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8INF433

8INF433. Algorithmes parallèles. Algorithmes parallèles. Trois nouvelles instructions: s pawn s ync parallèle. Exemple: Fibonacci. P-Fib (n) if n<=1 return n else x = spawn P-Fib(n-1) y = P-Fib(n-2) sync return x+y. Exemple: Fibonacci.

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8INF433

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  1. 8INF433 Algorithmes parallèles

  2. Algorithmes parallèles Trois nouvelles instructions: • spawn • sync • parallèle

  3. Exemple: Fibonacci P-Fib(n) if n<=1 return n else x = spawn P-Fib(n-1) y = P-Fib(n-2) sync return x+y

  4. Exemple: Fibonacci

  5. Mesures de performance • Travail (work): temps séquentiel • Durée (span): temps parallèle • TP:temps d’exécution sur P processeurs • T1: travail • T∞:durée • Loi du travail: TP ≥ T1/P • Loi de la durée: TP ≥ T∞ • Parallélisme: T1/T∞≥ T1/Tp(accélération)

  6. Analyse de P-Fib T1(n) = θ(ϕn) où ϕ est le nombre d’or T∞(n) = max(T∞(n-1), T∞(n-2)) + θ(1) = T∞(n-1) + θ(1) = θ(n) Parallélisme: θ(ϕn/n)

  7. Boucles parallèles Exemple: On veut multiplier une matrice M par un vecteur x Mat-Vec(M,x,n) parallèle for i=1 to n do y[i]=0 parallèle for i=1 to n do for j=1 to n do y[i] = y[i] + M[i,j]*x[j] return y

  8. Implémentation des boucles parallèles On implémente les boucles parallèles à l’aide de l’instruction spawn: Mat-Vec-Main-Loop(M,x,y,n,d,f) if (d==f) for j=1 to n do y[d] = y[d] + M[d,j]*x[j] else m = (d+f)/2 spawn Mat-Vec-Main-Loop(M,x,y,n,d,m) Mat-Vec-Main-Loop(M,x,y,n,m+1,f) sync

  9. Implémentation des boucles parallèles

  10. Analyse de Mat-Vec(A,x,n) • Travail: θ(n2) • Durée: Total: θ(n) • Parallélisme: θ(n2/n) = θ(n) Mat-Vec(M,x,n) parallèle for i=1 to n do y[i]=0 parallèle for i=1 to n do for j=1 to n do y[i] = y[i] + M[i,j]*x[j] return y Θ(lg n) Θ(lg n) + θ(n) Θ(1)

  11. Multiplication matricielle (1) P-Square-Matrix-Multiply(A,B,n) parallèle for i=1 to n do parallèle for j=1 to n do C[i,j]=0 for k=1 to n do C[i,j] = C[i,j] + A[i,k]*B[k,j] return C

  12. Multiplication matricielle (2)

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