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On-line adaptative
parallel prefix computation
Jean-Louis Roch, Daouda Traore, Julien Bernard
INRIA-CNRS Moais team - LIG Grenoble, France
Contents
I. Motivation II. Work-stealing scheduling of parallel algorithms III. Processor-oblivious parallel prefix computation
EUROPAR’2006 - Dresden, Germany - 2006, August 29th,
• Prefix problem : • input : a0, a1, …, an • output : 1, …, n with
Parallel prefix on fixed architecture
• Tight lower bound on p identical processors:Optimal time Tp = 2n / (p+1) but performs 2.n.p/(p+1) ops
[Nicolau&al. 1996]
Parallel requires twice more operations thansequential !!
performs only n operations
• Sequential algorithm : • for ([0] = a[0], i = 1 ; i <= n; i++ ) [ i ] = [ i – 1 ] * a [ i ] ;
Critical time = 2. log n but performs 2.n ops
[Ladner-Fisher-81]
• Fine grain optimal parallel algorithm :
Dynamic architecture : non-fixed number of resources, variable speeds
eg: grid, … but not only: SMP server in multi-users mode
The problemTo design a single algorithm that computes efficiently prefix( a ) on
an arbitrary dynamic architecture
Sequentialalgorithm
parallelP=2
parallelP=100
parallelP=max
...
Multi-user SMP server GridHeterogeneous network
?Which algorithm to choose ?
… …
- Model of heterogeneous processors with changing speed [Bender&al 02]
=> i(t) = instantaneous speed of processor i at time t (in #operations * per second ) Assumption : max(t) < constant . min(t)
Def: ave = average speed per processor for a computation with duration T
- Theorem 2 : Lower bound for the time of prefix computation on p processors with changing speeds :
Sketch of the proof: - extension of the lower bound on p identical processors [Faith82]
- based on the analysis on the number of performed operations.
Lower bound for prefix on processors with changing speeds
Changing speeds and work-stealing• Workstealing schedule on-line adapts to processors availability
and speeds [Bender-02]
• Principle of work-stealing= “greedy” schedule but distributed and randomized
• Each processor manages locally the tasks it creates• When idle, a processor steals the oldest ready task on a remote -non idle-
victim processor (randomly chosen)
«Depth »
W = #ops on a critical path
(parallel time on resources)
« Work »
W1= #total
operations performed
[Bender-Rabin02]
Work-stealing and adaptation
«Depth »
W = #ops on a critical path
(parallel time on resources)
« Work »
W1= #total
operations performed
• Interest: if W1 fixed and W small, near-optimal adaptative schedulewith good probability on p processors with average speeds ave
• Moreover : #steals = #task migrations < p.W [Blumofe 98 Narlikar 01 Bender 02]
• But lower bounds for prefix : • Minimal work W1 = n W = n
• Minimal depth W < 2 log n W1 > 2n
• With work-stealing, how to reach the lower bound ?
• General approach: by coupling two algorithms :• a sequential algorithm with optimal number of operations Ws • and a fine grain parallel algorithm with minimal critical time W but
parallel work >> Ws
• Folk technique : parallel, than sequential • Parallel algorithm until a certain « grain »; then use the sequential one• Drawback with changing speeds :
• Either too much idle processors or too much operations
• Work-preserving speed-up technique [Bini-Pan94] sequential, then parallel Cascading [Jaja92] =Careful interplay of both algorithms to build one with
both W small and W1 = O( Wseq ) • Use the work-optimal sequential algorithm to reduce the size • Then use the time-optimal parallel algorithm to decrease the time
Drawback : sequential at coarse grain and parallel at fine grain
How to get both work W1 and depth W small?
Alternative : concurrently sequential and parallel
SeqCompute
Extract_parLastPartComputation
SeqCompute
Based on the work-stealing and the Work-first principle : Execute always a sequential algorithm to reduce parallelism overhead
use parallel algorithm only if a processor becomes idle (ie workstealing) by extracting parallelism from a sequential computation (ie adaptive granularity)
Hypothesis : two algorithms : • - 1 sequential : SeqCompute
- 1 parallel : LastPartComputation : at any time, it is possible to extract parallelism from the remaining computations of the sequential algorithm
– Self-adaptive granularity based on work-stealing
Alternative : concurrently sequential and parallel
SeqCompute
SeqCompute
preempt
Alternative : concurrently sequential and parallel
SeqCompute
SeqCompute
merge/jump
complete
Seq
Parallel
Sequential
0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12
Work-stealer 1
MainSeq.
