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Parallel Algorithms

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A Pram Algorithm. Every Processor knows its own index (usually indicated by variable i) ... Optimal Parallel Algorithms. NC -- The class of algorithms that run ... – PowerPoint PPT presentation

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Title: Parallel Algorithms


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 Mi Into x Read Min Into y x x
y Write x into Mi
5
Binary Fan-In
Read Mi into Largest Write Mi into
Min Delta 1 For k 1 to élg nù Read
MiDelta into x Largest
Maximum(x,Largest) Write Largest into Mi
Delta Delta 2 End For
6
Parallel Addition
Read Mi into Total Write 0 into Min Delta
1 For k 1 to élg nù Read MiDelta
into x Total x Total Write Total
into Mi Delta Delta 2 End For
7
Pointer Jumping
Read Mi Into Total For k 1 to élg nù
Read Nexti into Ptr If Ptr ¹ 0 Then
Read MPtr Into x Total Total x
Write Total into Mi Read
NextPtr Into NewPtr Write NewPtr into
Nexti End If End For
8
Initialization of Nexti
If i n Then Write 0 Into Nexti Else
Write i1 Into Nexti 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 Ri Read Mi into x Read Mj
into y If x lt y Then Write False Into
Ri Else If y lt x Then Write False Into
Rj 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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