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Lecture 5: Part 1
Performance Laws:
Speedup and Scalability
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Sequential Execution Time
Execution time = Seconds = Instructions x Cycles x Seconds
(Te) Program Program Instruction Cycle
Execution time = Seconds = Instructions x Cycles x Seconds
(Te) Program Program Instruction Cycle
500 MHz CPU: 500 x 106 clock cycles/secMy program consists of operations:
addition: 500 x 106 (1 cycle per inst)mul: 180 x 106 (3 cycles/inst)div: 120 x 106 (2 cycles/inst)data move: 300 x 106 (1 cycle/inst)
Expected execution time: 2 ns x (500x1+180x3+120x2+300x1)x 106
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Basic Performance Metrics
MIPS = (instructions/second)x 10-6
MFLOPS = (floating point ops/second)x 10-6
CPI = Average cycles per instruction Throughput: number of results per second Workload: W, number of Ops. required to
complete the program Speed: W/TE
Speedup (S)= Te / Timprove
Efficiency (using P processors) = Speedup / P
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Part I: Speedup
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Speedup
General concepts of Speedup in parallel computing: How much faster an application runs on parallel
computer ? What benefits derive from the use of parallelism?
General agreement : speedup = serial time/parallel time
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Speedup
Fixed-Workload Speedup (Amdahl’s Law) Fixed-Time Speedup (Gustafson’s Law) Fixed-Memory Speedup (Sun and Ni)
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Amdahl's Law
Speedup due to enhancement E:
ExTime w/o E Speedup(E) = ------------- ExTime w/ E
Suppose that enhancement E accelerates a fraction F of the task by a factor S, and the remainder of the task is unaffected
ExTime: execution time
1/S
F F/S
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Amdahl’s Law
ExTimenew = ExTimeold x (1 - Fractionenhanced) + Fractionenhanced
Speedupoverall =ExTimeold
ExTimenew
Speedupenhanced
=
1
(1 - Fractionenhanced) + Fractionenhanced
Speedupenhanced
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Amdahl’s Law
Floating point instructions improved to run 2X; but only 10% of actual instructions are FP
Speedupoverall =
ExTimenew =
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Amdahl’s Law
Floating point instructions improved to run 2X; but only 10% of actual instructions are FP
Speedupoverall = 1
0.95= 1.053
ExTimenew = ExTimeold x (0.9 + .1/2) = 0.95 x ExTimeold
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Amdahl’s Law
Some applications need real-time response Amdahl’s law shows the upper bound of the
achievable speedup for a given problem size. Limit on the achievable speedup:
W: total workload
of W must be executed sequentially
Sp = W / ( W+(1- )(W/P))
= P / (1+(P-1) ) 1 / as P
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Fixed-Time Speedup
Gustafson’s Law (1988) Scaling the problem size along with the increase of
machine size within the same execution time Scaling for higher accuracy Parameters:
W: workload done by a single node W’ = W+(1- )WP = work done by P nodes
Sp = ( W+(1- )WP) / W
= +(1-)P Speedup is linear function of P
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Fixed-Memory Speedup
Sun and Ni’s Law: Memory bounding (1993) To solve the largest possible problem, limited
only by the available memory space. As P increases, use up all the increased memory
by scaling the problem size also See Kai’s book (Section 3.6.3)
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More on Speedup
Can you measure the sequential time? What if the memory is too small ?
Is the sequential machine using the same processor as the one in parallel machine ?
Can the sequential time be shorter than the parallel time ?
Is speedup really important ? Or only the execution time is important ?
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More on Speedup Speedup larger than “P” (Superlinear) ?
What if the disk swapping effects happen in sequential execution?
More caches/memory used in the parallel machines (PxC, PxM)
Both sequential and parallel machines use the same OS/compiler ?
Is the data distribution/collection time included in the parallel time? Some parallel machines provide parallel I/O !!
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More on Speedup
What if we use different algorithms for sequential and parallel solutions Take a look at the parallel sorting assignment.
What is the maximum speedup for parallel sorting?
