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Transcript of Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 1 High-Contrast Algorithm Behavior...
![Page 1: Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 1 High-Contrast Algorithm Behavior Observation, Hypothesis, and Experimental Design Matthias.](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb35503460f94bba6e9/html5/thumbnails/1.jpg)
Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 1
High-Contrast Algorithm BehaviorObservation, Hypothesis, and Experimental Design
Matthias F. Stallmann and Franc Brglez
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 2
Summary
Two competing algorithms:
int-dual - all-integer dual simplex
old idea, only limited success
cplex - industrial strength optimizer
Restricted domain (logic minimization).
Extreme differences across instances:
time out for one algorithm vs. seconds for the other.
Want to “profile” the instances, or at least explain the
differences.
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 3
Outline
Observation in a limited domain Initial experiments: design Initial experiments: results A hypothesis More experimental evidence Theoretical explanation Summary and future work
algorithmstreatedas black boxes
int-dual cplex
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 4
a typical problem instance
x1 + x4 >= 1
x2 + x3 + x4 >= 1
x1 + x2 + x4 >= 1
x1 – x4 >= 0
x2 – x4 >= 0
x3 – x4 >= 0
minimize x1 + x2 + x3 + x4
subject to x1 x2 x3 x4
constraint matrix pictorial representation
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 5
A pattern?
alg N/33 med. mean stdev
Int-dual
29 15.9 N/a N/a
cplex 0 N/a N/a N/a
Benchmark: e64.b (logic synthesis) -- a “canonical” picture
# succ. completions, one hour
original + 32 random row/col permutations
dedicated fast processor
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 6
A pattern: Less pronounced
alg N/33 med. mean stdev
Int-dual 33 14.6 21.1 15.1
cplex 33 3.9 6.0 6.1
Benchmark: rot.b
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 7
A pattern: Not much of one
alg N/33 med. mean stdev
Int-dual 0 N/a N/a N/a
cplex 33 3.6 3.5 0.9
Benchmark: max1024
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 8
The pattern
int-dual cplexBenchmark: e64.b Benchmark: max1024
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 9
Process for generating pictures
crossing minimization
(moves nonzeros close to the diagonal)may still look “random”
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 10
this is as good as it gets...
This is max1024 after crossing minimization
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 11
Conversion to block structure does not improve performance
crossing minimization
int-dual performance:
not necessarily
improved
underlying
block structure...
random
permutations
}better
aggregate
performance
for int-dual
e64 as posted: 27.5 after minimization: 55.0
recall that median = 15.9
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 12
Basic idea
More underlying structure => better for int-dual Less underlying structure => better for cplex
Why is this interesting?e.g., optimum for e64.b was unknown prior to int-dual
What do we mean by structure? More precise (and testable) hypothesis?
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 13
Outline Observation in a limited domain Initial experiments: design Initial experiments: results A hypothesis More experimental evidence Theoretical explanation Summary and future work
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 14
From observation to controlled experiment
Behavior “in the wild”:
algorithms on industrial
instances
Behavior “in captivity”:
algorithms on carefully
designed instances
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 15
Instances with pure blocks
Create K copies of a small instance, arrange on diagonal,
permute rows and columns randomly 32 times to obtain
a class of equivalent instances.
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 16
Construction of blocks with added rows
xxxx
two pure blocks
add new row(s) with random nonzeros below both blocks
xxxx
repeat
xxxx
apply same process to
“new” blocks
Create 32 instances with different random choices
of the nonzeros; randomly permute each one
row factor
= # of rows
added each step
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 17
Blocks used in results reported here
st15 - instance based on Steiner triples (15 vars, 35 const)
maincont - small logic synthesis instances (61 vars, 50 const)
f51m - slightly larger logic synthesis instance (175 vars, 187 const)
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 18
Outline Observation in a limited domain Initial experiments: design Initial experiments: results A hypothesis More experimental evidence Theoretical explanation Summary and future work
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 19
Details and reporting
Class of 32 random related instances Intel(R) Xeon(TM) CPU 3.20GHz, 2 GB memory,
2048 KB cache Report runtime only (seconds, time out = 600) Here we report median only, but note number of
successful completions, if not 32
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 20
results for st15, pure blocks
blocks 1 2 4 8 16
int-dual 0.1 0.3 1.0 4.1 29.8
cplex 0.02 0.9 > 10 hours
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 21
results for maincont, pure blocks
blocks 1 2 4 8 16 32
int-dual < 0.1 < 0.1 < 0.1 0.1 0.3 1.6
cplex < 0.1 < 0.1 < 0.1 0.2 0.5 > 600
29/32 completions
> 600 between 24 and 28 blocks
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 22
results for f51m, pure blocksblocks 1 2 4 8 16
int-dual 0.1 0.4 2.1 12.4 65.7
cplex 0.3 59.0 > 600 > 600 > 600
int-dual is roughly between O(n^2) and O(n^3)
cplex is wildly exponential
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 23
added rows?
blank = neither dominates, X = neither finishes in 1 hour
I = int-dual dominates, C = cplex dominates
next page...
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 24
results for maincont, added rows
row fact. 1 2 3 4 5 6 7 8
N/32 27 14 5 1
id 0.1 0.1 0.2 0.7 2.7 >600 >600 >600
cplex 0.2 0.3 0.4 1.4 9.6 20.9 71.1 90.0
N/32 31 23 31 31 30 27
8 blocks with increasing row factor
Wild distributions!mean = 59.9
stdev = 140.5
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 25
this is not going well
not much hope of a statisticallymeaningful comparison betweenint-dual and cplex.
too many variables: will anything hold up
not clear what to measure
domain already very limited
...
