This presentation covers the basics of parallel computing. Beginning with a
brief overview and some concepts and terminology associated with parallel
computing, the topics of parallel memory architectures and programming models
are then explored. These topics are followed by a discussion on a number of
issues related to designing parallel programs. The last portion of the
presentation is spent examining how to parallelize several different types
of serial programs.
Level/Prerequisites: None
Overview
What is Parallel Computing?
Traditionally, software has been written for serial
computation:
To be run on a single computer having a single Central Processing
Unit (CPU);
A problem is broken into a discrete series of instructions.
Instructions are executed one after another.
Only one instruction may execute at any moment in time.
In the simplest sense, parallel computing is the simultaneous
use of multiple compute resources to solve a computational problem.
To be run using multiple CPUs
A problem is broken into discrete parts that can be solved concurrently
Each part is further broken down to a series of instructions
Instructions from each part execute simultaneously on different CPUs
The compute resources can include:
A single computer with multiple processors;
An arbitrary number of computers connected by a network;
A combination of both.
The computational problem usually demonstrates characteristics such as
the ability to be:
Broken apart into discrete pieces of work that can be solved
simultaneously;
Execute multiple program instructions at any moment in time;
Solved in less time with multiple compute resources than with a single
compute resource.
Parallel computing is an evolution of serial computing that
attempts to emulate what has always been the state of affairs in the natural
world: many complex, interrelated events happening at the same time, yet
within a sequence. Some examples:
Planetary and galactic orbits
Weather and ocean patterns
Tectonic plate drift
Rush hour traffic in LA
Automobile assembly line
Daily operations within a business
Building a shopping mall
Ordering a hamburger at the drive through.
Traditionally, parallel computing has been considered to be
"the high end of computing" and has been motivated by numerical
simulations of complex systems and "Grand Challenge Problems" such as:
weather and climate
chemical and nuclear reactions
biological, human genome
geological, seismic activity
mechanical devices - from prosthetics to spacecraft
electronic circuits
manufacturing processes
Today, commercial applications are providing an equal or greater driving
force in the development of faster computers.
These applications require the processing of large
amounts of data in sophisticated ways. Example applications include:
parallel databases, data mining
oil exploration
web search engines, web based business services
computer-aided diagnosis in medicine
management of national and multi-national corporations
advanced graphics and virtual reality, particularly in the entertainment
industry
networked video and multi-media technologies
collaborative work environments
Ultimately, parallel computing is an attempt to maximize the infinite but
seemingly scarce commodity called time.
Overview
Why Use Parallel Computing?
The primary reasons for using parallel computing:
Save time - wall clock time
Solve larger problems
Provide concurrency (do multiple things at the same time)
Other reasons might include:
Taking advantage of non-local resources - using available compute
resources on a wide area network, or even the Internet when local
compute resources are scarce.
Cost savings - using multiple "cheap" computing resources instead
of paying for time on a supercomputer.
Overcoming memory constraints - single computers have very finite
memory resources. For large problems, using the memories
of multiple computers may overcome this obstacle.
Limits to serial computing - both physical and practical reasons pose
significant constraints to simply building ever faster serial computers:
Transmission speeds - the speed of a serial computer is directly
dependent upon how fast data can move through hardware.
Absolute limits are the speed of light (30 cm/nanosecond) and the
transmission limit of copper wire (9 cm/nanosecond). Increasing
speeds necessitate increasing proximity of processing elements.
Limits to miniaturization - processor technology is allowing an
increasing number of transistors to be placed on a chip. However,
even with molecular
or atomic-level components, a limit will be reached on how small
components can be.
Economic limitations - it is increasingly expensive to make a single
processor faster. Using a larger number of moderately fast
commodity processors to
achieve the same (or better) performance is less expensive.
The future: during the past 10 years, the trends indicated by ever faster
networks, distributed systems, and multi-processor computer architectures
(even at the desktop level) suggest that parallelism is the future
of computing.
Concepts and Terminology
von Neumann Architecture
For over 40 years, virtually all computers have followed a common machine
model known as the von Neumann computer. Named after the Hungarian
mathematician John von Neumann.
A von Neumann computer uses the stored-program concept.
The CPU executes a stored program that specifies
a sequence of read and write operations on the memory.
Basic design:
Memory is used to store both program and data instructions
Program instructions are coded data which tell the computer to
do something
Data is simply information to be used by the program
A central processing unit (CPU) gets instructions and/or data from
memory, decodes the instructions and then sequentially
performs them.
Concepts and Terminology
Flynn's Classical Taxonomy
There are different ways to classify parallel computers. One of the more
widely used classifications, in use since 1966, is called Flynn's Taxonomy.
Flynn's taxonomy distinguishes multi-processor computer architectures
according
to how they can be classified along the two independent dimensions of
Instruction and Data. Each of these dimensions
can have only one of two possible states: Single or
Multiple.
The matrix below defines the 4 possible classifications according to Flynn.
S I S D
Single Instruction, Single Data
S I M D
Single Instruction, Multiple Data
M I S D
Multiple Instruction, Single Data
M I M D
Multiple Instruction, Multiple Data
Single Instruction, Single Data (SISD):
A serial (non-parallel) computer
Single instruction: only one instruction stream is
being acted on by the CPU during any one clock cycle
Single data: only one data stream is being used as input during any one clock cycle
Deterministic execution
This is the oldest and until recently, the most prevalent form of computer
Examples: most PCs, single CPU workstations and mainframes
Single Instruction, Multiple Data (SIMD):
A type of parallel computer
Single instruction: All processing units execute the same instruction at any given clock cycle
Multiple data: Each processing unit can operate on a different data element
This type of machine typically has an instruction dispatcher, a very
high-bandwidth internal network, and a very large array of very
small-capacity instruction units.
Best suited for specialized problems characterized by a high degree of
regularity,such as image processing.
Synchronous (lockstep) and deterministic execution
Two varieties: Processor Arrays and Vector Pipelines
Vector Pipelines: IBM 9000, Cray C90, Fujitsu VP, NEC SX-2,
Hitachi S820
Multiple Instruction, Single Data (MISD):
A single data stream is fed into multiple processing units.
Each processing unit operates on the data independently via independent
instruction streams.
Few actual examples of this class of parallel computer have ever existed.
One is the experimental Carnegie-Mellon C.mmp computer (1971).
Some conceivable uses might be:
multiple frequency filters operating on a single signal stream
multiple cryptography algorithms attempting to crack a single coded
message.
