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Questions and Answers
What is the time complexity of the sequential algorithm T1 for the matrix-vector multiplication?
What is the time complexity of the sequential algorithm T1 for the matrix-vector multiplication?
- O(N)
- O(N^3)
- O(N^2) (correct)
- O(N log N)
What does the variable 'p' represent in the performance of matrix-vector multiplication?
What does the variable 'p' represent in the performance of matrix-vector multiplication?
- The maximum value of elements in matrix A
- The number of cores used (correct)
- The efficiency of the algorithm
- The size of the matrix
In the SPMD sum on PRAM example, what is the condition for the global read and write operation in the loop?
In the SPMD sum on PRAM example, what is the condition for the global read and write operation in the loop?
- If i = n/2^h
- If i < n/2^h (correct)
- If i < n^h
- If i >= n/2^h
What is the efficiency formula represented by Ep in the context of the performance analysis?
What is the efficiency formula represented by Ep in the context of the performance analysis?
What does 'O(N^2/P)' signify in the performance metrics?
What does 'O(N^2/P)' signify in the performance metrics?
Which components allow processors to communicate in a PRAM system?
Which components allow processors to communicate in a PRAM system?
What happens during the write conflict in a PRAM?
What happens during the write conflict in a PRAM?
What is the key feature of Exclusive Read (ER) PRAM?
What is the key feature of Exclusive Read (ER) PRAM?
Which phase is NOT part of the computation steps in PRAM?
Which phase is NOT part of the computation steps in PRAM?
What does 'Concurrent Write (CW)' allow in a PRAM?
What does 'Concurrent Write (CW)' allow in a PRAM?
Which of the following statements about PRAM processors is TRUE?
Which of the following statements about PRAM processors is TRUE?
What defines the structure of a PRAM system?
What defines the structure of a PRAM system?
During a read operation in PRAM, what type of memory location can processors access simultaneously?
During a read operation in PRAM, what type of memory location can processors access simultaneously?
What is the primary purpose of a machine model in parallel computing?
What is the primary purpose of a machine model in parallel computing?
What type of machine model is defined as a system of infinitely many identical processors?
What type of machine model is defined as a system of infinitely many identical processors?
Which of the following statements is true about RAM in the context of machine models?
Which of the following statements is true about RAM in the context of machine models?
Which operation is not included in the instruction set of the RAM model?
Which operation is not included in the instruction set of the RAM model?
In a PRAM model, what is assumed about the processors?
In a PRAM model, what is assumed about the processors?
What characterizes the time complexity in the RAM model?
What characterizes the time complexity in the RAM model?
What is a key feature of the PRAM model regarding memory?
What is a key feature of the PRAM model regarding memory?
In parallel computing, what does the term 'write conflict' refer to?
In parallel computing, what does the term 'write conflict' refer to?
What is the purpose of the statement 'IF 𝑙 = 1' in STEP 3?
What is the purpose of the statement 'IF 𝑙 = 1' in STEP 3?
What does the variable 𝑇1 represent in the performance of matrix multiplication?
What does the variable 𝑇1 represent in the performance of matrix multiplication?
If the condition 'IF 𝑙 ≤ 𝑛/2ℎ' is bypassed, what could be the consequence?
If the condition 'IF 𝑙 ≤ 𝑛/2ℎ' is bypassed, what could be the consequence?
How is the cost of the matrix multiplication represented?
How is the cost of the matrix multiplication represented?
What is one of the key characteristics of BOUNDED NUMBER OF SHARED MEMORY CELLS in PRAM?
What is one of the key characteristics of BOUNDED NUMBER OF SHARED MEMORY CELLS in PRAM?
What does the MVM algorithm's Step 3 primarily focus on?
What does the MVM algorithm's Step 3 primarily focus on?
In the context of the matrix-vector multiplication example, what is meant by 'embarrassingly parallel'?
In the context of the matrix-vector multiplication example, what is meant by 'embarrassingly parallel'?