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 26
all is not lost: let’s stick with what we know
int-dual performs well on pure blocks
cplex performs badly on pure blocks
int-dual gets worse as more random nonzeros are added(has been observed for columns, too)
id 0.1 0.1 0.2 0.7 2.7 >600 >600 >600
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 27
Outline Observation in a limited domain Initial experiments: design Initial experiments: results A hypothesis More experimental evidence Theoretical explanation Summary and future work
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 28
A viable hypothesis
int-dual runtime is predictable on pure-blocks instances and grows polynomially with increasing number of blocks
cplex runtime on pure-blocks instances becomes erratic as soon as an instance-dependent threshold is reached
int-dual runtime increases and/or becomes erratic as the number of nonzeros added to a pure-blocks instance is increasederratic = “heavy tail” distribution (stdev >> mean >> median)
or time outs and overflows
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 29
why this is good
not dependent on comparison between the algorithms
not dependent on statistics extreme contrasts are what we observed
initially -- so we need to deal with them appears to be true in a variety of
circumstances even though instances are artificial, may
explain why int-dual has limited use
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Outline Observation in a limited domain Initial experiments: design Initial experiments: results A hypothesis More experimental evidence Theoretical explanation Summary and future work
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 31
f51m, added rows
row fact. 1 2 3 4 5 6 7 8
N/32 23 22 19 12 15 10 7
id 3.8 72.6 52.6 210.2 >600 >600 >600 >600
cplex >600 >600 260.2 70.2 17.4 11.0 9.3 6.4
1 9 23 31 N/32
distributions that are not erratic
3 blocks, increasing row factor
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Outline Observation in a limited domain Initial experiments: design Initial experiments: results A hypothesis More experimental evidence Theoretical explanation Summary and future work
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 33
All-Integer Dual Simplex
vertex cover of a triangle
x2 x3
x1
0 1 1 1
–1 –1 –1 0
–1 –1 0 –1
–1 0 –1 –1
x1 x3x2
e1
e2
e3
x1 + x2 >= 1
min x1 + x2 + x3
–e1= –x1 – x2
tableau
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 34
a pivot step
0 1 1 1
–1 –1 –1 0
–1 –1 0 –1
–1 0 –1 –1
x1 x3x2
e1
e2
e3
choose constraint to satisfy
choose variable to do the trick
–1 1 0 1
1 –1 1 0
0 –1 1 –1
–1 0 –1 –1
x3x2
e2
e3
swap them
now x1=1, e1=0
e1
x1
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 35
example (continued)
–1 1 0 1
1 –1 1 0
0 –1 1 –1
–1 0 –1 –1
–1 1 0 1
0 –1 1 –1
–1 –1 1 –2
1 0 –1 1
and another, but this one involves division by –2 ...
another pivot...
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 36
oops: solution is not integral
–1 1 0 1
0 –1 1 –1
–1 –1 1 –2
1 0 –1 1
–3/2 1/2 –1/2 –1/2
1/2 –1/2 3/2 1/2
1/2 1/2 –1/2 1
1/2 –1/2 –3/2 1/2
and we get fractions. The LP solution is
x1 = 1/2, x2 = 1/2, x3 = 1/2
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 37
Maintaining an integer tableau
–1 1 0 1
0 –1 1 –1
–1 –1 1 –2
1 0 –1 1
–1 –1 0 –1
x2
x1
x3e1
ct
e2
e3
–2 0 0 1
1 0 1 –1
1 1 1 –2
0 –1 –1 1
1 1 0 –1
x2
x1
cte1
x3
e2
e3
add a new constraint that must be met by all integer solutions
x1 = x3 = 1, cost = 2
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 38
int-dual: another look
0 1 1 1
–1 –1 –1 0
–1 –1 0 –1
–1 0 –1 –1
–1 1 0 1
1 –1 1 0
0 –1 1 –1
–1 0 –1 –1
0 in pivot row means no change in pivot column
0 in pivot column means no change in pivot row
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 39
int-dual on “pure blocks”
1 1 1 0 0 0
1 1 1 0 0 0
1 1 1 0 0 0
0 0 0 2 2 2
0 0 0 2 2 2
0 0 0 2 2 2
2 0 0 2 0 2
0 1 1 0 1 0
2 0 0 2 0 2
0 1 1 0 1 0
0 1 1 0 1 0
2 0 0 2 0 2
this phenomenon is not
affected by random permutation
element of second blocknot changed by pivotin first block
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why does int-dual work well on pure blocks?
because each block is solved independently
progress is made in whatever block contains the pivot
One out of three parts (of the hypothesis) explained
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Stallmann/Brglez, High contrast… ExpCS, 2007/06/14, San Diego 41
part 2: cplex does badly on pure blocks
branching tree for
one block
tree for
two blocks
Note: this is a gross oversimplification
cplex is a “branch and bound” algorithm
x
y
0
0
1
1
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Summary
Some observations about algorithm behavior on industrial benchmarks
Hypothesis specializes observations to a limited, verifiable domain
Extensive experiments consistent with hypothesis
Partial theoretical explanation for hypothesis Aside: int-dual dates to 1960’s; experiments in
early 1970’s deemed it good for small instances of set cover (but impractical otherwise)
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Future work better understanding of both algorithms to
explain what’s going on “pure” branch and bound instead of cplex experiments to explore variations on variables
(how to add rows, columns too, add –1 entries) observations in other domains (particular where
there has been success with int-dual) other approaches to claiming one algorithm
better than another where distributions are crazy
can we learn from outliers, i.e., easy and hard instances of a class
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Thanks...
Xiao Yu Li, now at Amazon, started us down this path
Eric Sills, NCSU High Performance Computing Center, helped with hardware and software availability
Source code for software and
(eventually) instance classes and results athttp://people.engr.ncsu.edu/mfms/Software