Multiple Instruction, Multiple Data (MIMD):
Currently, the most common type of parallel computer. Most modern
computers fall into this category.
Multiple Instruction: every processor may be executing a different
instruction stream
Multiple Data: every processor may be working with a different data
stream
Execution can be synchronous or asynchronous, deterministic or
non-deterministic
Examples: most current supercomputers, networked parallel computer
"grids" and multi-processor SMP computers - including some types of PCs.
Concepts and Terminology
Some General Parallel Terminology
Like everything else, parallel computing has its own "jargon". Some of the
more commonly used terms associated with parallel computing are listed below.
Most of these will be discussed in more detail later.
Task
A logically discrete section of computational work. A task is typically a
program or program-like set of instructions that is executed by a processor.
Parallel Task
A task that can be executed by multiple processors safely (yields correct
results)
Serial Execution
Execution of a program sequentially, one statement at a time. In the
simplest sense, this is what happens on a one processor machine. However,
virtually all parallel tasks will have sections of a parallel program that
must be executed serially.
Parallel Execution
Execution of a program by more than one task, with each task being able to
execute the same or different statement at the same moment in time.
Shared Memory
From a strictly hardware point of view, describes a computer architecture
where all processors have direct (usually bus based) access to common
physical memory. In a programming sense, it describes a model where
parallel tasks all have the same "picture" of memory and can directly
address and access the same logical memory locations regardless
of where the physical memory actually exists.
Distributed Memory
In hardware, refers to network based memory access for physical memory that
is not common. As a programming model, tasks can only logically "see"
local machine memory and must use communications to access memory on other
machines where other tasks are executing.
Communications
Parallel tasks typically need to exchange data. There are several ways this can be
accomplished, such as through a shared memory bus or over a network, however the actual
event of data exchange is commonly referred to as communications regardless of the method
employed.
Synchronization
The coordination of parallel tasks in real time, very often associated with
communications. Often implemented by establishing a synchronization point within an
application where a task
may not proceed further until another task(s) reaches the same or logically equivalent
point.
Synchronization usually involves waiting by at least one task, and can therefore cause
a parallel application's wall clock execution time to increase.
Granularity
In parallel computing, granularity is a qualitative measure of the ratio
of computation to communication.
Coarse: relatively large amounts of computational work
are done between communication events
Fine: relatively small amounts of computational work are
done between communication events
Observed Speedup
Observed speedup of a code which has been parallelized, defined as:
wall-clock time of serial execution
wall-clock time of parallel execution
One of the simplest and most widely used indicators for a parallel program's performance.
Parallel Overhead
The amount of time required to coordinate parallel tasks, as opposed to
doing useful work. Parallel overhead can include factors such as:
Task start-up time
Synchronizations
Data communications
Software overhead imposed by parallel compilers, libraries, tools,
operating system, etc.
Task termination time
Massively Parallel
Refers to the hardware that comprises a given parallel system - having many processors.
The meaning of many keeps increasing, but currently BG/L pushes this number
to 6 digits.
Scalability
Refers to a parallel system's (hardware and/or software) ability to demonstrate
a proportionate increase in parallel speedup with the addition of more processors.
Factors that contribute to scalability include:
Hardware - particularly memory-cpu bandwidths and network communications
Application algorithm
Parallel overhead related
Characteristics of your specific application and coding
Parallel Computer Memory Architectures
Shared Memory
General Characteristics:
Shared memory parallel computers vary widely, but generally have in common
the ability for all processors to access all memory as global address space.
Multiple processors can operate independently but share the same memory
resources.
Changes in a memory location effected by one processor are visible to all
other processors.
Shared memory machines can be divided into two main classes based upon
memory access times: UMA and NUMA.
Uniform Memory Access (UMA):
Most commonly represented today by Symmetric Multiprocessor (SMP)
machines
Identical processors
Equal access and access times to memory
Sometimes called CC-UMA - Cache Coherent UMA.
Cache coherent means if one processor updates a location in shared
memory, all
the other processors know about the update. Cache coherency is
accomplished at the hardware level.
Non-Uniform Memory Access (NUMA):
Often made by physically linking two or more SMPs
One SMP can directly access memory of another SMP
Not all processors have equal access time to all memories
Memory access across link is slower
If cache coherency is maintained, then may also be called CC-NUMA -
Cache Coherent NUMA
Advantages:
Global address space provides a user-friendly programming perspective
to memory
Data sharing between tasks is both fast and uniform due to the proximity
of memory to CPUs
Disadvantages:
Primary disadvantage is the lack of scalability between memory and CPUs.
Adding more CPUs can geometrically increases traffic on the shared
memory-CPU path, and for cache coherent systems, geometrically increase
traffic associated with cache/memory management.
Programmer responsibility for synchronization constructs that insure
"correct" access of global memory.
Expense: it becomes increasingly difficult and expensive to design and
produce shared memory machines with ever increasing numbers of
processors.
Parallel Computer Memory Architectures
Distributed Memory
General Characteristics:
Like shared memory systems, distributed memory systems vary widely but
share a common characteristic. Distributed memory systems require a
communication network to connect inter-processor memory.
Processors have their own local memory. Memory addresses in one
processor do not map to another processor, so there is no concept of
global address space across all processors.
Because each processor has its own local memory, it operates
independently. Changes it makes to its local memory have no effect
on the memory of other processors. Hence, the concept of cache
coherency does not apply.
When a processor needs access to data in another processor, it is
usually the task of the programmer to explicitly define how and when
data is communicated. Synchronization between tasks is likewise the
programmer's responsibility.
The network "fabric" used for data transfer varies widely, though it can
can be as simple as Ethernet.
Advantages:
Memory is scalable with number of processors. Increase the number of
processors and the size of memory increases proportionately.
Each processor can rapidly access its own memory without interference
and without the overhead incurred with trying to maintain cache
coherency.
Cost effectiveness: can use commodity, off-the-shelf processors and
networking.
Disadvantages:
The programmer is responsible for many of the details associated with
data communication between processors.
It may be difficult to map existing data structures, based on global
memory, to this memory organization.
