Le modèle BSP B ulk- S ynchronous P arallel

1. Frédéric Gava Le modèle BSP Bulk-Synchronous Parallel

2. Background Implicit Explicit BSP Automatic parallelization skeletons Data-parallelism Parallel extensions Concurrent programming Parallel programming

3. The BSP model Unit of synchronization P/M P/M P/M P/M P/M Network BSP architecture: Characterized by: • pNumber of processors • rProcessors speed • LGlobal synchronization • gPhase of communication (1 word at most sent of received by each processor)

4. Model of execution wi Super-step i ghi L wi+1 Super-step i+1 ghi+1 L Beginning of the super-step i Local computing on each processor Global (collective) communications between processors Global synchronization : exchanged data available for the next super-step Cost(i) = (max0x<p wxi) + hig + L

5. Exemple d’une machine BSP

6. Modèle de coût • Coût(programme)=somme des coûts des super-étapes • BSP computation: scalable, portable, predictable • BSP algorithm design: minimising W (temps de calculs), H (communications), S (nombre de super-étapes)‏ • Coût(programme) = W + g*H+ S*L • g et L sont calculables (benchmark) d’où possibilité de prédiction • Main principles: • Load-balancing minimises W • data locality minimises H • coarse granularity minimises S • In genral, data locality good, network locality bad! • Typically, problem size n>>>p (slackness)‏ • Input/output distribution even, but otherwise arbitrary

7. A libertarian model • No master : • Homogeneous power of the nodes • Global (collective) decision procedure instead • No god : • Confluence (no divine intervention) • Cost predictable • Scalable performances • Practiced but confined

8. Advantages and drawbacks • Advantages : • Allows cost prediction and deadlock free • Structuring execution and thus bulk-sending ; it can be very efficient (sending one file of 1000 bytes performs better than sending 1000 file of 1 byte) in many architectures (multi-cores, clusters, etc.) • Abstract architecture = portable • … ? • Drawbacks : • Some algorithmic patterns don’t feet well in the BSP model : pipeline etc. • Some problem are difficult (impossible) to feet to a Coarse-grain execution model (fine-grained parallelism) • Abstract architecture = don’t take care of some efficient possibilities of some architecture (cluster of multi-core, grid) and thus need other libraries or model of execution • … ?

9. Example : broadcast Direct broadcast (one super-step): 0 1 2 BSP cost = png + L Broadcast with2 super-steps: BSP cost = 2ng + 2L

10. Algorithmes BSP • Matrices : multiplication, inversion, décomposition, algèbre linéaire, etc. • Matrices creuses : idem. • Graphes : plus court chemin, décomposition, etc. • Géométrie : diagramme de Voronoi, intersection de polygones, etc. • FFT : Fast Fournier Transformation • Recherches de motifs • Etc.

11. Parallel prefixes On a processor On another processor • If we suppose associative operator (+) • a+(b+c)=(a+b)+c or better • a+(b+(c+d))=(a+b) + (c+d) • Example :

12. Parallel Prefixes • Classical log(p) super-steps method : 0 1 2 3 Cost = log(p) × ( Time(op)+Size(d)×g+L)

13. Parallel Prefixes • Divide-and-conquer method : 0 1 2 3

14. Our parallel machine • Cluster of PCs • Pentium IV 2.8 Ghz • 512 Mb RAM • A front-end Pentium IV 2.8 Ghz, 512 Mb RAM • Gigabit Ethernet cards and switch, • Ubuntu 7.04 as OS

15. Our BSP Parameters ‘g’

16. Our BSP Parameters ‘L’

17. How to read bench • There are many manners to publish benchs : • Tables • Graphics • The goal is to say « it is a good parallel method, see my benchs » but it is often easy to arrange the presentation of the graphics to hide the problems • Using graphics (from the simple to hide to the hardest) : • Increase size of data and see for some number of processors • Increase number of processors to a typical size of data • Acceleration, i.e, Time(seq)/Time(par) • Efficienty , i.e, Acceleration/Number of processors • Increase number of processors and size of the data

18. Increase number of processors

19. Acceleration

20. Efficienty

21. Increase data and processors

22. Super-linear acceleration ? • Better than theoretical acceleration. Possible if data feet more on caches memories than in the RAM due to a small number of data on each processor. • Why the impact of caches ? Mainly, each processor has a little of memory call cache. Access to this memory is (all most) twice faster that RAM accesses. • Take for example, multiplication of matrices

23. “Fast” multiplication • A straight-forward C implementation of res=mult(A,B) (of size N*N) can look like this : for (i = 0; i < N; ++i) for (j = 0; j < N; ++j) for (k = 0; k < N; ++k) res[i][j] += a[i][k] * b[k][j]; • Considerer the following equation : where “T” is the transposition of matrix “b”

24. “Fast” multiplication • One can implement this equation in C as : double tmp[N][N]; for (i = 0; i < N; ++i) for (j = 0; j < N; ++j) tmp[i][j] = b[j][i]; for (i = 0; i < N; ++i) for (j = 0; j < N; ++j) for (k = 0; k < N; ++k) res[i][j] += a[i][k] * tmp[j][k]; where tmp is the transpose of “b” • This new multiplication if 2 time fasters. With other caches optimisations, one can have a 64 faster programs without modifying really the algorithm.