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Definitions of Speedup
Diverse definitions of serial and parallel execution times [Sahni:1996] Relative Real Absolute Asymptotic Asymptotic relative
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Parameters Used in Speedup Definitions I = problem instance P = number of processors Q = parallel program n = size of I
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Relative Speedup
serial time : execution time of the parallel program on a single node of the parallel computer. (mpirun -np 1)
Relative speedup (I,P) = time to solve I using program Q and 1 processor ÷ time to solve I using Q program and P processors
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Relative Speedup
Depends on the characteristics of the instance I being solved as well as the size P of the parallel computer
Same OS, same node architecture, using same compiler
Extra overheads in serial time (distribution/collection).
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Real Speedup
Parallel time vs. the fastest serial algorithm or program running on a single node of the parallel computer
Real speedup (I,P) = time to solve I using best serial program and 1 processor ÷ time to solve I using Q program and P processors
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Problems on Real Speedup The fastest algorithm might not be
known. No single algorithm might be fastest in
all instances for some applications In practice, we use the runtime of the
most frequently used sequential algorithm.
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Absolute Speedup
Parallel time vs. the fastest sequential algorithm run on the fastest serial computer
Absolute speedup (I,P) = time to solve I using best serial program and 1 fastest processor ÷ time to solve I using Q program and P processors
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Absolute Speedup
Can also use the sequential algorithm most often used in practice.
Time-variant: researchers keep designing new algorithms
Speedup could be less than 1
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Asymptotic Real Speedup
Compares the execution time of the best serial algorithm for a particular problem with the asymptotic complexity of the parallel algorithm
Assuming the the parallel computer has all the processors it can use
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Asymptotic Real Speedup
Asymptotic real speedup (n) = asymptotic complexity of best serial algorithm ÷ asymptotic complexity of Q using as many processors as needed
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Asymptotic Real Speedup
For problem such as sorting where the asymptotic complexity is not uniquely characterized by the instance size n, the worst-case complexity is used -- O(n2), O(n log n)
Unbounded number of processors.
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Asymptotic Relative Speedup
Uses the asymptotic time complexity of the parallel algorithm when run on a single processor.
Asymptotic relative speedup = asymptotic complexity of Q using 1 processor ÷ asymptotic complexity of Q using as many processors as needed
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Matrix Multiplication (n X n) Serial time: O(n3) Parallel time (on n3/log n processors
hypercube): O(log n) time Asymptotic Relative Speedup: O(n3/log n)
Others are measured Speedup Relative Real Absolute
Asymptotic Relative Speedup
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Part II: Scalability
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Scalability
Algorithmic Scalability: the available parallelism increases at
least linearly with problem size. Architectural (Size) Scalability:
the architecture continues to yield the same performance per processor, as the number of processors is increased and as the problem size is increased.
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Architectural Scalability
Architecturalscalability
Interconnection Network: latency/bandwidth
I/O performance
OS support(lock, processessynchronizationoverheads)
Processor SpeedNo. of issues, depthof pipelining
Memory Subsystem:cache/memory speed
Performance grows in all aspects
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Discussion of Architectural Scalability: bus-based SMPs
Bus (single set of wires) : fixed bandwidth shared by all processors (P) Bandwidth accessible by each processor decreases as P
increases Physical constraints: fixed number of slots Electrical constraints:
bus loading, wire length determine frequency power
Cost scaling: X (bus is the core) OS scaling : scheduling, locking, I/O
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Scalable Interconnection Network
Bandwidth (P)? Latency (P)? Cost (P)? More bandwidth,
smaller L greater cost SGI XL: 1.5 M HK$
(1.2 GB/s) ALR SMP: 0.1 M HK$
(533 MB/s)
Network (bus ? multistage,mesh, torus)
PE PE PE PE
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Bottom Line
There is NO “truly scalable” machine Goal is to design for a given “range of scale”
one (instruction level) two to tens (SMP):
Top-end: Sun Enterprise 1000 : 64 processors, COMPACQ 64 Alpha processors
tens to a few thousands (Distributed-Memory Multicomputers: T3E, SP2, Paragon, ...)
tens of thousands (ASCI machines: cluster of SMPs) Techniques at one scale may not be cost effect at another
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Bottom Line
Even though we can meet the engineering requirements to physically scale over the range of interest, we must ensure that the communication and synchronization operations required to support target programming models also scale and are cost effective.