Non-uniform memory access (NUMA) times
Parallel Computer Memory Architectures
Hybrid Distributed-Shared Memory
Summarizing a few of the key characteristics of
shared and distributed memory machines:
Comparison of Shared and Distributed Memory Architectures
Architecture
CC-UMA
CC-NUMA
Distributed
Examples
SMPs Sun Vexx DEC/Compaq SGI Challenge IBM POWER3
SGI Origin Sequent HP Exemplar DEC/Compaq
IBM POWER4 (MCM)
Cray T3E Maspar IBM SP2
Communications
MPI Threads OpenMP shmem
MPI Threads OpenMP shmem
MPI
Scalability
to 10s of processors
to 100s of processors
to 1000s of processors
Draw Backs
Memory-CPU bandwidth
Memory-CPU bandwidth Non-uniform access times
System administration Programming is hard to develop and maintain
Software Availability
many 1000s ISVs
many 1000s ISVs
100s ISVs
The largest and fastest computers in the world today employ both shared
and distributed memory architectures.
The shared memory component is usually a cache coherent SMP machine.
Processors on a given SMP can address that machine's memory as global.
The distributed memory component is the networking of multiple SMPs.
SMPs know only about their own memory - not the memory on another SMP.
Therefore, network communications are required to move data from one
SMP to another.
Current trends seem to indicate that this type of memory architecture
will continue to prevail and increase at the high end of computing for
the foreseeable future.
Advantages and Disadvantages: whatever is common to both shared and
distributed memory architectures.
Parallel Programming Models
Overview
There are several parallel programming models in common use:
Shared Memory
Threads
Message Passing
Data Parallel
Hybrid
Parallel programming models exist as an abstraction above hardware
and memory architectures.
Although it might not seem apparent, these models are NOT specific
to a particular type of machine or memory architecture. In fact, any
of these models can (theoretically) be implemented on any underlying
hardware. Two examples:
Shared memory model on a distributed memory machine:
Kendall Square Research (KSR) ALLCACHE approach.
Machine memory was physically
distributed, but appeared to the user as a single shared memory
(global address space). Generically, this approach is referred to as
"virtual shared memory". Note: although KSR is no longer in business,
there is no reason to suggest that a similar implementation will not
be made available by another vendor in the future.
Message passing model on a shared memory machine: MPI on SGI Origin.
The SGI Origin employed the CC-NUMA type of shared memory architecture,
where every task has direct access to global memory. However, the
ability to
send and receive messages with MPI, as is commonly done over a network
of distributed memory machines, is not only implemented but is very
commonly used.
Which model to use is often a combination of what is available and personal
choice. There is no "best" model, although there certainly are better
implementations of some models over others.
The following sections describe each of the models mentioned above, and
also discuss some of their actual implementations.
Parallel Programming Models
Shared Memory Model
In the shared-memory programming model, tasks share a common address space,
which they read and write asynchronously.
Various mechanisms such as locks / semaphores may be used to control
access to the shared memory.
An advantage of this model from the programmer's point of view is that the
notion of data "ownership" is lacking, so there is no need to specify
explicitly the communication of data between tasks. Program
development can often be simplified.
An important disadvantage in terms of performance is that it becomes
more difficult to understand and manage data locality.
Implementations:
On shared memory platforms, the native compilers translate
user program variables into actual memory addresses, which are global.
No common distributed memory platform implementations currently exist.
However, as mentioned previously in the Overview section, the KSR
ALLCACHE approach provided a shared memory view of data even though
the physical memory of the machine was distributed.
Parallel Programming Models
Threads Model
In the threads model of parallel programming, a single process can have
multiple, concurrent execution paths.
Perhaps the most simple analogy that can be used to describe threads is the
concept of a single program that includes a number of subroutines:
The main program a.out is scheduled to run by the
native operating system. a.out loads and acquires all of the
necessary system and user resources to run.
a.out performs some serial work, and then creates
a number of tasks (threads) that can be scheduled and run by the
operating system concurrently.
Each thread has local data, but also, shares the entire resources of
a.out. This saves the overhead associated with
replicating a program's resources for each thread. Each thread also
benefits from a global memory view because it shares the memory space
of a.out.
A thread's work may best be described as a subroutine within
the main program. Any thread can execute any subroutine at the
same time as other threads.
Threads communicate with each other through global memory (updating
address locations). This requires synchronization constructs to insure
that more than one thread is not updating the same global address at
any time.
Threads can come and go, but a.out remains present
to provide the necessary shared resources until the
application has completed.
Threads are commonly associated with shared memory architectures and
operating systems.
Implementations:
From a programming perspective, threads implementations commonly
comprise:
A library of subroutines that are called from within
parallel source code
A set of compiler directives imbedded in either serial
or parallel source code
In both cases, the programmer is responsible for determining all
parallelism.
Threaded implementations are not new in computing. Historically,
hardware vendors have implemented their own proprietary versions of
threads. These implementations differed substantially from each other
making it difficult for programmers to develop portable threaded
applications.
Unrelated standardization efforts have resulted in
two very different implementations of threads:
POSIX Threads and OpenMP.
POSIX Threads
Library based; requires parallel coding
Specified by the IEEE POSIX 1003.1c standard (1995).
C Language only
Commonly referred to as Pthreads.
Most hardware vendors now offer Pthreads in addition to their
proprietary threads implementations.
Very explicit parallelism; requires significant programmer
attention to detail.
OpenMP
Compiler directive based; can use serial code
Jointly defined and endorsed by a group of major computer hardware
and software vendors.
The OpenMP Fortran API was released October 28, 1997. The C/C++ API
was released in late 1998.
Portable / multi-platform, including Unix and Windows NT platforms
Available in C/C++ and Fortran implementations
Can be very easy and simple to use - provides for "incremental
parallelism"
Microsoft has its own implementation for threads, which is not related
to the UNIX POSIX standard or OpenMP.
Parallel Programming Models
Message Passing Model
The message passing model demonstrates the following characteristics:
A set of tasks that use their own local memory during computation.
Multiple tasks can reside on the same physical machine as well
across an arbitrary number of machines.
Tasks exchange data through communications by sending and
receiving messages.
Data transfer usually requires cooperative operations to be performed
by each process. For example, a send operation must have a matching
receive operation.
Implementations:
From a programming perspective, message passing implementations commonly
comprise a library of subroutines that are imbedded in source code.
The programmer is responsible for determining all parallelism.
Historically, a variety of message passing libraries have been
available since the 1980s. These implementations differed substantially
from each other making it difficult for programmers to develop portable
applications.
In 1992, the MPI Forum was formed with the primary goal of establishing
a standard interface for message passing implementations.
Part 1 of the Message Passing Interface (MPI) was released in
1994. Part 2 (MPI-2) was released in 1996.
Both MPI specifications are available on the web at
www.mcs.anl.gov/Projects/mpi/standard.html.
MPI is now the "de facto" industry
standard for message passing, replacing virtually all other
message passing implementations used for production work.
