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Multi-Tasking Models and Algorithms. General Concepts (Part I). Outline for Multi-Tasking Models. Note : Items in black are in this slide set (Part I). Preliminaries Common Decomposition Methods Characteristics of Tasks and Interactions Mapping Techniques for Load Balancing

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Multi-Tasking Models and Algorithms

General Concepts

(Part I)

Outline for multi tasking models l.jpg
Outline for Multi-Tasking Models

Note: Items in black are in this slide set (Part I).

  • Preliminaries

  • Common Decomposition Methods

  • Characteristics of Tasks and Interactions

  • Mapping Techniques for Load Balancing

  • Some Parallel Algorithm Models

    • The Data-Parallel Model

    • The Task Graph Model

    • The Work Pool Model

    • The Master-Slave Model

    • The Pipeline or Producer-Consumer Model

    • Hybrid Models

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Outline (cont.)

  • Algorithm examples for most of preceding algorithm models.

    • This part currently missing & need to add next time.

    • Some could be added as examples under Task/Channel model

  • Task-Channel (Computational) Model

  • Asynchronous Communication and Performance Evaluation

    • Modeling Asynchronous Communicaiton

    • Performance Metrics and Asynchronous Communications

    • The Isoefficiency Metric & Scalability

  • Future revision plans for preceding material.

  • BSP (Computational) Model

    • Slides posted separately on course website

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  • Michael Quinn, Parallel Programming in C with MPI and OpenMP, McGraw Hill, 2004.

    • Particularly, Chapters 3 and 7 plus algorithm examples.

    • Textbook slides for this book

  • Ananth Grama, Anshul Gupta, George Karypis, and Vipin Kumar, Introduction to Parallel Computing, 2nd Edition, Addison Wesley, 2003.

    • Particularly, Chapter 3 (available online)

    • Also, Section 2.5 (Asynchronous Communications)

    • Slides by the Authors’

  • Barry Wilkinson and Michael Allen, “Parallel Programming: Techniques and Applications


  • Using Networked Workstations and Parallel Computers ”, Second Edition, Prentice Hall, 2005.

  • Ian Foster, Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering, Addison Wesley, 1995, Online at

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Change in Chapter Title

  • This chapter consists of three sets of slides.

  • This chapter was formerly called

    Strictly Asynchronous Models

  • The name has now been changed to

    Multi-Tasking Models

  • However, the old name still occurs regularly in the internal slides.

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Specifying Asynchronous Algorithms

  • Identifying parts that can be done concurrently

    • Tasks

  • Mapping of the tasks onto multiple processors

    • Processes vs processors

  • Distributing the input, output, and intermediate results across different processors

  • Management of access to shared data

    • Either input or intermediate

  • Synchronization of the processors at various stages of the parallel execution

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Finding Concurrent Pieces of Work

  • Decomposition

    • The process of dividing the computation into smaller pieces of work called tasks

  • Tasks are programmer defined and are considered to be indivisible.

    • Tasks may be of arbitrary sizes

  • Simultaneous execution of multiple tasks is the key to reducing time required

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Example: Dense Matrix-Vector Multiplication

  • Tasks can be of different size

    • Granularity of Task

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Task-Dependency Graph

  • In most cases, there are dependencies between the different tasks

    • Certain task(s) can only start once some other task(s) have finished

      • Example: Producer-consumer relationships

  • These dependencies are represented using a DAG called a task-dependency graph

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Task-Dependency Graph (cont)

  • A task-dependency graph is a directed acyclic graph in which the nodes represent tasks and the directed edges indicate the dependences between them

    • The task corresponding to a node can be executed when all tasks connected to this node by incoming edges have been completed.

    • The number and size of the tasks that the problem is decomposed into determines the granularity of the decomposition.

