cs546perf

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X. Sun (IIT) CS546 Lecture 5 Page 1

Performance Evaluation of

Parallel Processing

Xian-He Sun

Illinois Institute of Technology

[email protected]

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Outline

• Performance metrics – Speedup

– Efficiency

– Scalability• Examples

• Reading: Kumar – ch 5

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Performance Evaluation(Improving performance is the goal)

• Performance Measurement

– Metric, Parameter

• Performance Prediction – Model, Application-Resource

• Performance Diagnose/Optimization

– Post-execution, Algorithm improvement,

Architecture improvement, State-of-the-art,Scheduling, Resource management/Scheduling

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Parallel Performance Metrics(Run-time is the dominant metric)

• Run-Time (Execution Time)

• Speed: mflops, mips, cpi

• Efficiency: throughput

• Speedup

• Parallel Efficiency

• Scalability: The ability to maintain performance gain when

system and problem size increase• Others: portability, programming ability,etc

TimeExecutionParallelTimeExecutionorUniprocess pS

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Models of Speedup • Speedup

• Scaled Speedup

– Parallel processing gain over sequentialprocessing, where problem size scales up with

computing power (having sufficientworkload/parallelism)

TimeExecutionParallel

TimeExecutionorUniprocess pS

Performance Evaluation of Parallel Processing

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Speedup

• Ts =time for the best serial algorithm

• Tp=time for parallel algorithm using pprocessors

p

s p

T

T S

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Example

Processor 1

time

100

time

1 2 3 4

25 25 25 25 time

1 2 3 4

35 35 35 35

(a) (b) (c)

ationparallelizperfect

,0.4

25

100 pS

10iscostsynchbut

balancingloadperfect

,85.235

100 pS

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Example (cont.)

time

1 2 3 4

30 20 40 10time

1 2 3 4

50 50 50 50

(d) (e)

imbalanceloadbutsynchno

,5.240

100 pS

costsynchandimbalanceload

,0.250

100 pS

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What Is “Good” Speedup?

• Linear speedup:

• Superlinear speedup

• Sub-linear speedup:

pS p

pS p

pS p

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Speedup

p

speedup

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Sources of Parallel Overheads

• Interprocessor communication

• Load imbalance

• Synchronization

• Extra computation

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Degradations of Parallel Processing

Unbalanced Workload

Communication Delay

Overhead Increases with the Ensemble Size

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Degradations of Distributed Computing

Unbalanced Computing Power and Workload

Shared Computing and Communication Resource

Uncertainty, Heterogeneity, and Overhead Increases

with the Ensemble Size

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Causes of Superlinear Speedup

• Cache size increased

• Overhead reduced

• Latency hidden

• Randomized algorithms• Mathematical inefficiency of the serial algorithm

• Higher memory access cost in sequential

processing

• X.H. Sun, and J. Zhu, "Performance Considerations of Shared Virtual Memory Machines,"

IEEE Trans. on Parallel and Distributed Systems, Nov. 1995

http://www.cs.iit.edu/~scs/psfiles/ieee95.pdf

http://www.cs.iit.edu/~scs/psfiles/ieee95.pdf

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• Fixed-Size Speedup (Amdahl’s law)

– Emphasis on turnaround time

– Problem size, W , is fixed


W

W S p

Solvingof TimeParallel

Solvingof TimeorUniprocess

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Amdahl’s Law

• The performance improvement that can be gained bya parallel implementation is limited by the fraction oftime parallelism can actually be used in anapplication

• Let = fraction of program (algorithm) that is serialand cannot be parallelized. For instance: – Loop initialization – Reading/writing to a single disk – Procedure call overhead

• Parallel run time is given by

s p T ) p

α

(αT 1

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Amdahl’s Law

•Amdahl’s law

gives a limit on speedup in terms of

p p

T T

T S

p

T T T

ss

s p

ss p

1

1

)1(

)1(

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Enhanced Amdahl’s Law

pas

T

T T

p

T T

T Speedup

overhead overhead

FS

1

11

1 1

)1(

• To include overhead

• The overhead includes parallelism and interaction

overheads

Amdahl’s law: argument against massively parallel systems

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• Fixed-Size Speedup (Amdahl Law, 67)