Most, if not all of the popular parallel computing platforms
offer at least one implementation of MPI. A few offer a full
implementation of MPI-2.
For shared memory architectures, MPI implementations usually don't
use a network for task communications. Instead, they use shared
memory (memory copies) for performance reasons.
Parallel Programming Models
Data Parallel Model
The data parallel model demonstrates the following characteristics:
Most of the parallel work focuses on performing operations on a
data set. The data set is typically organized into a common
structure, such as an array or cube.
A set of tasks work collectively on the same data structure, however,
each task works on a different partition of the same data structure.
Tasks perform the same operation on their partition of work, for
example, "add 4 to every array element".
On shared memory architectures, all tasks may have access to the data
structure through global memory. On distributed memory architectures
the data structure is split up and resides as "chunks" in the local
memory of each task.
Implementations:
Programming with the data parallel model is usually accomplished by
writing
a program with data parallel constructs. The constructs can be calls to
a data parallel subroutine library or, compiler directives recognized by
a data parallel compiler.
Fortran 90 and 95 (F90, F95): ISO/ANSI standard extensions to
Fortran 77.
Contains everything that is in Fortran 77
New source code format; additions to character set
Additions to program structure and commands
Variable additions - methods and arguments
Pointers and dynamic memory allocation added
Array processing (arrays treated as objects) added
Recursive and new intrinsic functions added
Many other new features
Implementations are available for most common parallel platforms.
High Performance Fortran (HPF): Extensions to Fortran 90 to
support data
parallel programming.
Contains everything in Fortran 90
Directives to tell compiler how to distribute data added
Assertions that can improve optimization of generated code added
Data parallel constructs added (now part of Fortran 95)
Implementations are available for most common parallel platforms.
Compiler Directives: Allow the programmer to specify the
distribution and alignment of data. Fortran implementations are
available for most common parallel platforms.
Distributed memory implementations of this model usually have the
compiler convert the program into standard code with calls to a message
passing library (MPI usually) to distribute the data to all the
processes. All message passing is done invisibly to the programmer.
Parallel Programming Models
Other Models
Other parallel programming models besides those previously mentioned
certainly exist, and will continue to evolve along with the ever
changing world of computer hardware and software. Only three of
the more common ones are mentioned here.
Hybrid:
In this model, any two or more parallel programming models
are combined.
Currently, a common example of a hybrid model is the combination
of the message passing model (MPI) with either the threads model
(POSIX threads) or the shared memory model (OpenMP). This hybrid
model lends itself well to the increasingly common hardware
environment of networked SMP machines.
Another common example of a hybrid model is combining data
parallel with message passing. As mentioned in the
data parallel model section previously, data parallel
implementations (F90, HPF) on distributed memory architectures
actually use message passing to transmit data between tasks,
transparently to the programmer.
Single Program Multiple Data (SPMD):
SPMD is actually a "high level" programming model that can be
built upon any combination of the previously mentioned parallel
programming models.
A single program is executed by all tasks simultaneously.
At any moment in time, tasks can be executing the same or different
instructions within the same program.
SPMD programs usually have the necessary logic programmed into them to
allow different tasks to branch or conditionally execute only those
parts of the program they are designed to execute. That is, tasks
do not necessarily have to execute the entire program - perhaps only a
portion of it.
All tasks may use different data
Multiple Program Multiple Data (MPMD):
Like SPMD, MPMD is actually a "high level" programming model that can
be built upon any combination of the previously mentioned parallel
programming models.
MPMD applications typically have multiple executable object files
(programs). While the application is being run in parallel, each
task can be executing
the same or different program as other tasks.
All tasks may use different data
Designing Parallel Programs
Automatic vs. Manual Parallelization
Designing and developing parallel programs has characteristically been a
very manual process. The programmer is typically responsible for
both identifying and actually implementing parallelism.
Very often, manually developing parallel codes is a time consuming,
complex, error-prone and iterative process.
For a number of years now, various tools have been available to assist
the programmer with converting serial programs into parallel programs.
The most common type of tool used to automatically parallelize a serial
program is a parallelizing compiler or pre-processor.
A parallelizing compiler generally works in two different ways:
Fully Automatic
The compiler analyzes the source code and
identifies opportunities for parallelism.
The analysis includes
identifying inhibitors to parallelism and possibly a cost
weighting on whether or not the parallelism would actually
improve performance.
Loops (do, for) loops are the most frequent target for
automatic parallelization.
Programmer Directed
Using "compiler directives" or possibly compiler flags,
the programmer explicitly tells the compiler how to
parallelize the code.
May be able to be used in conjunction with some degree of
automatic parallelization also.
If you are beginning with an existing serial code and have time
or budget constraints, then automatic parallelization may be
the answer. However, there are several important caveats that
apply to automatic parallelization:
Wrong results may be produced
Performance may actually degrade
Much less flexible than manual parallelization
Limited to a subset (mostly loops) of code
May actually not parallelize code if the analysis suggests there
are inhibitors or the code is too complex
Most automatic parallelization tools are for Fortran
The remainder of this section applies to the manual method of
developing parallel codes.
Designing Parallel Programs
Understand the Problem and the Program
Undoubtedly, the first step in developing parallel software is to
first understand the problem that you wish to solve in parallel.
If you are starting with a serial program, this necessitates
understanding the existing code also.
Before spending time in an attempt to develop a parallel solution
for a problem, determine whether or not the problem is one that can
actually be parallelized.
Example of Parallelizable Problem:
Calculate the potential energy for each of several thousand
independent conformations of a molecule.
When done, find the minimum energy conformation.
This problem is able to be solved in parallel. Each of the
molecular conformations is independently determinable.
The calculation of the minimum energy conformation is also a
parallelizable problem.
Example of a Non-parallelizable Problem:
Calculation of the Fibonacci series (1,1,2,3,5,8,13,21,...) by use of
the formula:
F(k + 2) = F(k + 1) + F(k)
This is a non-parallelizable problem because the calculation of the
Fibonacci sequence as shown would entail
dependent calculations rather than independent ones.
The calculation of the k + 2 value uses those of
both k + 1 and k. These three terms cannot be calculated
independently and therefore, not in parallel.
Identify the program's hotspots:
Know where most of the real work is being done.
The majority of scientific and technical programs usually
accomplish most of their work in a few places.
Profilers and performance analysis tools can help here
Focus on parallelizing the hotspots and ignore those sections
of the program that account for little CPU usage.
