Machine Models in Computer Science

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Questions and Answers

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?

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?

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?

$E_p = \frac{T_1}{p T_p}$ Signup and view all the answers

What does 'O(N^2/P)' signify in the performance metrics?

The time complexity for parallel execution with P processors Signup and view all the answers

Which components allow processors to communicate in a PRAM system?

Shared memory cells Signup and view all the answers

What happens during the write conflict in a PRAM?

Only one processor can write successfully Signup and view all the answers

What is the key feature of Exclusive Read (ER) PRAM?

Processors can read from distinct memory locations simultaneously Signup and view all the answers

Which phase is NOT part of the computation steps in PRAM?

Compare values from different processors Signup and view all the answers

What does 'Concurrent Write (CW)' allow in a PRAM?

All processors to write to any memory location simultaneously Signup and view all the answers

Which of the following statements about PRAM processors is TRUE?

Each processor can access shared memory in constant time Signup and view all the answers

What defines the structure of a PRAM system?

Unbounded collection of RAM processors and shared memory Signup and view all the answers

During a read operation in PRAM, what type of memory location can processors access simultaneously?

Distinct memory locations exclusively Signup and view all the answers

What is the primary purpose of a machine model in parallel computing?

To simplify reasoning about algorithms and analyze performance. Signup and view all the answers

What type of machine model is defined as a system of infinitely many identical processors?

Parallel Random Access Machine (PRAM) Signup and view all the answers

Which of the following statements is true about RAM in the context of machine models?

Space complexity depends on the number of memory cells used. Signup and view all the answers

Which operation is not included in the instruction set of the RAM model?

Complex Data Structures Signup and view all the answers

In a PRAM model, what is assumed about the processors?

All processors are identical and work simultaneously. Signup and view all the answers

What characterizes the time complexity in the RAM model?

It is calculated based on the number of instructions executed. Signup and view all the answers

What is a key feature of the PRAM model regarding memory?

It allows an unbounded number of local memory cells. Signup and view all the answers

In parallel computing, what does the term 'write conflict' refer to?

Conflicts arising when multiple processors attempt to write to the same memory location. Signup and view all the answers

What is the purpose of the statement 'IF 𝑙 = 1' in STEP 3?

To initialize the matrix 𝐶𝑖,𝑗 with the first computed value. Signup and view all the answers

What does the variable 𝑇1 represent in the performance of matrix multiplication?

The time taken to compute all elements in the matrix in a single-thread environment. Signup and view all the answers

If the condition 'IF 𝑙 ≤ 𝑛/2ℎ' is bypassed, what could be the consequence?

Potential incorrect summation in parallel computations. Signup and view all the answers

How is the cost of the matrix multiplication represented?

Cost is expressed as 𝐶𝑜𝑠𝑡 = $𝑛^3 imes ext{LOG} ext{n}$. Signup and view all the answers

What is one of the key characteristics of BOUNDED NUMBER OF SHARED MEMORY CELLS in PRAM?

Input data set can exceed the capacity, necessitating distribution. Signup and view all the answers

What does the MVM algorithm's Step 3 primarily focus on?

Computing operations for each processor Signup and view all the answers

In the context of the matrix-vector multiplication example, what is meant by 'embarrassingly parallel'?

Calculations that can be performed simultaneously without dependence Signup and view all the answers

What is the significance of global reads in the MVM algorithm?

They allow simultaneous data access and reduce read time Signup and view all the answers

What is the primary purpose of Step 4 in the MVM algorithm?

To transfer computed elements without conflicts Signup and view all the answers

In the matrix-vector multiplication example, how is matrix A partitioned for processing?

Into separate blocks assigned to each processor Signup and view all the answers

What is the role of processor $P_i$ in the matrix-vector multiplication process?

It reads its assigned block and computes outputs Signup and view all the answers

What is the effect of using 32 processors on the computation of matrix-vector multiplication?

It distributes computation, thereby improving efficiency Signup and view all the answers

What operation is performed in Step 3 of the MVM algorithm to achieve processing?

Computing the product of a row and vector Signup and view all the answers

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.