Wp

W1

Wp Wp Wp Wp

W1 W1 W1 W1

1 2 3 4 5

Number of Processors (p)

Amount

of

Work

Tp

T1

Tp Tp Tp

T1 T1

Tp

T1 T1

1 2 3 4 5


Elapsed

Time

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Amdahl’s Law

• The speedup that is achievable on p processors is:

• If we assume that the serial fraction is fixed, then the

speedup for infinite processors is limited by 1/

• For example, if =10%, then the maximum speedup is

10, even if we use an infinite number of processors

p

T

T S

p

s p

1

1

1lim p p S

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Comments on Amdahl’s Law

• The Amdahl’s fraction in practice depends on the problem size

n and the number of processors p • An effective parallel algorithm has:

• For such a case, even if one fixes p , we can get linear speedups

by choosing a suitable large problem size

• Scalable speedup

• Practically, the problem size that we can run for a particularproblem is limited by the time and memory of the parallelcomputer

n pn as0),(

n p pn p

p

T

T S

p

s p as

),()1(1

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• Fixed-Time Speedup (Gustafson, 88)

° Emphasis on work finished in a fixed time

° Problem size is scaled from W to W '

° W' : Work finished within the fixed time with parallel

processing

W

W


'Solvingof TimeorUniprocess

'Solvingof TimeParallel

'Solvingof TimeorUniprocess'

W

W S p

W

W '

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Gustafson’s Law (Without Overhead)

a 1-a time

p (1-a)p

ps

s

t t

t

pW

pW W Work

pWork SpeedupFT )1(1()1()(

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• Fixed-Time Speedup (Gustafson)

Wp

W1 Wp

Wp

Wp Wp

W1

W1

W1

W1

1 2 3 4 5


Amount

of

Work

Tp

T1

Tp Tp Tp

T1 T1

Tp

T1 T1

1 2 3 4 5


Elapsed

Time

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Converting ’s between Amdahl’s

and Gustafon’s laws

Based on this observation,

Amdahl’s and Gustafon’s laws

are identical.

p p

pGG

A)1(1)1(

G

G

A

p

).1(

1

1

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Memory Constrained Scaling:

Sun and Ni’s Law • Scale the largest possible solution limited by

the memory space. Or, fix memory usage perprocessor – (ex) N-body problem

• Problem size is scaled from W to W*• W* is the work executed under memorylimitation of a parallel computer

• For simple profile, and G(n) is the increase of

parallel workload as the memory capacityincreases p times.)(* * M pGW

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Sun & Ni’s Law

timein Increase

work in Increase

TimeWork

pTime pWork Speedup MB

)1( / )1(

)( / )(

p pG

pG

TimeWork

pTime pWork Speedup MB

/ )()1(

)()1(

)1( / )1(

)( / )(

a 1-a

p (1-a)G(p)

time

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• Memory-Bounded Speedup (Sun & Ni, 90)

° Emphasis on work finished under current physical

limitation

° Problem size is scaled from W to W *

°

W *

: Work executed under memory limitation withparallel processing

*

**

Solvingof TimeParallel


W

W S p

• X.H. Sun, and L. Ni , "Scalable Problems and Memory-Bounded Speedup,"

Journal of Parallel and Distributed Computing, Vol. 19, pp.27-37, Sept. 1993 (SC90).

http://www.cs.iit.edu/~scs/psfiles/scalable93.pdf




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• Memory-Boundary Speedup (Sun & Ni)

