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Multi-Tasking Models and Algorithms

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### Multi-Tasking Models and Algorithms

### Multi-Tasking Models and Algorithms

### Task-Channel Model Agglomeration

### Task-Channel Model Data Structure

### Parallel and Distributed Algorithms the Boundary Value Problem(CS 6/76501)

### Communication Costs in Asychronous Computing the Boundary Value Problem

### Performance Evaluation Metrics the Boundary Value Problemwith Asynchronous Communication Costs

Task-Channel (Computational) Model

&

Asynchronous Communication

(Part II)

Outline for Multi-Tasking Models

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

- 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

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

References

- 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)
- Textbook Authors’ slides

- Barry Wilkinson and Michael Allen, “Parallel Programming: Techniques and Applications
- http://www-unix.mcs.anl.gov/dbpp/text/book.html

- 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

Primary References for Part II

- Michael Quinn, Parallel Programming in C with MPI and OpenMP, McGraw Hill, 2004.
- Also slides by author for this textbook.

- Ian Foster, Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering, Addison Wesley, 1995, Online at
- http://www-unix.mcs.anl.gov/dbpp/text/book.html

- Ananth Grama, Anshul Gupta, George Karypis, and Vipin Kumar, Introduction to Parallel Computing, 2nd Edition, Addison Wesley, 2003.
- Also, slides created by authors of this textbook

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.

The Task/Channel Model

Outline: Task/Channel Model

- Task/channel model of Ian Foster
- Used by both Foster and Quinn in their textbooks
- Is a model for a general style of computation; i.e., a computational model, not an algorithm model

- Algorithm design methodology
- Recommended algorithmic choice tree for problems
- Case studies
- Boundary value problem
- Finding the maximum

Relationship of Task/Channel Model to Algorithm Models

- In designing algorithms for problems, the Task Graph algorithm model discussed in textbook by Grama, et. al. uses both
- the task dependency graph where dependencies usually result from communications between two tasks.
- the task interaction graph also captures interactions between tasks such as data sharing.

- The Task Graph Algorithm model provides guidelines for creating one type of algorithm
- It does not attempt to model computational or communication costs.

Relationship of Task/Channel Model to Algorithm Models (cont.)

- The Task/Channel model is a computationalmodel, in that it attempts to capture a style of computation that can be used by certain types of parallel computers.
- It also uses the task dependency graph
- Also, it provides methods for analyzing computation time and communication time.

- Use of Task/Channel model results in more than one algorithmic style being used to solve problems.
- e.g., task graph algorithms, data-parallel algorithms, master-slave algorithms, etc.

The Task/Channel Model(Ref: Chapter 3 in Quinn)

- This model is intended for MIMDs (i.e., multiprocessors and multicomputers) and not for SIMDs.
- Parallel computation = set of tasks
- Ataskconsists of a
- Program
- Local memory
- Collection of I/O ports

- Tasks interact by sending messages through channels
- A task can send local data values to other tasks via output ports
- A task can receive data values from other tasks via input ports.

- The local memory contains the program’s instructions and its private data

Task/Channel Model

- A channelis a message queue that connects one task’s output port with another task’s input port.
- Data values appear in input port in the same order in which they are placed in the channel’s output queue.
- A task is blockedif a task tries to receive a value at an input port and the value isn’t available.
- The blocked task must wait until the value is received.
- A process sending a message is never blocked even if previous messages it has sent on the channel have not been received yet.
- Thus, receiving is a synchronous operation and sending is an asynchronous operation.

Task/Channel Model

- Local accesses of private data are assumed to be easily distinguished from nonlocal data access done over channels.
- Thus, we should think of local accesses as being faster than nonlocal accesses.
- In this model:
- The execution time of a parallel algorithm is the period of time a task is active.
- The starting time of a parallel algorithm is when all tasks simultaneously begin executing.
- The finishing time of a parallel algorithm is when the last task has stopped executing.