Work-stealer 2
Adaptive Prefix on 3 processors
1
Steal request
Parallel
Sequential
Adaptive Prefix on 3 processors
0 a1 a2 a3 a4
Work-stealer 1
MainSeq. 1
Work-stealer 2
a5 a6 a7 a8
a9 a10 a11 a127
3
Steal request
2
6 i=a5*…*ai
Parallel
Sequential
Adaptive Prefix on 3 processors
0 a1 a2 a3 a4
Work-stealer 1
MainSeq. 1
Work-stealer 2
a5 a6 a7 a8
7
3 42
6 i=a5*…*ai
a9 a10 a11 a12
8
4
Preempt
10 i=a9*…*ai
8
8
Parallel
Sequential
Adaptive Prefix on 3 processors
0 a1 a2 a3 a4 8
Work-stealer 1
MainSeq. 1
Work-stealer 2
a5 a6 a7 a8
7
3 42
6 i=a5*…*ai
a9 a10 a11 a12
85
10 i=a9*…*ai9
6
11
8
Preempt 11
118
Parallel
Sequential
Adaptive Prefix on 3 processors
0 a1 a2 a3 a4 8 11 a12
Work-stealer 1
MainSeq. 1
Work-stealer 2
a5 a6 a7 a8
7
3 42
6 i=a5*…*ai
a9 a10 a11 a12
85
10 i=a9*…*ai9
6
11
12
10
7
118
Parallel
Sequential
Adaptive Prefix on 3 processors
0 a1 a2 a3 a4 8 11 a12
Work-stealer 1
MainSeq. 1
Work-stealer 2
a5 a6 a7 a8
7
3 42
6 i=a5*…*ai
a9 a10 a11 a12
85
10 i=a9*…*ai9
6
11
12
10
7
118
Implicit critical path on the sequential process
• Theorem 3: Execution time
• Sketch of the proof : Analysis of the operations performed by :
– The sequential main performs S operations on one processor
– The (p-1) work-stealers perform X = 2(n-S) operations with depth log X– Each non constant time task can potentially be splitted (variable speeds)
The coupling ensures both algorithms complete simultaneously Ts = Tp - O(log X)=> enables to bound the whole number X of operations performedand the overhead of parallelism = (S+X) - #ops_optimal
Analysis of the algorithm
Lower bound
Adaptive prefix : experiments1
Single-user context : processor-adaptive prefix achieves near-optimal performance : - close to the lower bound both on 1 proc and on p processors
- Less sensitive to system overhead : even better than the theoretically “optimal” off-line parallel algorithm on p processors :
Optimal off-line on p procs
Adaptive
Prefix sum of 8.106 double on a SMP 8 procs (IA64 1.5GHz/ linux)T
ime
(s)
#processors
Pure sequential
Single user context
Adaptive prefix : experiments 2
Multi-user context : Additional external charge: (9-p) additional external dummy processes are concurrently executed Processor-adaptive prefix computation is always the fastest 15% benefit over a parallel algorithm for p processors with off-line schedule,
Multi-user context : Additional external charge: (9-p) additional external dummy processes are concurrently executed Processor-adaptive prefix computation is always the fastest 15% benefit over a parallel algorithm for p processors with off-line schedule,
External charge (9-p external processes)
Off-line parallel algorithm for p processors
Adaptive
Prefix sum of 8.106 double on a SMP 8 procs (IA64 1.5GHz/ linux)
Tim
e (s
)
#processors
Multi-user context :
Conclusion
The interplay of an on-line parallel algorithm directed by work-stealing schedule is useful for the design of processor-oblivious algorithms
Application to prefix computation : - theoretically reaches the lower bound on heterogeneous processors
with changing speeds - practically, achieves near-optimal performances on multi-user SMPs
Generic adaptive scheme to implement parallel algorithms with provable performance
- work in progress : parallel 3D reconstruction [oct-tree scheme with deadline constraint]
Thank you !
QuickTime™ et undécompresseur codec YUV420
sont requis pour visionner cette image.
Interactive Distributed Simulation[B Raffin &E Boyer]
- 5 cameras, - 6 PCs
3D-reconstruction+ simulation+ rendering
->Adaptive scheme to maximize 3D-reconstruction precision within fixed timestamp[L Suares, B Raffin, JL Roch]
The Prefix race: sequential/parallel fixed/ adaptive
Race between 9 algorithms (44 processes) on an octo-SMPSMP
0 5 10 15 20 25
1
2
3
4
5
6
7
8
9
Execution time (seconds)
Série1
Adaptative 8 proc.
Parallel 8 proc.
Parallel 7 proc.
Parallel 6 proc.Parallel 5 proc.
Parallel 4 proc.
Parallel 3 proc.
Parallel 2 proc.
Sequential
On each of the 10 executions, adaptive completes first
Adaptive prefix : some experiments
Single user contextAdaptive is equivalent to:
- sequential on 1 proc - optimal parallel-2 proc. on 2 processors - … - optimal parallel-8 proc. on 8 processors
Multi-user contextAdaptive is the fastest15% benefit over a static grain algorithm
Multi-user contextAdaptive is the fastest15% benefit over a static grain algorithm
External charge
Parallel
Adaptive
Parallel
Adaptive
Prefix of 10000 elements on a SMP 8 procs (IA64 / linux)
#processorsT
ime
(s)
Tim
e (s
)
#processors
With * = double sum ( r[i]=r[i-1] + x[i] )
Single user Processors with variable speeds
Remark for n=4.096.000 doubles :- “pure” sequential : 0,20 s- minimal ”grain” = 100 doubles : 0.26s on 1 proc
and 0.175 on 2 procs (close to lower bound)
Finest “grain” limited to 1 page = 16384 octets = 2048 double