Wp

W1

Wp

Wp

Wp Wp

W1

W1

W1

W1

1 2 3 4 5


Amount

of

Work

Tp

T1

Tp Tp Tp

T1 T1

Tp

T1 T1

1 2 3 4 5


Elapsed

Time

– Work executed under memory limitation

– Hierarchical memory

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Characteristics

• Connection to other scaling models – G(p) = 1, problem constrained scaling

– G(p) = p, time constrained scaling

• With overhead

• G(p) > p, can lead to large increase inexecution time

– (ex) 10K x 10K matrix factorization: 800MB, 1 hr in

uniprocessorwith 1024 processors, 320K x 320K matrix, 32 hrs

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– ScalableMore accurate solution

Sufficient parallelism

Maintain efficiency

– Efficient in parallelcomputingLoad balance

Communication

– Mathematically

effectiveAdaptive

Accuracy

Why Scalable Computing

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• Memory-Bounded Speedup

° Natural for domain decomposition based computing

° Show the potential of parallel processing (In gerneal,

computing requirement increases faster with problem

size than that of communication)

° Impacts extend to architecture design: trade-off of

memory size and computing speed

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Why Scalable Computing (2)

• Appropriate for small machine

– Parallelism overheads begin to dominate benefitsfor larger machines

• Load imbalance

• Communication to computation ratio

– May even achieve slowdowns

– Does not reflect real usage, and inappropriate for

large machine• Can exaggerate benefits of improvements

Small Work

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Why Scalable Computing (3)

• Appropriate for big machine

– Difficult to measure improvement

– May not fit for small machine

• Can’t run

• Thrashing to disk

• Working set doesn’t fit in cache

– Fits at some p , leading to superlinear speedup

Large Work

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Demonstrating Scaling Problems

parallelism

overhead

superlinear

User want to scale problems as machines grow!

Small Ocean problem

On SGI Origin2000

Big equation solver problem

On SGI Origin2000

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How to Scale

• Scaling a machine – Make a machine more powerful

– Machine size• <processor, memory, communication, I/O>

– Scaling a machine in parallel processing• Add more identical nodes

• Problem size – Input configuration

– data set size : the amount of storage required to

run it on a single processor – memory usage : the amount of memory used by

the program

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Two Key Issues in Problem Scaling

• Under what constraints should the problembe scaled?

– Some properties must be fixed as the machinescales

• How should the problem be scaled? – Which parameters?

– How?

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Constraints To Scale

• Two types of constraints – Problem-oriented

• Ex) Time

– Resource-oriented

• Ex) Memory

• Work to scale

– Metric-oriented

• Floating point operation, instructions

– User-oriented

• Easy to change but may difficult to compare

• Ex) particles, rows, transactions

• Difficult cross comparison

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• Speedup

Time ExecutionParallel

Time Executionor UniprocessS p

SpeedSequential

SpeedParallel pS

Rethinking of Speedup

• Why it is called speedup but compare time

• Could we compare speed directly?

• Generalized speedup

• X.H. Sun, and J. Gustafson, "Toward A Better Parallel Performance Metric,"

Parallel Computing, Vol. 17, pp.1093-1109, Dec. 1991.

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Compute : Problem

• Consider parallel algorithm for computing the value of

=3.1415…through the following numericalintegration

dx

x

π

1

0 2

1

421

4

x

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Compute : Sequential Algorithm

computepi(){

h=1.0/n;

sum =0.0;

for (i=0;i<n;i++) {

x=h*(i+0.5); sum=sum+4.0/(1+x*x);

}

pi=h*sum;

}

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Compute : Parallel Algorithm

• Each processor computes on a set of about n/p

points which are allocated to each processor in acyclic manner

• Finally, we assume that the local values of areaccumulated among the p processors under

synchronization

01 2 3

01 2 3

01 2 3

01 2 3 0

1 2 3

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Compute : Parallel Algorithm

computepi(){

id=my_proc_id();

nprocs=number_of_procs():

h=1.0/n;

sum=0.0;

for(i=id;i<n;i=i+nprocs) { x=h*(i+0.5);

sum=sum+4.0/(1+x*x);

}

localpi=sum*h;

use_tree_based_combining_for_critical_section();

pi=pi+localpi; end_critical_section();