Foster’s Design Methodology

- Ian Foster has proposed a 4-step process for designing parallel algorithms for machines that fit the task/channel model.
- Foster’s online textbook is a useful resource here

- It encourages the development of scalable algorithms by delaying machine-dependent considerations until the later steps.
- The 4 design steps are called:
- Partitioning
- Communication
- Agglomeration
- Mapping

Partitioning

- Partitioning: Dividing the computation and data into pieces
- Domain decomposition –one approach
- Divide data into pieces
- Determine how to associate computations with the data
- Focus on the largest and most frequently accessed data structure

- Functional decomposition –another approach
- Divide computation into pieces
- Determine how to associate data with the computations
- This often yields tasks that can be pipelined.

Example Domain Decompositions

Think of the primitive tasks as processors.

In 1st, each 2D slice is mapped onto one processor of a system using 3 processors.

In second, a 1D slice is mapped onto a processor.

In last, an element is mapped onto a processor

The last leaves more primitive tasks and is usually preferred.

Partitioning Checklist for Evaluating the Quality of a Partition

- At least 10x more primitive tasks than processors in target computer
- Minimize redundant computations and redundant data storage
- Primitive tasks are roughly the same size
- Number of tasks an increasing function of problem size
- Remember – we are talking about MIMDs here which typically have a lot less processors than SIMDs.

Foster’s Methodology Partition

Communication Partition

- Determine values passed among tasks
- There are two kinds of communication:
- Local communication
- A task needs values from a small number of other tasks
- Create channels illustrating data flow

- Global communication
- A significant number of tasks contribute data to perform a computation
- Don’t create channels for them early in design

Communication (cont.) Partition

- Communications is part of the parallel computation overhead since it is something sequential algorithms do not have do.
- Costs larger if some (MIMD) processors have to be synchronized.

- SIMD algorithms have much smaller communication overhead because
- Much of the SIMD data movement is between the control unit and the PEs
- especially true for associative

- Parallel data movement along the interconnection network involves lockstep (i.e. synchronously) moves.

- Much of the SIMD data movement is between the control unit and the PEs

Communication Checklist for Judging the Quality of Communications

- Communication operations should be balanced among tasks
- Each task communicates with only a small group of neighbors
- Tasks can perform communications concurrently
- Task can perform computations concurrently

Foster’s Methodology Communications

What We Have Hopefully at This Point – and What We Don’t Have

- The first two steps look for parallelism in the problem.
- However, the design obtained at this point probably doesn’t map well onto a real machine.
- If the number of tasks greatly exceed the number of processors, the overhead will be strongly affected by how the tasks are assigned to the processors.
- Now we have to decide what type of computer we are targeting
- Is it a centralized multiprocessor or a multicomputer?
- What communication paths are supported
- How must we combine tasks in order to map them effectively onto processors?

Agglomeration Have

- Agglomeration: Grouping tasks into larger tasks
- Goals
- Improve performance
- Maintain scalability of program
- Simplify programming – i.e. reduce software engineering costs.

- In MPI programming, a goal is
- to lower communication overhead.
- often to create one agglomerated task per processor

- By agglomerating primitive tasks that communicate with each other, communication is eliminated as the needed data is local to a processor.

Agglomeration Can Improve Performance Have

- It can eliminate communication between primitive tasks agglomerated into consolidated task
- It can combine groups of sending and receiving tasks

Scalability Have

- We are manipulating a 3D matrix of size 8 x 128 x 256.
- Our target machine is a centralized multiprocessor with 4 CPUs.
- Suppose we agglomerate the 2nd and 3rd dimensions. Can we run on our target machine?
- Yes- because we can have tasks which are each responsible for a 2 x 128 x 256 submatrix.
- Suppose we change to a target machine that is a centralized multiprocessor with 8 CPUs. Could our previous design basically work.
- Yes, because each task could handle a 1 x 128 x 256 matrix.

Scalability Have This says the decision to agglomerate the 2nd and 3rd dimension in the long run has the drawback that the code portability to more CPUs is impaired.

- However, what if we go to more than 8 CPUs? Would our design change if we had agglomerated the 2nd and 3rd dimension for the 8 x 128 x 256 matrix?
- Yes.

