
.4
.2
.5
.1
.6
.1
.5
.2
.5
.7
.5
.4
.6
N0 = 4
.2
Next Fit Packing Algorithm
.2
.5
Ν
2
Ν
0
.7
.5
.2
.1
.4
.5
.6
N=6
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!!!
Next Fit Packing
Algorithm
ai-1
aj-1
..
.
..
.
a1
ai
a1+……..+ ai > 1
ai+……..+ aj > 1
aj+……..+ ak > 1
..
..
.
al+……..+ am > 1
ak-1
..
.
aj
…
..
.
ak
am-1
..
.
al
..
.
am
Let a1+ a2 + …….. = 
2  N–1
N
N

1
N0   

2
2
N
2
N
0
Next Fit Packing
Algorithm (simpler proof)
ai-1
aj-1
..
.
..
.
a1
ai
ak-1
..
.
aj
..
.
…
ak
am-1
..
.
al
..
.
am
s(B1)+s(B2) > 1
s(B2)+s(B3) > 1
Let a1+ a2 +. … = 
2 > N–
..
1..
2N0  2  N – 1
… … … … …
s(BN-1)+s(BN) > 1
2( s(B1)+s(B2)+…+ s(BN) ) > N – 1
First Fit Packing
Algorithm
.5
.7
.5
.2
.4
.2
.5
.1
.6
Next Fit Packing Algorithm
.2
.5
.7
.5
.2
.1
.4
.5
.5
.6
.6
First Fit Packing Algorithm
.5
.1
.2
.2
.5
Ν
 1.7
Ν
0
.7
.4
N=5
(Proof omitted)