      • Called fine-grained for a large nr of small tasks

      • Called coarse-grained for a small nr of large tasks

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Task-Dependency Graph (cont)

  • Key Concepts Derived from Task-Dependency Graph

    • Degree of Concurrency

      • The number of tasks that can be executed concurrently

        • We are usually most concerned about the average degree of concurrency

    • Critical Path

      • The longest vertex-weighted path in the graph

        • The weights inside nodes represent the task size

        • Is the sum of the weights of nodes along the path

    • The degree of concurrency and critical path length normally increase as granularity becomes smaller.

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Task-Interaction Graph

  • Captures the pattern of interaction between tasks

  • This graph usually contains the task-dependency graph as a subgraph.

    • True since there may be interactions between tasks even if there are no dependencies.

    • These interactions usually due to accesses of shared data

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Task Dependency and Interaction Graphs

  • These graphs are important in developing effective mapping of the tasks onto the different processors

  • Need to maximize concurrency and minimize overheads.

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Processes vs Processors

  • Process vs Processor

    • Considered distinct concepts in this chapter.

    • Process: A logical computing agent that performs tasks.

    • Processor: Hardware units that physically perform computation.

    • Usually a 1:1 correspondence between processors and processes.

    • However, this distinction provides additional flexibility

  • In order to obtain any speedup over sequential programming, parallel program must have several processes active at the same time, working on different tasks.

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Mapping Tasks to Processes

  • Mapping: The way that tasks are assigned to processes for execution.

  • Illustrated in Figures 3.5 and 3.7

  • Good maps attempt to

    • Maximize the use of concurrency by mapping independent tasks onto different processors.

    • Minimize total completion time by ensuring that tasks on the critical path are executed as quickly as they become available.

    • Map tasks with a high degree of mutual interaction to the same process.

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Decomposition Methods

  • Decomposition: Technique used to split the composition into a set of tasks.

  • Common Decomposition techniques

    • Data Decomposition

    • Recursive Decomposition

    • Exploratory Decomposition

    • Speculative Decomposition

    • Hybrid Decomposition

  • Data and Recursive decompositions are general methods.

  • Exploratory & Recursive decompositions special purpose.

task decomposition methods

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Recursive Decomposition

  • Suitable for problems that can be solved using the divide and conquer paradigm

  • Each of the subproblems generated by the divide step becomes a new task.

  • Results in natural concurrency, as different subproblems can be solved concurrently

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Another Example: Finding the Minimum

  • Note that we can obtain divide-and-conquer algorithms for problems that are usually solved by using other methods.

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Recursive Decomposition

  • How good are the decompositions produced?

    • Average Concurrency?

    • Length of critical path?

  • How do the quicksort and min-finding decompositions measure up?

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Data Decomposition

  • Used to derive concurrency for problems that operate on large amounts of data

  • The idea is to derive the tasks by focusing on the multiplicity of data

  • Data decomposition is often performed in two steps:

    • Step 1: Partition the data

    • Step 2: Induce a computational partitioning from the data partitioning.

  • Which data should we partition

    • Input/Output/Intermediate?

      • All of above

      • This leads to different data decomposition methods

  • How to induce a computational partitioning

    • Use the “owner-computes” rule

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Matrix-Matrix Example (cont)

Note tasks created by previous decomposition is not unique.

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Partitioning Intermediate Data

  • The partitioning of the matrix multiplication in Figure 3.10 into four tasks can be partitioned further by partitioning intermediate data.

    • See next slide

  • The matrices Di,j created are not computed in sequential algorithm and requires a change in sequential algorithm.

  • Additionally, the creation of Di,j matrices require additional storage space.

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“Owner-Computes" Rule

  • Used when data decomposition is used to partition the work into tasks.

  • This general principle requires that each partition performs all computations that involve the data it owns.

  • This is illustrated in the next two slides.

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Exploratory Decomposition

  • Used to decompose computations that correspond to a search of the space of solutions.

  • The search space is partitioned into smaller parts and these are concurrently searched until desired solution is found.

  • The next slide shows the initial configuration for the 15 puzzle and a sequence of moves leading to the final configuration.

