Improved results for a
memory allocation problem
Rob van Stee
University of Karlsruhe
Germany
Leah Epstein
University of Haifa
Israel
WADS 2007
WAOA 2007
Background
Allocate memory to parallel processors
Goals:
–
Each processor has sufficient memory
–
Not too much memory is wasted
Background
Problem
: memory requirement may
vary over time
If each processor can access only
one
memory
, this is inefficient
If
all
processors share a single
memory, there is heavy contention
Suggested model
Chung et al. [SPAA 2004] suggest a
new architecture
Each memory may be accessed by at
most
two
processors
This avoids the mentioned problems
Bin packing
They abstract this problem as a bin
packing problem
Bins = memories
Items = memory requirements of
processors
Thus, items may be larger than 1
Bin packing
Items may be split and distributed
among more than one bin
Condition: each bin contains at most
two
parts of items
Corresponds to: each memory may be
accessed by at most two processors
We also consider the model where
each bin may contain
k
parts of items
Bin packing
Processor
1
2
3
4
5
Requirement
Bins
Assignment
1,2
3,4
2
3,4,5
1
Notes
There may be many bins which are
not full
An item may be split more than once
No upper bound on how often each
item is split
Results
Chung et al. [SPAA 2004]:
–
(k=2) NP

hardness
–
(k=2) 3/2

approximation
Epstein & vS [WADS 2007]:
–
(k=2) 7/5

approximation
–
For general k:
Next Fit is (2

1/k)

approximation
NP

hardness in the strong sense
Results
Epstein & vS [WAOA 2007]:
A polynomial time approximation
scheme (PTAS) for every (constant) k
–
For every constant
>0, a polynomial time
algorithm that uses at most 1+
times the
minimal number of bins
–
at most
OPT additional bins of size 1
Results
Epstein & vS [WAOA 2007]:
A dual approximation scheme for every
(constant) k
–
For every constant
>0, a polynomial time
algorithm that uses
bins of size 1+
–
Larger bins, but no additional bins
Next Fit
in this context
If current bin contains
k
parts of items
or
is fulll
, open a new bin
Put as much as possible of the current
item in the current bin
Open additional bin(s) if needed (for
large items)
Next Fit: lower bound
One item of size
Mk

1
–
Next Fit uses
Mk

1
bins
M
(
k

1)
k
very small items
–
Next Fit puts
k
items in one bin, uses
M
(
k

1)
bins
Mk

1
M
(
k

1)
Next Fit: lower bound
Optimal solution needs only
Mk
bins
Ratio: (
M
(2
k

1)

1) /
Mk
2
–
1/
k
Mk

1
M
(
k

1)
Mk
Next Fit: upper bound
Block
= set of full bins, followed by a
bin with k items
Each bin >1 in a block was opened
because the previous one was full
Weights
Weight
of item with size s =
s
/k
There are at least
s
parts of item
with size s in any packing
OPT is at least the
total weight of all
the items
Next Fit: upper bound
Consider last bin in a block (
not the
last block
)
Has
k
items, so at least
k

1
items of
weight
1/
k
(these items are not split!)
Block has 1 bin, then
k
such items
Else, consider all items in block
except
the
k

1 last items
Next Fit: upper bound
S
= all items in block except
the
k

1
last items
How many items are in
S
? Unknown!
What is the minimal
weight
of
S
?
Weight = , minimized for
single
item
Next Fit: upper bound
S
= all items in block except
the
k

1
last items
For block of size
b
, total size of items
in S is strictly more than
b

1
Total weight is then at least
b
/
k
Weight in block at least
b
/
k
+(
k

1)/
k
Note: also true for block of size 1
The last block
Only one bin: weight at least 1/
k
(has
at least one item)
b
>1 bins: weight at least
b
/
k
Let size of block
i
be
b
i
, then Next Fit
uses bins
We know that OPT is at least the total
weight
Next Fit: upper bound
Let m be the
amount of blocks
Also, OPT > NF
–
m
(in each block, at
most one bin is
not completely full
)
Next Fit: upper bound
Combining this we find
23
This talk
We only discuss the algorithms for
k=2
General properties
PTAS
Dual PTAS
Representation of
solutions
A solution can be seen as a graph
Vertices = items
An edge between two items =
Parts of these two items share a bin
–
A loop means that a bin contains just
one part of an item
We can assume a forest (with loops)
–
Cycles can be easily removed without
changing the cost (Chung et al.,2004).
Example
Items:
Graph
Removal of cycle
Additional properties
There exists an optimal packing forest
where all items of size in (0,1/2] are leaves
or singletons
If the forest representation is minimal (no
tree can be split) then a packing is implied
from the tree
–
Can be done iteratively
–
Singletons are packed into separate bins
–
A leaf is packed with a maximum amount of its
neighbor, and removed from the forest
(Different) example
1.5
0.4
0.2
1
0.9
0.2
0.3
Packing process
0.8
0.4
0.6
1
0.9
0.3
0.1
0.2
PTAS for k=2
We would like to use standard
methods such as
–
Linear grouping and rounding
–
Enumeration of patterns
But
–
Patterns are trees
–
Items sizes are not bounded from above
New tricks
Optimal packing is adapted in order to
make it possible to imitate it
Trees are “truncated”
–
Partitioned into smaller trees
Large items are truncated
–
Partitioned into several items that are still
large, but bounded
The last two adaptations increase the
number of bins by a factor of 1+
Trying all trees
Type of a tree is (j,E)
–
j is number of edges
–
E is set of j