}

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Compute : Analysis

• Assume that the computation of is performed over n points

• The sequential algorithm performs 6 operations (twomultiplications, one division, three additions) per points on the x-axis. Hence, for n points, the number of operations executed in thesequential algorithm is:

nT s 6

for (i=0;i<n;i++) {

x=h*(i+0.5);

sum=sum+4.0/(1+x*x);

}

3 additions

2 multiplications

1 division

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Compute : Analysis

• The parallel algorithm uses p processors with staticinterleaved scheduling. Each processor computes ona set of m points which are allocated to each processin a cyclic manner

• The expression for m is given by if p

does not exactly divide n. The runtime for the parallelalgorithm for the parallel computation of the localvalues of is:

1 p

nm

00 )66(*6 t p

nt mT p

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Compute : Analysis

• The accumulation of the local values of

using atree-based combining can be optimally performed inlog2(p) steps

• The total runtime for the parallel algorithm for thecomputation of including the parallel computationand the combining is:

• The speedup of the parallel algorithm is:

))(log()66(*6 000 c p t t pt p

nt mT

) / 1)(log(66

6

0t t p p

nn

T T S

c p

s p

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Compute : Analysis

• The Amdahl’s fraction for this parallel algorithm canbe determined by rewriting the previous equation as:

• Hence, the Amdahl’s fraction (n,p) is:

• The parallel algorithm is effective because:

),()1(1

6

)log(1

pn p

pS

n

p pc

n

p

pS p p

)1(6

)log(

)1(),(

pn

p pc

n p

p pn

pn pn fixedforas0),(

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Finite Differences: Problem

• Consider a finite difference iterative method appliedto a 2D grid where:

t

ji

t

ji

t

ji

t

ji

t

ji

t

ji X X X X X X ,,1,11,1,

1

, )1()(

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Finite Differences: Serial Algorithm

finitediff(){

for (t=0;t<T;t++) {

for (i=0;i<n;i++) {

for (j=0;j<n;j++) {

x[i,j]=w_1*(x[i,j-1]+x[i,j+1]+x[i-1,j]+x[i+1,j]+w_2*x[i,j]; }

}

}

}

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Finite Differences: Parallel Algorithm

• Each processor computes on a sub-grid ofpoints

• Synch between processors after every iterationensures correct values being used for subsequent

iterations

p

n

p

n

p

n

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Finite Differences: Parallel Algorithm

finitediff(){

row_id=my_processor_row_id();

col_id=my_processor_col_id();

p=numbre_of_processors();

sp=sqrt(p);

rows=cols=ceil(n/sp); row_start=row_id*rows;

col_start=col_id*cols;

for (t=0;t<T;t++) {

for (i=row_start;i<min(row_start+rows,n);i++) {

for (j=col_start;j<min(col_start+cols,n);j++) {

x[i,j]=w_1*(x[i,j-1]+x[i,j+1]+x[i-1,j]+x[i+1,j]+w_2*x[i,j]; }

barrier();

}

}

}

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Finite Differences:Analysis

• The sequential algorithm performs 6 operations(2multiplications, 4 additions) every iteration per point on the grid.Hence, for an n*n grid and T iterations, the number of operationsexecuted in the sequential algorithm is:

0

2

6 t nT s

x[i,j]=w_1*(x[i,j-1]+x[i,j+1]+x[i-1,j]+x[i+1,j]+w_2*x[i,j];

2 multiplications

4 additions

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• The parallel algorithm uses p processors with staticblockwise scheduling. Each processor computes onan m*m sub-grid allocated to each processor in ablockwise manner

• The expression for m is given by Theruntime for the parallel algorithm is:

p

nm

0

2

0

2 )(66 t p

nt mT p

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• The barrier synch needed for each iteration can be optimally

performed in log(p) steps

• The total runtime for the parallel algorithm for the computationis:

• The speedup of the parallel algorithm is:

))(log(6))(log()(66 00

2

00

2

0

2

cc p t t pt p

n

t t pt p

n

t mT

) / 1)(log(6

6

0

2

2

t t p p

nn

T T S

c p

s p

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• The Amdahl’s fraction for this parallel algorithm can be

determined by rewriting the previous equation as:

• Hence, the Amdahl’s fraction (n.p) is:

• We finally note that

• Hence, the parallel algorithm is effective

),()1(1

6

)log(1

2

pn p

pS

n

p pc

pS p p

26)1(

)log(),(

n p

p pc pn

pfixedforas0),( n pn

E i S l

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Equation Solver

A[i,j] = 0.2 * (A[i, j] + A[i, j-1] + A[i-1, j] + a[i, j+1] + a[i+1, j])

n

nprocedure solve (A)

…

while(!done) do

diff = 0;

for i = 1 to n do

for j = 1 to n do

temp = A[i, j];

A[i, j] = … diff += abs(A[i,j] – temp);

end for

end for

if (diff/(n*n) < TOL) then done =1 ;

end whileend procedure

W kl d

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Workloads

• Basic properties – Memory requirement : O(n2)

– Computational complexity : O(n3), assuming the number ofiterations to converge to be O(n)

• Assume speedups equal to # of p

• Grid size – Fixed-size : fixed

– Fixed-time :

– Memory-bound :

n pk k pn 333

n pk k pn 22

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Memory Requirement of Equation Solver

3

2232 )(

p

n

p

pn

p

k

Fixed-time: 33

k pn

Fixed-size :

Memory-bound : pn 2

,

p

n2

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Time Complexity of Equation Solver

Fixed-time:

Fixed-size:

Memory-bound:

22k pn 33 )( pnk

Sequential time complexity

,

p

n3

3n

pn p

pn 33

)(

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Concurrency

Fixed-time:

Fixed-size :

Memory-bound: 22 k pn

2

n

Concurrency is proportional to the number of grid points

33k pn 3 22232 )( pn pnk

,

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Communication to Computation Ratio

n

p

p

n

p

n

p

n

CCR 22

2

1

Fixed-time :

Fixed-size : Memory-bound :

n p

p

pn

p

k

p

k p

k

CCR6

2322

2

)(11

n

p

pn

p

k

p

k

p

k

CCR1

)(

11

222

2

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Scalability

• The Need for New Metrics

• Comparison of performances with different workload

• Availability of massively parallel processing

• Scalability

Ability to maintain parallel processing gain when both

problem size and system size increase

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Parallel Efficiency

• The achieved fraction of total potential

parallel processing gain – Assuming linear speedup p is ideal case

• The ability to maintain efficiency whenproblem size increase

pS E p

p

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Maintain Efficiency

• Efficiency of adding n numbers in parallel

– For an efficiency of 0.80 on 4 procs, n=64



Efficiency for Various Data Sizes

0

0.2

0.4

0.6

0.8

1

1 4 8 16 32

number of processors

E f f i c i e n c y

n=64

n=192

n=320

n=512

E=1/(1+2plogp/n)

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• Ideally Scalable

T (m p, m W ) = T ( p, W )

– T: execution time

– W: work executed

– P: number of processors used

– m: scale up m times – work: flop count based on the best practical

serial algorithm

• Fact:

T (m

p, m

W ) = T ( p, W )if and only if

The Average Unit Speed Is Fixed

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– Definition:

The average unit speed is the achieved speed divided by

the number of processors

– Definition (Isospeed Scalability):An algorithm-machine combination is scalable if the

achieved average unit speed can remain constant with

increasing numbers of processors, provided the problem

size is increased proportionally

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• Isospeed Scalability (Sun & Rover, 91)

– W: work executed when p processors are employed

– W': work executed when p' > p processors are employed

to maintain the average speed

– Ideal case

– Scalability in terms of time

'

')',(

W p

W p p p yScalabilit

,'

' p

W pW

processors'on'work withtime

processorsonwork withtime

'',

' pW

pW

W T

W T p p

p

p

1)',( p p

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• Isospeed Scalability (Sun & Rover)