Identify bottlenecks in the program
Are there areas that are disproportionately slow, or cause
parallelizable work to halt or be deferred?
For example, I/O is usually something that slows a program down.
May be possible to restructure the program or use a different
algorithm to reduce or eliminate unnecessary slow areas
Identify inhibitors to parallelism. One common class of inhibitor
is data dependence, as demonstrated by the Fibonacci sequence
above.
Investigate other algorithms if possible. This may be the single most
important consideration when designing a parallel application.
Designing Parallel Programs
Partitioning
One of the first steps in designing a parallel program is to break the
problem into discrete "chunks" of work that can be distributed to
multiple tasks. This is known as decomposition or partitioning.
There are two basic ways to partition computational work among parallel
tasks: domain decomposition and
functional decomposition.
Domain Decomposition:
In this type of partitioning, the data associated with a problem
is decomposed. Each parallel task then works on a portion of
of the data.
There are different ways to partition data:
Functional Decomposition:
In this approach, the focus is on the computation that is to be
performed rather than on the data manipulated by the computation.
The problem is decomposed according to the work that must be done.
Each task then performs a portion of the overall work.
Functional decomposition lends itself well to problems that can be
split into different tasks. For example:
Ecosystem Modeling
Each program calculates the population
of a given group, where each group's growth depends on that of its
neighbors. As time progresses, each process calculates
its current state, then exchanges information with the neighbor
populations. All tasks then progress to calculate the state at the
next time step.
Signal Processing
An audio signal data set is passed
through four distinct computational filters. Each filter is a
separate process. The first segment of data must pass through the
first filter before progressing to the second. When it does, the
second segment of data passes through the first filter. By the time
the fourth segment of data is in the first filter, all four
tasks are busy.
Climate Modeling
Each model component can be thought of as a separate task.
Arrows represent exchanges of data between components during
computation: the atmosphere model generates wind velocity data
that are used by the ocean model, the ocean model generates sea
surface temperature data that are used by the atmosphere model,
and so on.
Combining these two types of problem decomposition is common and natural.
Designing Parallel Programs
Communications
Who Needs Communications?
The need for communications between tasks depends upon your problem:
You DON'T need communications
Some types of problems can be decomposed and executed in parallel
with virtually no need for tasks to share data. For example, imagine an
image processing operation where every pixel in a black and white image
needs to have its color reversed. The image data can easily be
distributed to multiple tasks that then act independently of each other
to do their portion of the work.
These types of problems are often called embarrassingly
parallel
because they are so straight-forward. Very little inter-task communication
is required.
You DO need communications
Most parallel applications are not quite so simple, and do require
tasks to
share data with each other. For example, a 3-D heat diffusion problem
requires a task to know the temperatures calculated by the tasks that have
neighboring data. Changes to neighboring data has a direct effect on that
task's data.
Factors to Consider:
There are a number of important factors to consider when designing your
program's inter-task communications:
Cost of communications
Inter-task communication virtually always implies overhead.
Machine cycles and resources that could be used for computation
are instead used to package and transmit data.
Communications frequently require some type of synchronization
between tasks, which can result in tasks spending time "waiting"
instead of doing work.
Competing communication traffic can saturate the available network
bandwidth, further aggravating performance problems.
Latency vs. Bandwidth
latency is the time it takes to send a minimal (0 byte)
message from point A to point B. Commonly expressed as microseconds.
bandwidth is the amount of data that can be communicated
per unit of time. Commonly expressed as megabytes/sec.
Sending many small messages can cause latency to dominate communication
overheads. Often it is more efficient to package small messages into a
larger message, thus increasing the effective communications bandwidth.
Visibility of communications
With the Message Passing Model, communications are explicit and
generally quite visible and under the control of the programmer.
With the Data Parallel Model, communications often occur
transparently to the programmer, particularly on distributed
memory architectures. The programmer may not even be able to
know exactly how inter-task communications are being accomplished.
Synchronous vs. asynchronous communications
Synchronous communications require some type of "handshaking"
between tasks that are sharing data. This can be explicitly
structured in code by the programmer, or it may happen at a
lower level unknown to the programmer.
Synchronous communications are often referred to as
blocking communications since other work must
wait until the communications have completed.
Asynchronous communications allow tasks to transfer data independently
from one another. For example, task 1 can prepare and send a
message to task 2, and then immediately begin doing other work.
When task 2 actually receives the data doesn't matter.
Asynchronous communications are often referred to as
non-blocking communications since other work can
be done while the communications are taking place.
Interleaving computation with communication is the single greatest
benefit for using asynchronous communications.
Scope of communications
Knowing which tasks must communicate with each other is critical during
the design stage of a parallel code. Both of the two scopings
described below can be implemented synchronously or asynchronously.
Point-to-point - involves two tasks with one task
acting as the sender/producer of data, and the other acting as
the receiver/consumer.
Collective - involves data sharing between more than
two tasks, which are often specified as being members in a common
group, or collective. Some common variations (there are more):
Efficiency of communications
Very often, the programmer will have a choice with regard to
factors that can affect communications performance. Only a
few are mentioned here.
Which implementation for a given model should be used? Using
the Message Passing Model as an
example, one MPI implementation may be faster on a given
hardware platform than another.
What type of communication operations should be used? As
mentioned previously, asynchronous communication operations
can improve overall program performance.
Network media - some platforms may offer more than one network
for communications. Which one is best?
Overhead and Complexity
Finally, realize that this is only a partial list of things to consider!!!
Designing Parallel Programs
Synchronization
Types of Synchronization:
Barrier
Usually implies that all tasks are involved
Each task performs its work until it reaches the barrier. It then
stops, or "blocks".
When the last task reaches the barrier, all tasks are synchronized.
What happens from here varies. Often, a serial section of work must
be done. In other cases, the tasks are automatically released to
continue their work.
Lock / semaphore
Can involve any number of tasks
Typically used to serialize (protect) access to global data
or a section of code. Only one task at a time may use (own) the
lock / semaphore / flag.
The first task to acquire the lock "sets" it. This task can then
safely (serially) access the protected data or code.
Other tasks can attempt to acquire the lock but must wait until the
task that owns the lock releases it.
Can be blocking or non-blocking
Synchronous communication operations
Involves only those tasks executing a communication operation
When a task performs a communication operation, some form of
coordination is required with the other task(s) participating in
the communication. For example, before a task can perform a
send operation, it must first receive an acknowledgment from the
receiving task that it is OK to send.
Discussed previously in the Communications section.
