Classification of the PRAM model

Advanced Topics in Algorithms and Data Structures

Classification of the PRAM Classification of the PRAM modelmodel

• In the PRAM model, processors communicate by reading from and writing to the shared memory locations.


Classification of the PRAM model

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



In every clock cycle,• In the Exclusive Read Exclusive

Write (EREW) PRAM, each memory location can be accessed only by one processor.

• In the Concurrent Read Exclusive Write (CREW) PRAM, multiple processor can read from the same memory location, but only one processor can write.



• In the Concurrent Read Concurrent Write (CRCW) PRAM, multiple processor can read from or write to the same memory location.



• It is easy to allow concurrent reading. However, concurrent writing gives rise to conflicts.

• If multiple processors write to the same memory location simultaneously, it is not clear what is written to the memory location.



• In the Common CRCW PRAM, all the processors must write the same value.

• In the Arbitrary CRCW PRAM, one of the processors arbitrarily succeeds in writing.

• In the Priority CRCW PRAM, processors have priorities associated with them and the highest priority processor succeeds in writing.



• The EREW PRAM is the weakest and the Priority CRCW PRAM is the strongest PRAM model.

• The relative powers of the different PRAM models are as follows.



• An algorithm designed for a weaker model can be executed within the same time and work complexities on a stronger model.



•We say model A is less powerful compared to model B if either:•the time complexity for solving a

problem is asymptotically less in model B as compared to model A. or,

•if the time complexities are the same, the processor or work complexity is asymptotically less in model B as compared to model A.



An algorithm designed for a stronger PRAM model can be simulated on a weaker model either with asymptotically more processors (work) or with asymptotically more time.


Adding n numbers on a PRAM




• This algorithm works on the EREW PRAM model as there are no read or write conflicts.

• We will use this algorithm to design a matrix multiplication algorithm on the EREW PRAM.


For simplicity, we assume that n = 2p for some integer p.

Matrix multiplication


Matrix multiplication• Each can be computed in par

allel.• We allocate n processors for computing

ci,j. Suppose these processors are P1, P2,…,Pn.

• In the first time step, processor computes the product ai,m x bm,j.

• We have now n numbers and we use the addition algorithm to sum these n numbers in log n time.

, , 1 ,i jc i j n

, 1mP m n


Matrix multiplication• Computing each takes n proc

essors and log n time.• Since there are n2 such ci,j s, we need ove

rall O(n3) processors and O(log n) time.• The processor requirement can be redu

ced to O(n3 / log n). Exercise !• Hence, the work complexity is O(n3)

, , 1 ,i jc i j n


Matrix multiplication• However, this algorithm requires concur

rent read capability.• Note that, each element ai,j (and bi,j) parti

cipates in computing n elements from the C matrix.

• Hence n different processors will try to read each ai,j (and bi,j) in our algorithm.


For simplicity, we assume that n = 2p for some integer p.

Matrix multiplication


Matrix multiplication• Hence our algorithm runs on the CREW P

RAM and we need to avoid the read conflicts to make it run on the EREW PRAM.

• We will create n copies of each of the elements ai,j (and bi,j). Then one copy can be used for computing each ci,j .


Matrix multiplicationCreating n copies of a number in O (log n) time using O (n) processors on the EREW PRAM.•In the first step, one processor reads

the number and creates a copy. Hence, there are two copies now.

•In the second step, two processors read these two copies and create four copies.


Matrix multiplication•Since the number of copies doubles

in every step, n copies are created in O(log n) steps.

•Though we need n processors, the processor requirement can be reduced to O (n / log n).Exercise !


Matrix multiplication• Since there are n2 elements in the

matrix A (and in B), we need O (n3 / log n) processors and O (log n) time to create n copies of each element.

• After this, there are no read conflicts in our algorithm. The overall matrix multiplication algorithm now take O (log n) time and O (n3 / log n) processors on the EREW PRAM.


Matrix multiplication• The memory requirement is of course

much higher for the EREW PRAM.

Classification of the PRAM model

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