  • The subsequent slide shows how the state a state space search leads to the solution.

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Exploratory Decomposition

  • Not general purpose

  • After sufficient branches are generated, each node can be assigned the task to explore further down one branch

  • As soon as one task finds a solution, the other tasks can be terminated.

  • It can result in speedup and slowdown anomalies

    • The work performed by the parallel formulation of an algorithm can be either smaller or greater than that performed by the serial algorithm.

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Exploratory Decomposition

  • Not general purpose

  • Can result in speedup anomalies

    • Either engineered slow-down or superlinear speedup.

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Speculative Decomposition

  • Used to extract concurrency in problems in which the next step is one of several actions that can only be determined when the current task finishes.

  • While the current task is executing, other tasks can perform the computation of the multiple branches in parallel

  • This decomposition method guarantees some wasteful computation.

  • An alternate version is to explore only the most promising branch

    • Or most promising branches

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Speculative Decomposition

  • Difference from exploratory decompostion

    • In speculative decomposition, the input at a branch leading to multiple tasks is unknown.

    • In exploratory decomposition, the output of the multiple tasks originating at the branch is unknown.

  • Speculative decomposition can lead to more, less, or the same amount of work compared to the serial program.

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Speculative Execution

  • If predictions are wrong

    • Work is wasted

    • Work may need to be undone

      • State-restoring overhead

      • Memory/computations

  • However, it may be the only way to extract concurrency!

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Characteristics of Tasks

  • Task Generation

    • Static: All tasks are known before execution of algorithm starts.

      • Data decomposition usually results in static tasks

      • Example: Matrix Multiplication

  • Task Sizes

    • Relative amount of time to complete it

    • Uniform tasks: All require the same time

    • Non-uniform tasks: Execution time varies significantly.

  • Size of Data needed by a Task

    • Data must be available to process performing task

    • The size & location of this data may determine best process to perform task.

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Some Task Interaction Characteristics

  • Static vs Dynamic Interactions

    • Static interactions occur at predetermined times and involved predetermined tasks.

      • Ex: Matrix multiplication

    • Otherwise, interaction is dynamic

      • 15 puzzle – Tasks that finish their work can pick up an unexplored state from queue of another busy task.

  • Regular vs Irregular Interactions

    • Regular if has some structure that can be used to obtain efficient implementation

    • Otherwise, irregular.

      • Ex: In sparse matrix-vector multiplication, must scan row of matrix to find out which of the vector entries are needed

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Some Task Interactions Characteristics (cont)

  • Read-only vs Read-Write Data Sharing

    • Read-only: Task only needs to read data shared with other tasks

      • Ex: Matrix multiplication in Fig 3:10

    • Read-Write: Multiple tasks need to read and write to some shared data.

      • Using heuristic search solution to solve 15 puzzle.

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Mapping Tasks to Processors

  • A good mapping strives to achieve the following conflicting goals:

    • Reducing the amount of time processor spend interacting with each other.

    • Reducing the amount of total time that some processors are active while others are idle.

  • Good mappings attempt to reduce the parallel processing overheads

    • If Tp is the parallel runtime using p processors and Ts is the sequential runtime (for the same algorithm), then the

      the total overheadTo is p×Tp – Ts.

    • This is the work that is done by the parallel system that is beyond that required for the serial system.

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Mapping Tasks to Processors (cont)

  • Two Main sources of overheads

    • Load inbalance

      • Results in process inactivity during execution

    • Inter-process communications

      • Coordination

      • Synchronization

      • Data-sharing

  • Goal of mapping tasks to processes is to minimize the overheads.

    • Goal of minimizing both of above overheads are often in conflict with each other.

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Why Mappings can be Complicated

  • Mappings need to consider the task-dependency graph

    • Are tasks available a priority?

      • Static vs dynamic task generation

    • Computation requirements factors

      • Are they uniform or non-uniform

      • Do we know tasks a priority

    • How much data is associated with each task

  • Mappings need to consider the task-interaction graph to determine the interactions between tasks

    • Are they static or dynamic

    • Do we know about them a priori

    • Are they data instance dependent

    • Are they regular or irregular

    • Are they read-only or read-write?