Designing Parallel Programs
Data Dependencies
Definition:
A dependence exists between program statements when
the order of statement execution affects the results of the program.
A data dependence results from multiple use of the same
location(s) in storage by different tasks.
Dependencies are important to parallel programming because they are one
of the primary inhibitors to parallelism.
The value of A(J-1) must be computed before the value of A(J),
therefore A(J) exhibits a data dependency on A(J-1).
Parallelism is inhibited.
If Task 2 has A(J) and task 1 has A(J-1),
computing the correct value of A(J) necessitates:
Distributed memory architecture - task 2 must obtain the value
of A(J-1) from task 1 after task 1 finishes its computation
Shared memory architecture - task 2 must read A(J-1) after
task 1 updates it
Loop independent data dependence
task 1 task 2
------ ------
X = 2 X = 4
. .
. .
Y = X**2 Y = X**3
As with the previous example, parallelism is inhibited.
The value of Y is dependent on:
Distributed memory architecture - if or when the value of X is
communicated between the tasks.
Shared memory architecture - which task last stores the value of X.
Although all data dependencies are important to identify when designing
parallel programs, loop carried dependencies are particularly important
since loops are possibly the most common target of parallelization efforts.
How to Handle Data Dependencies:
Distributed memory architectures - communicate required data at
synchronization points.
Shared memory architectures -synchronize read/write operations between
tasks.
Designing Parallel Programs
Load Balancing
Load balancing refers to the practice of distributing work among tasks
so that all tasks are kept busy all of the time.
It can be considered a minimization of task idle time.
Load balancing is important to parallel programs for performance
reasons. For example, if all tasks are subject to a barrier
synchronization point, the slowest task will determine the overall
performance.
How to Achieve Load Balance:
Equally partition the work each task receives
For array/matrix operations where each task performs similar
work, evenly distribute the data set among the tasks.
For loop iterations where the work done in each iteration
is similar, evenly distribute the iterations across the tasks.
If a heterogeneous mix of machines with varying performance
characteristics are being used, be sure to use some type of performance
analysis tool to detect any load imbalances. Adjust work accordingly.
Use dynamic work assignment
Certain classes of problems result in load imbalances even if data
is evenly distributed among tasks:
Sparse arrays - some tasks will have actual data to work on
while others have mostly "zeros".
Adaptive grid methods - some tasks may need to refine their
mesh while others don't.
N-body simulations - where some particles may migrate
to/from their original task domain to another task's; where
the particles owned by some tasks require more work than
those owned by other tasks.
When the amount of work each task will perform is intentionally
variable, or is unable to be predicted, it may be helpful to use
a scheduler - task pool approach. As each task finishes
its work, it queues to get a new piece of work.
It may become necessary to design an algorithm which detects and handles
load imbalances as they occur dynamically within the code.
Designing Parallel Programs
Granularity
Computation / Communication Ratio:
In parallel computing, granularity is a qualitative measure of the ratio
of computation to communication.
Periods of computation are typically separated from periods of
communication by synchronization events.
Fine-grain Parallelism:
Relatively small amounts of computational work are done between
communication events
Low computation to communication ratio
Facilitates load balancing
Implies high communication overhead and less opportunity for
performance enhancement
If granularity is too fine it is possible that the overhead
required for communications and synchronization between tasks
takes longer than the computation.
Coarse-grain Parallelism:
Relatively large amounts of computational work are done between
communication/synchronization events
High computation to communication ratio
Implies more opportunity for performance increase
Harder to load balance efficiently
Which is Best?
The most efficient granularity is dependent on the algorithm and the
hardware environment in which it runs.
In most cases the overhead associated with communications and
synchronization is high relative to execution speed
so it is advantageous to have coarse granularity.
Fine-grain parallelism can help reduce overheads due to load imbalance.
Designing Parallel Programs
I/O
The Bad News:
I/O operations are generally regarded as inhibitors to parallelism
Parallel I/O systems are immature or not available for all platforms
In an environment where all tasks see the same filespace, write
operations will result in file overwriting
Read operations will be affected by the fileserver's ability to handle
multiple read requests at the same time
I/O that must be conducted over the network (NFS, non-local) can cause
severe bottlenecks
The Good News:
Some parallel file systems are available. For example:
GPFS: General Parallel File System for AIX (IBM)
Lustre: for Linux clusters (Cluster File Systems, Inc.)
PVFS/PVFS2: Parallel Virtual File System for Linux clusters
(Clemson/Argonne/Ohio State/others)
PanFS: Panasas ActiveScale File System for Linux clusters (Panasas,
Inc.)
HP SFS: HP StorageWorks Scalable File Share. Lustre based parallel file
system (Global File System for Linux) product from HP
The parallel I/O programming interface specification for MPI has been
available since 1996 as part of MPI-2. Vendor and "free" implementations
are now commonly available.
Some options:
If you have access to a parallel file system, investigate using
it. If you don't, keep reading...
Rule #1: Reduce overall I/O as much as possible
Confine I/O to specific serial portions of the job, and then use
parallel communications to distribute data to parallel tasks.
For example, Task 1 could read an input file and then communicate
required data to other tasks. Likewise, Task 1 could perform
write operation after receiving required data from all other tasks.
For distributed memory systems with shared filespace, perform I/O in
local, non-shared filespace.
For example, each processor may have /tmp filespace which can used.
This is usually much more efficient than performing I/O over the
network to one's home directory.
Create unique filenames for each tasks' input/output file(s)
Designing Parallel Programs
Limits and Costs of Parallel Programming
Amdahl's Law:
Amdahl's Law
states that potential program
speedup is defined by the fraction of code (P) that can be parallelized:
1
speedup = --------
1 - P
If none of the code can be parallelized, P = 0 and the speedup = 1 (no
speedup). If all of the code is parallelized, P = 1 and the speedup is
infinite (in theory).
If 50% of the code can be parallelized, maximum speedup = 2, meaning
the code will run twice as fast.
Introducing the number of processors performing the parallel fraction of
work, the relationship can be modeled by:
1
speedup = ------------
P + S
---
N
where P = parallel fraction, N = number of processors and S = serial
fraction.
It soon becomes obvious that there are limits to the scalability of
parallelism. For example, at P = .50, .90 and .99 (50%, 90% and 99% of
the code is parallelizable):
speedup
--------------------------------
N P = .50 P = .90 P = .99
----- ------- ------- -------
10 1.82 5.26 9.17
100 1.98 9.17 50.25
1000 1.99 9.91 90.99
10000 1.99 9.91 99.02
However, certain problems demonstrate increased performance by increasing
the problem size. For example:
We can increase the problem size by doubling the grid dimensions and
halving the time step. This results in four times the number of grid
points and twice the number of time steps. The timings then look like:
Problems that increase the percentage of parallel time with their size
are more scalable than problems with a fixed percentage of
parallel time.
