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Parallel Algorithms

Parallel Algorithms. Parallel Models. Hypercube Butterfly Fully Connected Other Networks Shared Memory v.s. Distributed Memory SIMD v.s. MIMD. The PRAM Model. P arallel R andom A ccess M achine All processors act in lock-step Number of processors is not limited

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Parallel Algorithms

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  1. Parallel Algorithms

  2. Parallel Models • Hypercube • Butterfly • Fully Connected • Other Networks • Shared Memory v.s. Distributed Memory • SIMD v.s. MIMD

  3. The PRAM Model • Parallel Random Access Machine • All processors act in lock-step • Number of processors is not limited • All processors have local memory • One global memory accessible to all processors • Processors must read and write global memory

  4. A Pram Algorithm • Every Processor knows its own index (usually indicated by variable i) • Vector Sum: Read M[i] Into x; Read M[i+n] Into y; x := x + y; Write x into M[i];

  5. Binary Fan-In Read M[i] into Largest; Write M[i] into M[i+n]; Delta := 1; For k := 1 to élg nù Read M[i+Delta] into x; Largest := Maximum(x,Largest); Write Largest into M[i]; Delta := Delta * 2; End For

  6. Parallel Addition Read M[i] into Total; Write 0 into M[i+n]; Delta := 1; For k := 1 to élg nù Read M[i+Delta] into x; Total := x + Total; Write Total into M[i]; Delta := Delta * 2; End For

  7. Pointer Jumping Read M[i] Into Total; For k := 1 to élg nù Read Next[i] into Ptr If Ptr ¹ 0 Then Read M[Ptr] Into x; Total := Total + x; Write Total into M[i]; Read Next[Ptr] Into NewPtr Write NewPtr into Next[i] End If End For

  8. Initialization of Next[i] If i = n Then Write 0 Into Next[i]; Else Write i+1 Into Next[i]; End If

  9. Calculate Node Depth I If there is a Left Child 1 -1 To “1” of Left Child 0 From “-1” of Left Child

  10. Calculate Node Depth 2 If there is no left child 1 -1 0

  11. Calculate Node Depth 3 If there is a Right Child 1 -1 From “-1” of Right Child 0 To “1” of Right Child

  12. Calculate Node Depth 4 If there is no right child 1 -1 0

  13. Concurrent Reads & Writes • EREW - Exclusive Read, Exclusive Write • CREW - Common Read, Exclusive Write • CRCW - Common Read, Common Write • All common writes must write the same thing • Highest Priority Processor wins contest • CREW is more powerful than EREW • CRCW is more powerful than CREW

  14. Finding Max • Square Array of Processors Indexed by i,j Write True into R[i]; Read M[i] into x; Read M[j] into y; If x < y Then Write False Into R[i]; Else If y < x Then Write False Into R[j]; End If

  15. CRCW V.S. CREW • CRCW Max runs in constant time • CREW Max runs in lg n time • CRCW cannot be any better than lg p faster than EREW

  16. EREW V.S. CREW • Finding Roots by Shortcutting Pointers • CREW Runs in lg lg n Time • EREW Runs in lg n Time

  17. Optimal Parallel Algorithms • NC -- The class of algorithms that run in Q(logmn) time using Q(nk) processors • General Boolean Functions Cannot be Computed any Faster than Q(lg n) • Q(lg n) is optimal for computing the sum of n integers

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