Last modified: August 03, 2024

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Designing Parallel Programs

Designing parallel programs involves breaking down computational tasks into smaller, concurrent units to be executed simultaneously. This approach leverages the power of multiple processors to enhance performance and efficiency. Key steps in this process include partitioning, communication, agglomeration, and mapping.

Partitioning

Partitioning is a fundamental concept in computational problem-solving, where a large problem is divided into smaller, more manageable tasks. This approach allows tasks to be executed concurrently, improving efficiency and performance. In essence, partitioning is about breaking a problem into smaller pieces.

#
      +-----------------------+
      |     Big Problem       |
      +-----------------------+
                 |
     +-----------+-----------+
     |           |           |
+---------+ +---------+ +---------+
| Part 1  | | Part 2  | | Part 3  |
+---------+ +---------+ +---------+

Goals

I. Balance the Workload Among Tasks

II. Minimize Dependencies Between Tasks

Types of Decomposition

I. Domain Decomposition

Domain decomposition involves splitting the data into partitions, with each partition processed by a separate task. This method is highly effective in scenarios where the data can be naturally divided into distinct segments.

Example:

In a weather simulation, the geographical area can be divided into smaller regions, with each region processed independently.

II. Functional Decomposition

Functional decomposition divides the computation into separate functional tasks, each performing a distinct part of the overall computation. This method focuses on breaking down the operations rather than the data.

Example:

In a web application, different tasks such as user authentication, data retrieval, and UI rendering can be handled by separate services.

IIII. Cyclic Decomposition

Cyclic decomposition is primarily used in the context of permutations and group theory. It breaks down a permutation into a collection of disjoint cycles, where each cycle represents a sequence of elements that are permuted among themselves.

Example:

Consider the permutation of the set ${1, 2, 3, 4}$ given by:

$$ \sigma = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 1 & 4 \end{pmatrix} $$

This permutation can be decomposed into disjoint cycles as follows:

$$ \sigma = (1 \to 2 \to 3 \to 1)(4) $$

Here, the cycle $(1 → 2 → 3 → 1)$ indicates that 1 is sent to 2, 2 is sent to 3, and 3 is sent back to 1. The cycle $(4)$ indicates that 4 remains fixed.

IV. Block Decomposition

Block decomposition is a technique used in matrix theory and linear algebra. It involves partitioning a matrix into smaller submatrices (blocks). This can simplify computations and is useful for analyzing the structure of the matrix.

Example:

Consider a matrix $A$ of size $4 \times 4$:

$$ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix} $$

This matrix can be decomposed into blocks:

$$ A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} $$

where each block is a submatrix:

$$ A_{11} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, \quad A_{12} = \begin{pmatrix} a_{13} & a_{14} \\ a_{23} & a_{24} \end{pmatrix}, \quad A_{21} = \begin{pmatrix} a_{31} & a_{32} \\ a_{41} & a_{42} \end{pmatrix}, \quad A_{22} = \begin{pmatrix} a_{33} & a_{34} \\ a_{43} & a_{44} \end{pmatrix} $$

Block decomposition helps in performing matrix operations more efficiently by working with smaller submatrices.

Communication

Communication in parallel computing involves the exchange of data between tasks. This process is essential when tasks depend on each other's results to proceed. Efficient communication is crucial for maintaining performance and minimizing delays.

Goals

I. Minimize the Volume and Frequency of Communication

II. Optimize the Use of Network Resources:

I apologize for the omission. Here is the full text with ASCII graphics and examples for each type of communication, including the example for point-to-point communication.

Types of Communication

I. Local Communication

Local communication refers to data exchange between tasks that reside on the same physical processor. This type of communication is generally faster due to lower latency and higher bandwidth within a single processor.

Example:

In a multi-core processor, tasks running on different cores can share data through shared memory.

+-----------+     +-----------+
| Core 1    | <-> | Core 2    |
| Task A    |     | Task B    |
+-----------+     +-----------+
      |               |
      +-------+-------+
              |
       Shared Memory

II. Global Communication

Global communication involves data exchange between tasks located on different physical processors. This type of communication often involves higher latency and lower bandwidth due to the physical distance and network constraints.

Example:

In a distributed computing system, tasks running on separate machines need to communicate over a network.

+-------------+       +-------------+
| Processor 1 | <----> | Processor 2 |
| Task A      |       | Task B      |
+-------------+       +-------------+
      |                     |
      +---------------------+
             Network

III. Point-to-Point Communication

Point-to-point communication is direct data exchange between pairs of tasks. It is the simplest form of communication, where one task (sender) sends data directly to another task (receiver).

+---------+     +---------+
| Sender  | --> | Receiver |
+---------+     +---------+

IV. Collective Communication

Collective communication involves multiple tasks and includes operations such as broadcast, scatter, and gather.

Scatter Communication

In scatter communication, a single process sends different pieces of data to multiple processes. Each process receives a unique piece of the data.

+-----------------------+
      |         ABC           |
      +-----------------------+
                 |
     +-----------+-----------+
     |           |           |
+---------+ +---------+ +---------+
|    A    | |    B    | |    C    |
+---------+ +---------+ +---------+

Broadcast Communication

In broadcast communication, a single process sends the same piece of data to all other processes.

+-----------------------+
      |         ABC           |
      +-----------------------+
                 |
     +-----------+-----------+
     |           |           |
+---------+ +---------+ +---------+
|   ABC   | |   ABC   | |   ABC   |
+---------+ +---------+ +---------+

Communication Modes

I. Synchronous Blocking Communication

II. Asynchronous Nonblocking Communication

Key Concepts in Communication

I. Overhead

II. Latency

III. Bandwidth

Agglomeration

Agglomeration is the process of combining smaller tasks and data partitions into larger tasks. This technique aims to reduce communication overhead and improve overall efficiency by grouping tasks that frequently interact and ensuring the aggregated tasks fit within the processor’s memory.

Goals

I. Improve Computational Granularity:

II. Minimize Communication:

Granularity

Granularity in parallel computing refers to the ratio of computation to communication in a task or set of tasks. It indicates the amount of work done between communication events.

$$ \text{Granularity} = \frac{\text{computation}}{\text{communication}} $$

Types of Parallelism

I. Fine-Grained Parallelism

II. Coarse-Grained Parallelism

Practical Considerations

I. Task Grouping

II. Memory Management

III. Optimization Balance

Mapping

Mapping is the process of assigning agglomerated tasks to specific processors in a parallel computing environment. Effective mapping is crucial for balancing the computational load and minimizing communication overhead.

Goals

I. Balance the Computational Load Across Processors

II. Minimize Communication Overhead

III. Minimize the Total Execution Time

Strategies

I. Static Mapping

II. Dynamic Mapping

III. Hierarchical Mapping

IV. Task Clustering

V. Graph Partitioning

Practical Considerations

I. Task Placement

II. Load Balancing

III. Communication Patterns

Does Not Apply To

Example Workflow in Designing Parallel Programs

I. Partitioning

II. Communication

III. Agglomeration

IV. Mapping

Key Principles

Tools and Techniques

Table of Contents

  1. Partitioning
    1. Goals
    2. Types of Decomposition
  2. Communication
    1. Goals
    2. Types of Communication
    3. Communication Modes
    4. Key Concepts in Communication
  3. Agglomeration
    1. Goals
    2. Granularity
    3. Types of Parallelism
    4. Practical Considerations
  4. Mapping
    1. Goals
    2. Strategies
    3. Practical Considerations
    4. Does Not Apply To
  5. Example Workflow in Designing Parallel Programs
  6. Key Principles
  7. Tools and Techniques