Thm13 - PRAM Models

8/18/2019 Thm13 - PRAM Models

1/22

PRAM Models

Advanced Algorithms & Data Structures

Lecture Theme 13

Prof. Dr. Th. OttmannSummer Semester 2006


2/22

2

• InthePRAM model,processorscommunicatebyreadingfromandwritingtothesharedmemorylocations.

• The power of a PRAM depends on the kind of access to the shared

memory locations.

Classification of the PRAM model


3/22

3

Ineveryclockcycle,• IntheExclusiveReadExclusiveWrite

(EREW)PRAM,eachmemorylocation

canbeaccessedonlybyoneprocessor.

• IntheConcurrentReadExclusiveWrite

(CREW)PRAM,multipleprocessorcanreadfromthesamememorylocation,but

onlyoneprocessorcanwrite.



4/22

4

• IntheConcurrentReadConcurrentWrite(CRCW)PRAM,multipleprocessorcan

readfromorwritetothesamememory

location.



5/22

5

• Itiseasytoallowconcurrentreading.However,concurrentwritinggivesriseto

conflicts.

• Ifmultipleprocessorswritetothesame

memorylocationsimultaneously,itisnot

clearwhatiswrittentothememorylocation.



6/22

6

• IntheCommonCRCWPRAM,alltheprocessorsmustwritethesamevalue.

• Inthe ArbitraryCRCWPRAM,oneofthe

processorsarbitrarilysucceedsinwriting.

• InthePriorityCRCWPRAM,processors

haveprioritiesassociatedwiththemandthehighestpriorityprocessorsucceedsin

writing.



7/22

7

• TheEREWPRAMistheweakestandthePriorityCRCWPRAMisthestrongest

PRAMmodel.

• TherelativepowersofthedifferentPRAM

modelsareasfollows.



8/22

8

• Analgorithmdesignedforaweakermodelcanbeexecutedwithinthesame

timeandworkcomplexitiesona

strongermodel.



9/22

9

• Wesaymodel A islesspowerfulcomparedtomodelB ifeither:

• thetimecomplexityforsolvinga

problemisasymptoticallylessinmodel

B ascomparedtomodel A.or,

• ifthetimecomplexitiesarethesame,theprocessororworkcomplexityis

asymptoticallylessinmodelB as

comparedtomodel A.



10/22


11/22

11

Addingn numbersonaPRAM

Adding n numbers on a PRAM


12/22

12

• ThisalgorithmworksontheEREWPRAMmodelastherearenoreadorwrite

conflicts.

• Wewillusethisalgorithmtodesigna

matrixmultiplicationalgorithmonthe

EREWPRAM.

Adding n numbers on a PRAM


13/22

13

Forsimplicity,weassumethatn =2p forsomeinteger p.

Matrix multiplication


14/22

14

• Eachcanbecomputedin

parallel.

• Weallocaten processorsforcomputingci,j.

SupposetheseprocessorsareP1 , P2 ,…,Pn.• Inthefirsttimestep,processor

computestheproductai,m x bm,j.

• Wehavenown numbersandweusethe

additionalgorithmtosumthesen numbers

inlogn time.

, , 1 ,i jc i j n≤ ≤

, 1mP m n≤ ≤



15/22

15

• Computingeachtakesn

processorsandlogn time.

• Sincetherearen2 suchci,j s,weneed

overallO (n3

) processorsandO (logn) time.• Theprocessorrequirementcanbe

reducedtoO (n3/logn).Exercise!

• Hence,theworkcomplexityisO (n3)

, , 1 ,i jc i j n≤ ≤



16/22

16

• However,thisalgorithmrequires

concurrentreadcapability.

• Notethat,eachelementai,j (andbi,j)

participatesincomputingn elementsfromtheC matrix.

• Hencen differentprocessorswilltryto

readeachai,j (andbi,j)inouralgorithm.



17/22

17

Forsimplicity,weassumethatn =2p forsomeinteger p.



18/22

18

• HenceouralgorithmrunsontheCREW

PRAM andweneedtoavoidthereadconflictstomakeitrunontheEREW

PRAM.

• Wewillcreaten copiesofeachofthe

elementsai,j (andbi,j).Thenonecopycan

beusedforcomputingeachci,j .



19/22

19

Creatingn copiesofanumberinO (logn)

timeusingO (n) processorsontheEREWPRAM.

• Inthefirststep,oneprocessorreadsthenumberandcreatesacopy.Hence,there

aretwocopiesnow.

• Inthesecondstep,twoprocessorsreadthesetwocopiesandcreatefourcopies.



20/22

20

• Sincethenumberofcopiesdoublesin

everystep,n copiesarecreatedinO (logn)steps.

• Thoughweneedn processors,theprocessorrequirementcanbereducedto

O(n /logn).

Exercise!



21/22

21

• Sincetherearen2 elementsinthematrix A

(andin B),weneedO(n3 /logn)processorsandO(logn) timetocreaten

copiesofeachelement.

• Afterthis,therearenoreadconflictsinour

algorithm.Theoverallmatrixmultiplication

algorithmnowtakeO(logn) timeandO(n3 /logn) processorsontheEREW

PRAM.



22/22

22

• Thememoryrequirementisofcourse

muchhigherfortheEREWPRAM.


Thm13 - PRAM Models

Documents

Transcript of Thm13 - PRAM Models