Best-fit bin packing in
O(n log n) time
• The search tree will contain one element for
each bin which is in use and has non-zero
available capacity
• Notice! It is possible that for two or more
bins to have the same available capacity
Best-fit bin packing in
O(n log n) time
• Example
• Suppose that when object i is to be packed,
there are nine bins (a through i) in use.
• Available capacity of these bins are 1, 3, 12,
6, 8, 1, 20, 6 and 5, respectively.
• The nine bins may be stored in a binary tree
with duplicates, using as key the available
capacity of a bin.
Best-fit bin packing in
O(n log n) time
• A possible binary search tree.
h
6
b
c
3
12
a
i
1
5
d
g
20
6
e
f
1
8
Best-fit bin packing in
O(n log n) time
• Object I that is to be packed requires s[i] =
4.
• Root h fits, proceeds
to the left subtree.
h
6
b
c
3
12
a
i
1
5
d
g
20
6
e
f
1
8
Best-fit bin packing in
O(n log n) time
• Object I that is to be packed requires s[i] =
4.
h
6
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
8
Best-fit bin packing in
O(n log n) time
• Object I that is to be packed requires s[i] =
4.
• b fails, proceeds to
the right subtree of b.
h
6
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
8
Best-fit bin packing in
O(n log n) time
• Object I that is to be packed requires s[i] =
4.
h
6
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
8
Best-fit bin packing in
O(n log n) time
• Object I that is to be packed requires s[i] =
4.
h
• i fits, proceeds to the
6
left subtree of i.
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
• i doesn’t have any left child,
it is the best candidate.
8
Best-fit bin packing in
O(n log n) time
• Another example, object I that is to be
packed requires s[i] = 7.
h
• h fails, proceeds to
6
the right subtree of h.
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
8
Best-fit bin packing in
O(n log n) time
• Another example, object I that is to be
packed requires s[i] = 7.
h
6
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
8
Best-fit bin packing in
O(n log n) time
• Another example, object I that is to be
packed requires s[i] = 7.
h
• c fits, proceeds to the
6
left subtree of c.
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
8
Best-fit bin packing in
O(n log n) time
• Another example, object I that is to be
packed requires s[i] = 7.
h
6
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
8
Best-fit bin packing in
O(n log n) time
• Another example, object I that is to be
packed requires s[i] = 7.
h
• d fails, proceeds to
6
the right subtree of d.
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
8
Best-fit bin packing in
O(n log n) time
• Another example, object I that is to be
packed requires s[i] = 7.
h
6
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
8
Best-fit bin packing in
O(n log n) time
• Another example, object I that is to be
packed requires s[i] = 7.
h
• e fits, proceeds to the
6
left subtree of e.
b
c
3
12
a
i
1
5
d
g
20
6
f
e
1
• e has no left child, so e is the 8
best candidate.
Best-fit bin packing in
O(n log n) time
• When we find the best bin, we can delete it
from the search tree, reduce its capacity by
s[i], and reinsert it (unless its remaining
capacity is zero).
• If we do not find a bin with enough
capacity, we can start a new bin.