25. More complicated examples

26. N-body problem

27. Presentation • We have a set of body • coordinate in 2D or 3D • point masse • The classic N-body problem is to calculate the gravitational energy of N point masses that is : • Quadratique complexity… • In practice, N is very big and sometime, it is impossible to keep the set in the main memory

28. Parallel methods Cost of the systolic method : • Each processor has a sub-part of the original set • Parallel method one each processsor : • compute local interactions • compute interactions with other point masses • parallel prefixes of the local interactions • For 2) simple parallel methods : • using a total exchange of the sub-sets • using a systolic loop

29. Systolic loop 0 1 2 3

30. Benchs and BSP predictions

31. Benchs and BSP predictions

32. Benchs and BSP predictions

33. Parallel methods • There exist many better algorithms than this one • Especially, considering when computing all interactions is not needed (distancing molecules) • One classic algorithm is to divide the space into-subspace, and computed recursively the n-body on each sub-space (so have sub-sub-spaces) and only consider, interactions between these sub-spaces. Stop the recursion when there are at most two molecules in the sub-space • That introduces n*log(n) computations

34. Sieve of Eratosthenes

35. Presentation • Classic : find the prime number by enumeration • Pure functional implementation using list • Complexity : n×log(n)/log(log(n)) • We used : • elim:int listintint list which deletes from a list all the integers multiple of the given parameter • final elim:int listint listint list iterates elim • seq_generate:intintint list which returns the list of integers between 2 bounds • select:intint listint list which gives the first prime numbers of a list.

36. Parallel methods 11,12,13,14,15 16,17,18,19,20 21,22,23,24,25 • Simple Parallel methods : • using a kind of scan • using a direct sieve • using a recursive one • Different partitions of data • per block (for scan) : • cyclic distribution : 11,14,17,20,23 12,15,18,21,24 13,16,19,22,25

37. Scan version • Method using a scan : • Each processor computes a local sieve (the processor 0 contains thus the first prime numbers) • then our scan is applied and we eliminate on processor i the integers that are multiple of integers of processors i−1, i−2, etc. • Cost : as a scan (logarithmic)

38. Direct version • Method : • each processor computes a local sieve • then integers that are less to are globally exchanged and a new sieve is applied to this list of integers (thus giving prime numbers) • each processor eliminates, in its own list, integers that are multiples of this first primes

39. Inductive version • Recursive method by induction over n : • We suppose that the inductive step gives the th first primes • we perform a total exchange on them to eliminates the non-primes. • End of this induction comes from the BSP cost: we end when n is small enough so that the sequential methods is faster than the parallel one • Cost :

40. Benchs and BSP predictions

41. Benchs and BSP predictions

42. Benchs and BSP predictions

43. Benchs and BSP predictions

44. Parallel sample sorting

45. Presentation • Each processor has listed set of data (array, list, etc.) • The goal is that : • data on each processor are ordored. • data on processor i are smaller than data on processor i+1 • good balancing • Parallel sorting is not very efficient due to too many communications • But usefull and more efficient than gather all the data in one processor and then sort them

46. Tri Parallèle BSP

47. Tiskin’s Sampling Sort Local sort 1,2,7,11,14,16,20 4,6,9,12,13,18,21 3,5,8,10,15,17,19 Select first samples (p+1 elements with at last first and last ones) 1,2,7,11,14,16,20 4,6,9,12,13,18,21 3,5,8,10,15,17,19 Total exchange of first sample 1,7,14,20,4,9,13,21,3,8,15,19 1,7,14,20,4,9,13,21,3,8,15,19 1,7,14,etc. Local sort of samples (each processor) 1,3,4,7,8,9,13,14,15,19,20,21 1,3,4,7,8,9,13,14,15,19,20,21 1,3,4,7,etc. 0 1 2 1,11,16,7,14,2,20 18,9,13,21,6,12,4 15,5,19,3,17,8,10

48. Tiskin’s Sampling Sort Select second samples (p+1 elements with at last first and last ones) 1,3,4,7,8,9,13,14,15,19,20,21 1,3,4,7,etc. 1,3,4,7,8,9,13,14,15,19,20,21 Interval for processor 0 Interval for processor 1 Interval for processor 2 1,2,7,11,14,16,20 4,6,9,12,13,18,21 3,5,8,10,15,17,19 1,2,7 4,6 3,5 11,14 9,12,13 8,10 16,20 18,21 15,17,19 Fusion of receveid and sorted elements 1,2,3,4,5,6,7 8,9,10,11,12,13,14 15,16,17,18,19,20,21

49. Benchs and BSP predictions

50. Benchs and BSP predictions