Complexity:
In general, parallel applications are much more complex than corresponding
serial applications, perhaps an order of magnitude. Not only do you have
multiple instruction streams executing at the same time, but you also have
data flowing between them.
The costs of complexity are measured in programmer time in virtually every
aspect of the software development cycle:
Design
Coding
Debugging
Tuning
Maintenance
Adhering to "good" software development practices is essential when
when working with parallel applications - especially if somebody besides
you will have to work with the software.
Portability:
Thanks to standardization in several APIs, such as MPI, POSIX threads,
HPF and OpenMP, portability issues with parallel programs are not as
serious as in years past. However...
All of the usual portability issues associated with serial programs
apply to parallel programs. For example, if you use vendor "enhancements"
to Fortran, C or C++, portability will be a problem.
Even though standards exist for several APIs, implementations will differ
in a number of details, sometimes to the point of requiring code
modifications in order to effect portability.
Operating systems can play a key role in code portability issues.
Hardware architectures are characteristically highly variable and can
affect portability.
Resource Requirements:
The primary intent of parallel programming is to decrease execution
wall clock time, however in order to accomplish this, more CPU time
is required. For example, a parallel code that runs in 1 hour on 8
processors actually uses 8 hours of CPU time.
The amount of memory required can be greater for parallel codes than
serial codes, due to the need to replicate data and for overheads
associated with parallel support libraries and subsystems.
For short running parallel programs, there can actually be a decrease
in performance compared to a similar serial implementation. The overhead
costs associated with setting up the parallel environment, task creation,
communications and task termination can comprise a significant portion of
the total execution time for short runs.
Scalability:
The ability of a parallel program's performance to scale is a result
of a number of interrelated factors. Simply adding more machines
is rarely the answer.
The algorithm may have inherent limits to scalability. At some point,
adding more resources causes performance to decrease. Most parallel
solutions demonstrate this characteristic at some point.
Hardware factors play a significant role in scalability. Examples:
Memory-cpu bus bandwidth on an SMP machine
Communications network bandwidth
Amount of memory available on any given machine or set of machines
Processor clock speed
Parallel support libraries and subsystems software can limit scalability
independent of your application.
Designing Parallel Programs
Performance Analysis and Tuning
As with debugging, monitoring and analyzing parallel program execution
is significantly more of a challenge than for serial programs.
A number of parallel tools for execution monitoring and program analysis
are available.
Some are quite useful; some are cross-platform also.
Work remains to be done, particularly in the area of scalability.
Parallel Examples
Array Processing
This example demonstrates calculations on 2-dimensional array
elements, with the computation on each array element being
independent from other array elements.
The serial program calculates one element at a time in sequential
order.
Serial code could be of the form:
do j = 1,n
do i = 1,n
a(i,j) = fcn(i,j)
end do
end do
The calculation of elements is independent of one another -
leads to an embarrassingly parallel situation.
The problem should be computationally intensive.
Array Processing Parallel Solution 1
Arrays elements are distributed so that each processor owns a
portion of an array (subarray).
Independent calculation of array elements insures there is no
need for communication between tasks.
Distribution scheme is chosen by other criteria, e.g. unit stride
(stride of 1) through the subarrays. Unit stride maximizes
cache/memory usage.
Since it is desirable to have unit stride through the subarrays, the
choice of a distribution scheme depends on the programming language.
See the Block - Cyclic Distributions Diagram
for the options.
After the array is distributed, each task
executes the portion of the loop corresponding to the data it owns.
For example, with Fortran block distribution:
do j = mystart, myend
do i = 1,n
a(i,j) = fcn(i,j)
end do
end do
Notice that only the outer loop variables are different from the serial
solution.
One Possible Solution:
Implement as SPMD model.
Master process initializes array, sends info to worker
processes and receives results.
Worker process receives info, performs its share of
computation and sends results to master.
Using the Fortran storage scheme, perform block
distribution of the array.
Pseudo code solution:
red highlights changes for
parallelism.
find out if I am MASTER or WORKER
if I am MASTER
initialize the array
send each WORKER info on part of array it owns
send each WORKER its portion of initial array
receive from each WORKER results
else if I am WORKER
receive from MASTER info on part of array I own
receive from MASTER my portion of initial array
# calculate my portion of array
do j = my first column,my last column
do i = 1,n
a(i,j) = fcn(i,j)
end do
end do
send MASTER results
endif
Array Processing Parallel Solution 2: Pool of Tasks
The previous array solution demonstrated static load balancing:
Each task has a fixed amount of work to do
May be significant idle time for faster or more lightly loaded
processors - slowest tasks determines overall performance.
Static load balancing is not usually a major concern if all tasks
are performing the same amount of work on identical machines.
If you have a load balance problem (some tasks work faster than
others), you may benefit by using a "pool of tasks"
scheme.
Pool of Tasks Scheme:
Two processes are employed
Master Process:
Holds pool of tasks for worker processes to do
Sends worker a task when requested
Collects results from workers
Worker Process: repeatedly does the following
Gets task from master process
Performs computation
Sends results to master
Worker processes do not know before runtime which portion of array
they will handle or how many tasks they will perform.
Dynamic load balancing occurs at run time: the faster tasks will
get more work to do.
Pseudo code solution:
red highlights changes for
parallelism.
find out if I am MASTER or WORKER
if I am MASTER
do until no more jobs
send to WORKER next job
receive results from WORKER
end do
tell WORKER no more jobs
else if I am WORKER
do until no more jobs
receive from MASTER next job
calculate array element: a(i,j) = fcn(i,j)
send results to MASTER
end do
endif
Discussion:
In the above pool of tasks example, each task calculated an individual
array element as a job. The computation to communication ratio is
finely granular.
Finely granular solutions incur more communication overhead in order
to reduce task idle time.
A more optimal solution might be to distribute more work with each job.
The "right" amount of work is problem dependent.