Agglomeration Checklist for Checking the Quality of the Agglomeration

- Locality of parallel algorithm has increased
- Replicated computations take less time than communications they replace
- Data replication doesn’t affect scalability
- Agglomerated tasks have similar computational and communications costs
- Number of tasks increases with problem size
- Number of tasks suitable for likely target systems
- Tradeoff between agglomeration and code modifications costs is reasonable

Foster’s Methodology Agglomeration

Mapping Agglomeration

- Mapping: The process of assigning tasks to processors
- Centralized multiprocessor: mapping done by operating system
- Distributed memory system: mapping done by user
- Conflicting goals of mapping
- Maximize processor utilization –i.e. the average percentage of time the system’s processors are actively executing tasks necessary for solving the problem.
- Minimize interprocessor communication

Mapping Example Agglomeration

(a) is a task/channel graph showing the needed communications over channels.

(b) shows a possible mapping of the tasks to 3 processors.

Mapping Example Agglomeration

If all tasks require the same amount of time and each CPU has the same capability, this mapping would mean the middle processor will take twice as long as the other two..

Optimal Mapping Agglomeration

- Optimality is with respect to processor utilization and interprocessor communication.
- Finding an optimal mapping is NP-hard.
- Must rely on heuristics applied either manually or by the operating system.
- It is the interaction of the processor utilization and communication that is important.
- For example, with p processors and n tasks, putting all tasks on 1 processor makes interprocessor communication zero, but utilization is 1/p.

A Mapping Decision Tree Agglomeration(Quinn, Pg 72)

- Static number of tasks
- Structured communication
- Constant computation time per task
- Agglomerate tasks to minimize communications
- Create one task per processor

- Variable computation time per task
- Cyclically map tasks to processors

- Constant computation time per task
- Unstructured communication
- Use a static load balancing algorithm

- Structured communication
- Dynamic number of tasks
- Frequent communication between tasks
- Use a dynamic load balancing algorithm

- Many short-lived tasks. No internal communication
- Use a run-time task-scheduling algorithm

- Frequent communication between tasks

Mapping Checklist to Judge the Quality of a Mapping Agglomeration

- Consider designs based on one task per processor and multiple tasks per processor.
- Evaluate static and dynamic task allocation
- If dynamic task allocation chosen, the task allocator (i.e., manager) is not a bottleneck to performance
- If static task allocation chosen, ratio of tasks to processors is at least 10:1

Task/Channel Case Studies Agglomeration

- Boundary value problem
- Finding the maximum
- The n-body problem (omitted)
- Adding data input (omitted)

Boundary Value Problem

Boundary Value Problem Agglomeration

Ice water

Insulation

Rod

Problem:

The ends of a rod of length 1 are in contact with ice water at 00 C. The initial temperature at distance x from the end of the rod is 100sin(x). (These are the boundary values.)

The rod is surrounded by heavy insulation. So, the temperature changes along the length of the rod are a result of heat transfer at the ends of the rod and heat conduction along the length of the rod.

We want to model the temperature at any point on the rod as a function of time.

- Over time the rod gradually cools. Agglomeration
- A partial differential equation (PDE) models the temperature at any point of the rod at any point in time.
- PDEs can be hard to solve directly, but a method called the finite difference method is one way to approximate a good solution using a computer.
- The derivative of f at a point s is defined by the limit: lim f(x+h) – f(x)
h0 h

- If h is a fixed non-zero value (i.e. don’t take the limit), then the expression is called a finite difference.

Finite differences approach differential quotients as Agglomerationh goes to zero.

Thus, we can use finite differences to approximate derivatives.

This is often used in numerical analysis, especially in numerical ordinary differential equations and numerical partial differential equations, which aim at the numerical solution of ordinary and partial differential equations respectively.

The resulting methods are called finite-difference methods.

An Example of Using a Finite Difference Method for an ODE (Ordinary Differential Equation)

Given f’(x) = 3f(x) + 2, the fact that

f(x+h) – f(x) approximates f’(x)

h

can be used to iteratively calculate an approximation to f’(x).

In our case, a finite difference method finds the temperature at a fixed number of points in the rod at various time intervals.

The smaller the steps in space and time, the better the approximation.