What is the significance of global reads in the MVM algorithm?
What is the significance of global reads in the MVM algorithm?
What is the primary purpose of Step 4 in the MVM algorithm?
What is the primary purpose of Step 4 in the MVM algorithm?
In the matrix-vector multiplication example, how is matrix A partitioned for processing?
In the matrix-vector multiplication example, how is matrix A partitioned for processing?
What is the role of processor $P_i$ in the matrix-vector multiplication process?
What is the role of processor $P_i$ in the matrix-vector multiplication process?
What is the effect of using 32 processors on the computation of matrix-vector multiplication?
What is the effect of using 32 processors on the computation of matrix-vector multiplication?
What operation is performed in Step 3 of the MVM algorithm to achieve processing?
What operation is performed in Step 3 of the MVM algorithm to achieve processing?
Study Notes
What is a Machine Model?
- A machine model describes an abstract "machine" that makes reasoning about algorithms easier
- Machine models are used to achieve complexity bounds and analyze maximum parallelism
RAM (Random Access Machine)
- An unbounded number of local memory cells
- Each memory cell can hold an integer of unbounded size
- RAMs include an instruction set consisting of simple operations, data operations, comparators, and branches
- All operations take unit time
- Time Complexity of RAMs is the number of instructions executed
- Space Complexity of RAMs is the number of memory cells used
PRAM (Parallel Random Access Machine)
- An abstract machine for designing algorithms applicable to parallel computers
- Consists of infinitely many RAMs (processors)
- Each processor has its own index
- All processors are identical and have the ability to recognize their own index
- PRAMs include input and output cells, as well as shared memory cells
Steps in PRAM Computation
- Consists of 5 phases carried out in parallel by all processors:
- Read a value from input cells
- Read a shared memory cell
- Internal computation
- Write into output cells
- Write into shared memory cells
PRAM Memory Access Conflicts
- Exclusive Read (ER): All processors can read simultaneously from distinct memory locations
- Exclusive Write (EW): All processors can write simultaneously to distinct memory locations
- Concurrent Read (CR): All processors can read simultaneously from any memory location
- Concurrent Write (CW): All processors can write simultaneously to any memory location
- PRAMs are classified based on their read/write capabilities: EREW, CREW, CRCW
Matrix Vector Multiply (MVM) Example
- Objective: Calculate Y = AX, where A is a matrix (N x N) and X is a vector (N x 1)
- Solution: Divide the matrix into P blocks (each block representing 𝑁/𝑝 rows) and assign a block to each processor
- Steps:
- Each processor concurrently reads X
- Each processor reads its assigned block of A
- Each processor calculates its local result (block * vector)
- Each processor writes its local result to Y
- Performance:
- T1(N2): O(N2) sequential time
- TP(N2): O(N2/P) parallel time
- Cost: O(P * N2/P) = O(N2)
- Speedup: P
- Efficiency: 1 (perfect efficiency)
Example 2: SPMD Sum A(1:N)
- Objective: Sum the elements of an array A(1:N)
- Steps:
- Each processor reads its assigned element of A
- Summation is performed iteratively in a tree-like structure, where each level combines the sums of two processors
- Performance:
- T1: O(N) sequential time
- TP: O(log N) parallel time
- Cost: O(N * log N)
- Efficiency: decreasing with growing P
Variations of PRAM
- Bounded Number of Shared Memory Cells (Small Memory PRAM): Input data set exceeds capacity of the shared memory, requiring data distribution among processors.
- Bounded Number of Processors (Small PRAM): When the number of threads exceeds available processors, processors interleave execution of multiple threads.
- Bounded Size of a Machine Word: Limits the size of data that can be processed in a single operation.
- Handling Access Conflicts: Constrains simultaneous access to shared memory cells.
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Description
This quiz covers essential concepts of machine models in computer science, including RAM and PRAM. Understand how these abstract machines facilitate the analysis of algorithms and their complexities. Test your knowledge on how these models function and their applications in parallel computing.