– W: work executed when p processors are employed – W': work executed when p' > p processors are employed

to maintain the average speed

– Ideal case'

'

)',( W p

W p

p p yScalabilit

,'

' p

W pW

1)',( p p

• X. H. Sun, and D. Rover, "Scalability of Parallel Algorithm-Machine Combinations,"

IEEE Trans. on Parallel and Distributed Systems, May, 1994 (Ames TR91)

http://www.cs.iit.edu/~scs/psfiles/scalability94.pdf




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The Relation of Scalability and Time

• More scalable leads to smaller time – Better initial run-time and higher scalability lead to

superior run-time

– Same initial run-time and same scalability lead to

same scaled performance – Superior initial performance may not last long if

scalability is low

• Range Comparison

• X.H. Sun, "Scalability Versus Execution Time in Scalable Systems,"

Journal of Parallel and Distributed Computing, Vol. 62, No. 2, pp. 173-192, Feb 2002.

Range Comparison Via Performance Crossing Point



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Range Comparison Via Performance Crossing Point

Assume Program I is oz times slower than program 2 at the initial state

Begin (Range Comparison)

p' = p;

Repeat

p' = p' + 1;

Compute the scalability of program 1 (p,p');

Compute the scalability of program 2 (p,p') ;

Until ( (p,p') > (p,p') or p' = the limit of ensemble size)

If (p,p') > (p,p') Then

p is the smallest scaled crossing point;

program 2 is superior at any ensemble size p†, p p† < p'

Else program 2 is superior at any ensemble size p†, p p† p’

End {if}

End {Range Comparison}

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• Range Comparison

Influence of Communication Speed Influence of Computing Speed

• X.H. Sun, M. Pantano, and Thomas Fahringer, "Integrated Range Comparison for Data-Parallel

Compilation Systems," IEEE Trans. on Parallel and Distributed Processing, May 1999.

http://www.cs.iit.edu/~scs/SCALA/range.pdf







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The SCALA (SCALability Analyzer) System

• Design Goals

– Predict performance

– Support program optimization

– Estimate the influence of hardware variations• Uniqueness

– Designed to be integrated into advanced compilersystems

– Based on scalability analysis

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• Vienna Fortran Compilation System

– A data-parallel restructuring compilation system

– Consists of a parallelizing compiler for VF/HPFand tools for program analysis and restructuring

– Under a major upgrade for HPF2

• Performance prediction is crucial forappropriate program restructuring

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The Structure of SCALA

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Prototype Implementation • Automatic range comparison for different data distributions

• The P 3 T static performance estimator

• Test cases: Jacobi and Redblack

No Crossing Point Have Crossing Point

Summary

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Summary

• Relation between Iso-speed scalability and iso-efficiency scalability – Both measure the ability to maintain parallel efficiency

defined as

– Where iso-efficiency’s speedup is the traditional speedupdefined as

– Iso-speed’s speedup is the generalized speedup defined as

– If the the sequential execution speed is independent ofproblem size, iso-speed and iso-efficiency is equivalent

– Due to memory hierarchy, sequential execution performancevaries largely with problem size

p

S E

p

p

SpeedSequential

SpeedParallel pS


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Summary

• Predict the sequential execution performancebecomes a major task of SCALA due to advancedmemory hierarchy

– Memory-LogP model is introduced for data access cost

• New challenge in distributed computing• Generalized iso-speed scalability

• Generalized performance tool: GHS

• K. Cameron and X.-H. Sun, "Quantifying Locality Effect in Data Access Delay: Memory logP,"

Proc. of 2003 IEEE IPDPS 2003, Nice, France, April, 2003.

• X.-H. Sun and M. Wu, "Grid Harvest Service: A System for Long-Term, Application-Level Task

Scheduling," Proc. of 2003 IEEE IPDPS 2003, Nice, France, April, 2003.

http://www.cs.iit.edu/~scs/psfiles/KSipdps03.PDF

http://www.cs.iit.edu/~scs/psfiles/SWipdps03.PDF








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