Rod Cools as Time Progresses (Ordinary Differential Equation)

A finite difference method computes these temperature approximations (vertical axis) at various points along the rod (horizontal axis) for different times between 0 and 3.

The Finite Difference Approximation Requires the Following Data Structure

A matrix is used where columns represent positions and rows represent time.

The element u(i,j) contains the temperature at position i on the rod at time j.

At each end of the rod the temperature is always 0. At time 0, the temperature at point x is 100sin(x)

Finite Difference Method Actually Used Data Structure

- We have seen that for small h, we may approximate f’(x) by
f’(x) ~ [f(x + h) – f(x)] / h

- It can be shown that in this case, for small h,
f’’(x) ~ [f(x + h) – 2f(x) + f(x-h)]

- Let u(i,j) represent the matrix element containing the temperature at position i on the rod at time j.
- Using above approximations, it is possible to determine a positive value r so that
u(i,j+1) ~ ru(i-1,j) + (1 – 2r)u(i,j) + ru(i+1,j)

- In the finite difference method, the algorithm computes the temperatures for the next time period using the above approximation.

Partitioning Step Data Structure

- This one is fairly easy to identify initially.
- There is one data item (i.e. temperature) per grid point in matrix.
- Let’s associate one primitive task with each grid point.
- A primitive task would be the calculation of u(i,j+1) as shown on the last slide.

Communication Step Data Structure

- Next, we identify the communication pattern between primitive tasks.
- Each interior primitive task needs three incoming and three outgoing channels because to calculate
u(i,j+1) = ru(i-1,j) + (1 – 2r)u(i,j) + ru(i+1,j)

the task needs u(i-1,j), u(i,j), and u(i+1,j).

– i.e. 3 incoming channels and

u(i,j+1) will be needed for 3 other tasks

- i.e. 3 outgoing channels.

- Tasks on the sides don’t need as many channels, but we really need to worry about the interior nodes.

Agglomeration Step Data Structure

We now have a task/channel graph below:

It should be clear this is not a good situation even if we had enough processors.

The top row depends on values from bottom rows.

Be careful when designing a parallel algorithm that you don’t think you have parallelism when tasks are sequential.

Collapse the Columns in the 1 Data Structurest Agglomeration Step

This task/channel graph represents each task as computing one temperature for a given position and time.

This task/channel graph represents each task as computing the temperature at a particular position for all time steps.

Mapping Step Data Structure

This graph shows only a few intervals. We are using one processor per task.

For the sake of a good approximation, we may want many more intervals than we have processors.

We go back to the decision tree on page 72 to see if we can do better when we want more intervals than we have available processors.

Note: On a SIMD with an interconnection network, we could probably stop here, as we could possibly have enough processors.

Use Decision Tree Pg 72 Data Structure

- The number of tasks is static once we decide on how many intervals we want to use.
- The communication pattern among the tasks is regular – i.e. structured.
- Each task performs the same computations.
- Therefore, the decision tree says to create one task per processor by agglomerating primitive tasks so that computation workloads are balanced and communication is minimized.
- So, we will associate a contiguous piece of the rod with each task by dividing the rod into n pieces of size h, where n is the number of processors we have.

Pictorially Data Structure

Our previous task/channel graph assumed 10 consolidated tasks, one per interval:

If we now assume 3 processors, we would now have:

Note this maintains the possibility of using some kind of nearest neighbor interconnection network and eliminates unnecessary communication.

What interconnection networks would work well?

Sequential execution time Data Structure

Notation:

- – time to update element u(i,j)
- n – number of intervals on rod
- There are n-1 interior positions

- m – number of time iterations
Then, the sequential execution time is

m (n-1)

Parallel Execution Time Data Structure

Notation (in addition to ones on previous slide):

- p – number of processors
- – time to send (receive) a value to (from) another processor
- In task/channel model, a task may only send and receive one message at a time, but it can receive one message while it is sending a message.
- Consequently, a task requires 2 time to send data values to its neighbors, but it can receive the two data values it needs from its neighbors at the same time.
- So, we assume each processor is responsible for roughly an equal-sized portion of the rod’s intervals.