Parallel Examples
PI Calculation
The value of PI can be calculated in a number of ways. Consider the
following method of approximating PI
Inscribe a circle in a square
Randomly generate points in the square
Determine the number of points in the square that are also in the circle
Let r be the number of points in the circle divided by the number of
points in the square
PI ~ 4 r
Note that the more points generated, the better the approximation
Serial pseudo code for this procedure:
npoints = 10000
circle_count = 0
do j = 1,npoints
generate 2 random numbers between 0 and 1
xcoordinate = random1 ; ycoordinate = random2
if (xcoordinate, ycoordinate) inside circle
then circle_count = circle_count + 1
end do
PI = 4.0*circle_count/npoints
Note that most of the time in running this program would be
spent executing the loop
Leads to an embarrassingly parallel solution
Computationally intensive
Minimal communication
Minimal I/O
PI Calculation Parallel Solution
Parallel strategy: break the loop into portions that can be
executed by the tasks.
For the task of approximating PI:
Each task executes its portion of the loop a number of times.
Each task can do its work without requiring any information
from the other tasks (there are no data dependencies).
Uses the SPMD model. One task acts as master and collects
the results.
Pseudo code solution:
red highlights changes for
parallelism.
npoints = 10000
circle_count = 0
p = number of tasks
num = npoints/p
find out if I am MASTER or WORKER
do j = 1,num
generate 2 random numbers between 0 and 1
xcoordinate = random1 ; ycoordinate = random2
if (xcoordinate, ycoordinate) inside circle
then circle_count = circle_count + 1
end do
if I am MASTER
receive from WORKERS their circle_counts
compute PI (use MASTER and WORKER calculations)
else if I am WORKER
send to MASTER circle_count
endif
Parallel Examples
Simple Heat Equation
Most problems in parallel computing require communication among
the tasks.
A number of common problems require communication with "neighbor"
tasks.
The heat equation describes the temperature change over time,
given initial temperature distribution and boundary conditions.
A finite differencing scheme is employed to solve the
heat equation numerically on a square region.
The initial temperature is zero on the boundaries and high in the middle.
The boundary temperature is held at zero.
For the fully explicit problem, a time stepping algorithm is used.
The elements of a 2-dimensional array represent the temperature at
points on the square.
The calculation of an element is dependent upon neighbor element
values.
A serial program would contain code like:
do iy = 2, ny - 1
do ix = 2, nx - 1
u2(ix, iy) =
u1(ix, iy) +
cx * (u1(ix+1,iy) + u1(ix-1,iy) - 2.*u1(ix,iy)) +
cy * (u1(ix,iy+1) + u1(ix,iy-1) - 2.*u1(ix,iy))
end do
end do
Simple Heat Equation Parallel Solution 1
Implement as an SPMD model
The entire array is partitioned and distributed as subarrays to all
tasks. Each task owns a portion of the total array.
Determine data dependencies
interior
elements belonging to a task are independent of other tasks
border
elements are dependent upon
a neighbor task's data, necessitating communication.
Master process sends initial info to workers, checks
for convergence and collects results
Worker process calculates solution, communicating as necessary with
neighbor processes
Pseudo code solution:
red highlights changes for parallelism.
find out if I am MASTER or WORKER
if I am MASTER
initialize array
send each WORKER starting info and subarray
do until all WORKERS converge
gather from all WORKERS convergence data
broadcast to all WORKERS convergence signal
end do
receive results from each WORKER
else if I am WORKER
receive from MASTER starting info and subarray
do until solution converged
update time
send neighbors my border info
receive from neighbors their border info
update my portion of solution array
determine if my solution has converged
send MASTER convergence data
receive from MASTER convergence signal
end do
send MASTER results
endif
Simple Heat Equation
Parallel Solution 2: Overlapping Communication and Computation
In the previous solution, it was assumed that blocking communications
were used by the worker tasks. Blocking communications wait for the
communication process to complete before continuing to the next
program instruction.
In the previous solution, neighbor tasks communicated border
data, then each process updated its portion of the array.
Computing times can often be reduced by using non-blocking
communication. Non-blocking communications allow work to be performed
while communication is in progress.
Each task could update the interior of its part of the solution
array while the communication of border data is occurring, and
update its border after communication has completed.
Pseudo code for the second solution:
red highlights changes for
non-blocking communications.
find out if I am MASTER or WORKER
if I am MASTER
initialize array
send each WORKER starting info and subarray
do until all WORKERS converge
gather from all WORKERS convergence data
broadcast to all WORKERS convergence signal
end do
receive results from each WORKER
else if I am WORKER
receive from MASTER starting info and subarray
do until solution converged
update time
non-blocking send neighbors my border info
non-blocking receive neighbors border info
update interior of my portion of solution array
wait for non-blocking communication complete
update border of my portion of solution array
determine if my solution has converged
send MASTER convergence data
receive from MASTER convergence signal
end do
send MASTER results
endif
Parallel Examples
1-D Wave Equation
In this example, the amplitude along a uniform, vibrating string is
calculated after a specified amount of time has elapsed.
The calculation involves:
the amplitude on the y axis
i as the position index along the x axis
node points imposed along the string
update of the amplitude at discrete time steps.
The equation to be solved is the one-dimensional wave equation:
Note that amplitude will depend on previous timesteps (t, t-1) and
neighboring points (i-1, i+1). Data dependence will mean that a
parallel solution will involve communications.
1-D Wave Equation Parallel Solution
Implement as an SPMD model
The entire amplitude array is partitioned and distributed as
subarrays to all tasks. Each task owns a portion of
the total array.
Load balancing: all points require equal work, so the points should
be divided equally
A block decomposition would have the work partitioned into the number
of tasks as chunks, allowing each task to own mostly contiguous data points.
Communication need only occur on data borders. The larger the block size
the less the communication.
Pseudo code solution:
find out number of tasks and task identities
#Identify left and right neighbors
left_neighbor = mytaskid - 1
right_neighbor = mytaskid +1
if mytaskid = first then left_neigbor = last
if mytaskid = last then right_neighbor = first
find out if I am MASTER or WORKER
if I am MASTER
initialize array
send each WORKER starting info and subarray
else if I am WORKER
receive starting info and subarray from MASTER
endif
#Update values for each point along string
#In this example the master participates in calculations
do t = 1, nsteps
send left endpoint to left neighbor
receive left endpoint from right neighbor
send right endpoint to right neighbor
receive right endpoint from left neighbor
#Update points along line
do i = 1, npoints
newval(i) = (2.0 * values(i)) - oldval(i)
+ (sqtau * (values(i-1) - (2.0 * values(i)) + values(i+1)))
end do
end do
#Collect results and write to file
if I am MASTER
receive results from each WORKER
write results to file
else if I am WORKER
send results to MASTER
endif