  • Depending on above characteristics, different mapping techniques are required with differing complexities and costs.

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    Simple & Complex Task Interactions Example

    • Consider the task-interaction graph for image dithering

      • The color of each pixel is determined as weighted average of its original color and values of neighboring pixels

      • If break image up into square regions and assign a different task to each, have simple task interactions

    • Consider sparse matrix-vector graph.

      • Assign i-th row and i-th vector value to i-th task.

      • If j-th entry in i-th row is non-zero, then i-th row must obtain the j-th vector value from j-th task (unless i=j).

      • Result is a complex task interaction graph.

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    Mapping Techniques for Load Balancing

    • Problem: The assignment of tasks who total computational requirements are the same does not automatically ensure load balanced.

    • Each processor below is assigned three tasks, but (a) is better than (b).

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    Load Balancing Techniques

    • Static Mapping

      • The tasks are distributed among the processors prior to execution

      • Applicable for tasks that are

        • Generated statically

        • Known and/or uniform computational requirements

      • Optimal mapping for non-uniform tasks is NP-hard so requires a heuristic mapping for acceptable solutions

    • Dynamic Mapping

      • The tasks are distributed among the processors during the execution of the algorithm

      • i.e., tasks & data are migrated during execution

    • Applicable for tasks that are either

      • Generated dynamically

      • Unknown computational requirements

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    Static Mapping – Array Distribution

    • Suitable for algorithms that

      • Use data decomposition

      • Their underlying data is in the form of arrays

        • i.e., input, output, or intermediate data

    • Block Distribution

    • Cyclic Distribution

    • Block-Cyclic Distribution

    • Randomized Distribution


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    1D Block Distributions

    • Partitioning a nm two-dimensional array along one dimension among p processes.

      • Process k can be given the k-th block of n/p consecutive rows.

        • i.e, row numbers kn/p, ... ,(k+1)n/p is given to process k.

        • If n/p is not an integer,

          • all processes except the last can be given a block of n/p rows and last process the remaining block of rows

          • Alternately, the initial rows could receive n/p rows, and the rest receive n/p -1 rows

      • Similarly, process k can be given the k-th block of m/p consecutive columns.

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    2D Block Distributions

    • We could partition along more than one dimension.

    • With a d-dimensional array, we can partition along up to d dimensions.

    • If we have p process and p = p1p2, the p2, n we could partition an nn block into p subblocks of size n/p1 n/p2 and assign one to each process.

    • The preceding 1D and 2D distributions are illustrated in the next slide.

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    Block-Cyclic Distribution

    • Variation of the block distribution method.

    • Can lead to a substantial more balanced work distribution.

    • Central idea is to partition an k-dimensional array into many more blocks than the number of processes.

    • Next, the partitions are assigned to processes (& associated tasks) in a round-robin manner

      • Every process gets several non-adjacent blocks

      • Some blocks may require substantially more work than others.

      • If partitioning is fine enough, then all processes have a sampling of tasks from all parts of the original k-dimensional array.

      • This increases the chances that the work for processes will balance out

      • It also increases the chances that each process will have a process that is ready to execute at any particular time.

    • A block-cyclic distribution example is given next

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    Randomized Block Distribution

    • When the distribution of work has some special pattern, block-cyclic distributions may fail to balance computation across processes.

    • This is illustrated on next slide for a sparse matrix where the shaded area indicates areas of non-zero entries.

    • Random block distribution can be used in situations like this to better balance the load on processes.

    • The array again is partitioned into many more blocks than the number of processes.

    • Each process receives an equal number of randomly selected blocks.

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    Random Block Distributions

    • Sometimes the contributions are performed only at certain portions of an array

      • Sparse matrix-matrix multiplication

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    Random Block Distributions

    • Better load balance can be achieved via a random block distribution

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    Graphics Partitioning

    • The array-based distribution schemes are good at balancing the communications and minimizing the interactions for a wide range of algorithms.