1 edges (so that it is a tree)
Pattern = type plus vector of length j
–
Element i in the vector indicates
group
of
node i
Constant
number of patterns
Only question: how many of each?
Dual PTAS for k=2
(briefly)
We use an integer parameter K, based
on
.
Items can be rounded down to a sizes
of the form i/K
We round some non

zero sizes into
zero!
By scaling, we get items of
integer
sizes
, and a bin size
K
Cutting
There is no reason to cut items at
non

integer point
For each item size (at most K), we
define cutting patterns
We “guess” how to cut the input items
–
Using enumeration
Finding a solution
We use a layered graph to find an optimal
packing for each option
–
If the cutting was applied, the problem becomes
much easier
–
However, we did not cut items larger than the
bin size
–
We cut a piece off each one of them, but do not
decide on all cutting points
The best (smallest number of bins) option is
chosen
Increasing items back to original sizes
increases bins slightly
Open problems
PTAS, dual PTAS for non

constant k
(FPTAS or dual FPTAS are most likely
impossible since the problem is
strongly NP

hard for every constant
value of k)
Generalize 7/5

approximation for
larger
k
7/5

approximation
We show an approximation algorithm
for k=2 (at most two parts of items
per bin allowed)
First it sorts the items
Then it packs them, starting with the
most difficult ones: size between ½
and 1
We use
Next Fit
as a subroutine
Step 1: Sorting
Items of size at most ½ are
small
–
Sort them in order of
increasing
size
Items larger than 1 are
large
–
Sort them in order of
decreasing
size
Other items are
medium

sized
–
Sort them in order of
decreasing
size
–
We pack these items first
Step 2: medium

sized
items
Pack
medium

sized
items one
by one:
–
If current item fits in one bin
with the smallest unpacked
small item, pack them together
–
Else, pack current item
together with two
largest
unpacked small items in two
bins
Step 3: all small items are
packed
We do this until we run out of either
the medium

sized or the small items
If all small items are packed, use Next
Fit on rest, start with medium

sized
items
Step 4: some small items
remain
Pack each small item in
its own bin
Pack large items into these bins using
Next Fit
Still use ordering from before:
–
Largest large items first
–
Smallest small items first
Step 5: small items left
We have now run out of either the
large or the small items
If bins with small items remain,
repack
these items two per bin
Step 6: large items left
Pack remaining large items using Next
Fit
Upper bound
We use the same weight definition as
before: weight of item with size s is
We know that OPT is at least the total
weight
How much weight do we pack in a
bin?
Amount of weight packed
Step 2:
Pack medium

sized items one by one:
–
If current item fits in one bin with the
smallest unpacked small item, pack them
together.
Weight ½+½ =
1
–
Else, pack current item together with two
largest unpacked small items in two bins.
Three items of weight ½ in two bins:
¾
Amount of weight packed
Step 3:
If all small items are packed, use Next Fit on
rest, start with medium

sized items
We assume
many
small items exist
Then, this step
does not occur
Amount of weight packed
Step 4:
Pack each small item in
its own bin
Pack large items into these bins using Next
Fit
Still use ordering from before:
–
Largest large items first
–
Smallest small items first
Amount of weight packed
Consider a large item which is packed
into
g
bins
It is packed together with
g
small
items (total weight
g
/2)
The large item has size strictly more
than (
g

1)/2, so weight at least
g
/4
Average weight per bin =
3/4
Amount of weight packed
Step 5:
We have now run out of either the
large or the small items
If bins with small items remain, repack
these items two per bin
Weight: ½ + ½ =
1
Packed weight: summary
Step 2: ½+½=1, or 3 times ½ in two
bins, average ¾
Step 4: average ¾
Step 5: ½+½=1
Gives upper bound of 4/3! (Since OPT is
at least total weight)
However…
There may be only one small item left
at the end of Step 5
–
„If bins with small items remain, repack
these items
two per bin
“
We may run out of small items in Step
2, so that Step 3 is executed
We may run out of small items in Step
4, so that Step 6 is executed
Upper bound
It turns out that these cases are the
most difficult
Proof is a number of pages
Eventually we find an upper bound of
7/5
in all cases
Open questions
Improve this approximation ratio
–
We can give a PTAS, but this is much less
efficient already for epsilon = 2/5
Generalize this algorithm for larger k
–
We have the upper bound of Next Fit,
which is 2

1/k, and a PTAS
–
Can we improve on Next Fit efficiently?
Thank you!
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