PRAM ModelsAdvanced Algorithms & Data Structures

Lecture Theme 13

Prof. Dr. Th. OttmannSummer Semester 2006

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

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

Classification of the PRAM model

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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.

Classification of the PRAM model

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

Classification of the PRAM model

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• 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.

Classification of the PRAM model

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• 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.

Classification of the PRAM model

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• 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.

Classification of the PRAM model

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

Classification of the PRAM model

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•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.

Classification of the PRAM model

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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.

Classification of the PRAM model

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Adding n numbers on a PRAM

Adding n numbers on a PRAM

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• 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.

Adding n numbers on a PRAM

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

Matrix multiplication

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• Each can be computed in parallel.

• 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

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• Computing each takes n processors and log n time.

• Since there are n2 such ci,j s, we need overall O(n3) processors and O(log n) time.

• The processor requirement can be reduced to O(n3 / log n). Exercise !

• Hence, the work complexity is O(n3)

, , 1 ,i jc i j n

Matrix multiplication

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• However, this algorithm requires concurrent read capability.

• Note that, each element ai,j (and bi,j) participates 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.

Matrix multiplication

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

Matrix multiplication

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• Hence our algorithm runs on the CREW PRAM 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 multiplication

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Creating 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

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•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

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• 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

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• The memory requirement is of course much higher for the EREW PRAM.

Matrix multiplication