    • However, many algorithms operate on sparse data structures and have patterns of interactions that are irregular & data dependant

    • Numerical simulations of physical phenomena involves computing the values of certain physical quantities at each mesh point.

    • Computation at a mesh point usually involves the data for that point and for points that are adjacent in the graph

    • Ideally, want to distribute mesh points in ways that balances the load & minimizes the amount of data that each process will need to access

    • Next example involves levels of a water contamination at each mesh node.

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    Graph Partitioning

    • A mapping can be achieved by directly partitioning the task interaction graph

      • E.g., Finite element mesh-based computations

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    Example: Sparse Matrix-Vector

    • Another instance of graph partitioning

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    Dynamic Load Balancing

    • Needed where static mapping produces unbalanced workload or else the task-dependency graph is dynamic

    • Centralized Schemes

      • All executable tasks are maintained by a special process or set of processes

      • If a special process manages the pool of available tasks, then it is called the master and the other processes that carry out the work are called slaves

      • When a process has no work, it takes a portion of the work from the central data structure or the master.

      • When a new task is generated, it is added to central data structure or is reported to the master.

      • Assigning too little work at a time can create a bottleneck.

      • In chunk scheduling, a process without work is given a group of tasks.

        • Too large a chunk can create load-imbalance.

        • Also, chunk sizes must be reduced near end of run.

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    Dynamic Load Balancing (cont)

    • Distributed Schemes

      • Executable tasks are distributed among processes

      • Processes exchange tasks at run time to balance work.

      • Each process can either send to or receive work from another process.

      • Some important issues each scheme must handle

        • How are sending & receiving processes paired

        • Does sender or received initiate work transfer

        • How much work is transferred each time. Must avoid work transfers being too small or too large.

        • When is work transfer initiated? When process is out of work or when process anticipates running out of work.

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    Parallel Algorithm Models

    • The Data-Parallel Model

    • The Task Graph Model

      • Closely related to Foster’s Task/Channel Model

      • Requires the task dependency graph that the Task/Channel model focuses on.

        • Dependencies usually result from communications between two tasks

      • Also requires the task-interaction graph, which also captures other interactions between tasks such as data sharing

    • The Work Pool Model

    • The Master-Slave Model

    • The Pipeline or Producer-Consumer Model

    • Hybrid Models

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    The Data Parallel Model

    • One of simplest models

    • Tasks are statically or semi-statically mapped to processes.

    • Each process performs similar operations on different data and is called data parallelism

    • Typically, computation is interspersed with interactions to synchronize or to get fresh data.

    • Decomposition is usually based on data partitioning.

      • Uniform data partitioning and static assignment produces load balance

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    The Data Parallel Model (cont)

    • Can be used for both shared memory & message passing paradigms.

    • Interaction overhead can be minimized by

      • choosing a locality preserving decomposition

      • Overlapping computation and communications, when possible

    • For most problems, the degree of parallelism increases with the size of the problem

      • Allows more processes to be used to solve larger problems

    • Example: Dense matrix multiplication

      • All tasks are identical in decomposition shown in Fig 3.10 but are applied to different data.

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    The Task Graph Model

    • The computations in a parallel algorithm can be viewed as a task-dependency graph.

    • Tasks are mapped to processes so that locality is promoted

      • Volume and frequency of interactions are reduced

    • Tasks usually mapped statically to help optimize the cost of data movement among tasks.

    • Typically used to solve problems in which the data related to a task is rather large compared to the amount of computation.

    • Asynchronous interaction methods are used to overlap interactions with computation

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    The Task Graph Model (cont.)

    • Examples of algorithms based on task graph model

      • Parallel Quicksort (Section 9.4.1)

      • Sparse Matrix Factorization

      • Multiple parallel algorithms derived from divide-and-conquer decompositions.

    • Task Parallelism

      • The type of parallelism that is expressed by the independent tasks in a task-dependency graph.

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    The Work Pool Model

    • Also called the “Task Pool Model”

    • Involves dynamic mapping of tasks onto processes for load balancing

    • Any task may be potentially be performed by any process

    • The mapping of tasks to processes can be centralized or decentralized.

    • Pointers to tasks may be stored in

      • a physically shared list, a priority queue, hash table, or tree

      • a physically distributed data structure.

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    The Work Pool Model (cont.)

    • When work is generated dynamically and a decentralized mapping is used, then a termination detection algorithm is required

    • When used with a message passing paradigm, normally the data required by the tasks is relatively small when compared to the computations

      • Tasks can be readily moved around without causing too much data interaction overhead

      • Granularity of tasks can be adjusted to obtain desired tradeoff between load imbalance and the overhead of adding and extracting tasks

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    The Work Pool Model (cont.)

    • Examples of algorithms based on the Work Pool Model

      • Chunk-Scheduling

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    Master-Slave Model or Master-Worker

    • Also called the Manager-Worker model

    • One or more master processes generate work and allocate it to workers

      • Managers can allocate tasks in advance if they can estimate the size of tasks or if a random mapping can avoid load-balancing problems

      • Normally, workers are assigned smaller tasks, as needed

    • Work can be performed in phases

      • Work in each phase is completed and workers synchronized before next phase is started.

    • Normally, any worker can do any assigned task

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    Master-Slave Model (cont)

    • Can be generalized to a multi-level manager-worker model

      • Top level managers feed large chunks of tasks to second-level managers

      • Second-level managers subdivide tasks to their workers and may also perform some of the work

    • Danger of manager becoming a bottleneck

      • Can happen if tasks are too small

      • Granularity of tasks should be chosen so that cost of doing work dominates cost of synchronization

      • Waiting time may be reduced if worker requests are non-deterministic.

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    Master-Slave Model (cont)

    • Examples of algorithms based on the Master-Slave Model

      • A master-slave example for centralized load-balancing is mentioned for centralized dynamic load balancing in Section 3.4.2 (page 130)

      • Several examples are given in textbook, Barry Wilkinson and Michael Allen, “Parallel Programming: Techniques and Applications Using Networked Workstations and Parallel Computers”, 1st or 2nd Edition,1999 & 2005, Prentice Hall.

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    Pipeline or Producer-Consumer Model

    • Similiar to the “linear array model” studied in Akl’s textbook.

    • A stream of data is passed through a succession of processes, each of which performs some task on it.

    • Called Stream Parallelism

    • With exception of process initiating the work for the pipeline,

      • Arrival of new data triggers the execution of a new task by a process in the pipeline.

      • Each process can be viewed as a consumer of the data items produced by the process preceding it

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    Pipeline or Producer-Consumer Model (cont)

    • Each process in pipeline can be viewed as a producer of data for the process following it.

  • The pipeline is a chain of producers and consumers

  • The pipeline does not need to be a linear chain. Instead, it can be a directed graph.

  • Process could form pipelines in form of

    • Linear or multidimensional arrays

    • Trees

    • General graphs with or without cycles

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    Pipeline or Producer-Consumer Model (cont)

    • Load balancing is a function of task granularity

      • With larger tasks, it takes longer to fill up the pipeline

        • This keeps tasks waiting

      • Too fine a granularity increases overhead, as processes will need to receive new data and initiate a new task after a small amount of computation

    • Examples of algorithms based on this model

      • A two-dimensional pipeline is used in the parallel LU factorization algorithm discussed in Section 8.3.1

      • An entire chapter is devoted to this model in previously mentioned textbook by Wilkinson & Allen.

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    Hybrid Models

    • In some cases, more than one model may be used in designing an algorithm, resulting in a hybrid algorithm

    • Parallel quicksort (Section 3.2.5 and 9.4.1) is an application for which a hybrid model is ideal.