Parallel Execution Time For Task/Channel Model Data Structure

- Then, the parallel execution time is for one iteration is
(n-1)/p +2

and an estimate of the parallel execution time for all m iterations is

m ( (n-1)/p +2)

where

- – time to update element u(i,j)
- n – number of intervals on rod
- m – number of time iterations
- p – number of processors
- – time to send (receive) a value to (from) another processor
Note that s means to round up to the nearest integer.

Comparisons ( Data Structuren = #intervals; m = time )

1For a SIMD, communications are quicker than for a message passing machine as a packet doesn’t have to be built.

Designing the Reduction Algorithm

Evaluating the Finite Difference Method (FDM) Solution for the Boundary Value Problem

- The FDM only approximates the solution for the PDE.
- Thus, there is an error in the calculation.
- Moreover, the FDM tells us what the error is.
- If the computed solution is x and the correct solution is c, then the percent error is |(x-c)/c| at a given interval m.
- Let’s enhance the algorithm by computing the maximum error for the FDM calculation.
- However, this calculation is an example of a more general calculation, so we will solve the general problem instead.

Reduction Calculation the Boundary Value Problem

- We start with any associative operator . A reduction is the computation of the expression
a0 a1 a2 … an-1

- Examples of associative operations:
- Add
- Multiply
- And, Or
- Maximum, Minimum

- On a sequential machine, this calculation would require how many operations?
n – 1 i.e. the calculation is Θ(n).

How many operations are needed on a parallel machine?

For notational simplicity, we will work with the operation +.

Partitioning the Boundary Value Problem

- Suppose we are adding n values.
- First, divide the problem as finely as possible and associate precisely one value to a task.
- Thus we have n tasks.

Communication

- We need channels to move the data together in a processor so the sum can be computed.
- At the end of the calculation, we want the total in a single processor.

Communication the Boundary Value Problem

- The brute force way would be to have one task receive all the other n-1 values and perform the additions.
- Obviously, this is not a good way to go. In fact, it will be slower than the sequential algorithm because of the communication overhead!
- Its time is (n-1)( + ) where is the communication cost to send and receive one element and is the time to perform the addition.
- The sequential algorithm is only (n-1)!

Parallel Reduction Evolution the Boundary Value ProblemLet’s Try

The timing is now (n/2)( + ) +

Parallel Reduction Evolution the Boundary Value ProblemBut, Why Stop There?

The timing is now (n/4)( + ) + 2

If We Continue With This Approach the Boundary Value Problem

- We have what is called a binomial tree communication pattern.
- It is one of the most commonly encountered communication patterns in parallel algorithm design.
- Now you can see why the interconnection networks we have seen are typically used.

The Hypercube and Binomial Trees the Boundary Value Problem

An 8 node bionomial tree is the Boundary Value Problem

a subgraph of the 8 node

hypercube

The Hypercube and Binomial TreesFinding Global Sum the Boundary Value ProblemUsing 16 Task/Channel Processors

Start with one number per processor.

Half send values and half receive and add.

4

2

0

7

-3

5

-6

-3

8

1

2

3

-4

4

6

-1

What If You Don’t Have a Power of 2? the Boundary Value Problem

- For example, suppose we have 2k + r numbers where r < 2k ?
- In the first step, r processors send values and r tasks receive values and add their values.
- Now r tasks become inactive and we proceed as before.
- Example: With 6 numbers.
- Send 2 numbers to 2 other tasks and add them.
- Now you have 4 tasks with numbers assigned.

- So, if the number of tasks n is a power of 2, reduction can be performed in log n communication steps. Otherwise, we need log n + 1.
- Thus, without loss of generality, we can assume we have a power of 2 for the communication steps.

Agglomeration and Mapping the Boundary Value Problem

- We will assume that the number of processors p is a power of 2.
- For task/channel machines, we’ll assume p << n (i.e. p is much less than n).
- Using the mapping decision tree on page 72, we see we should minimize communication and create one task per processor since we have
- Static number of tasks
- Structured communication
- Constant computation time per task

- Structured communication

Agglomeration to 4 Processors Initially the Boundary Value ProblemThis Minimizes Communication

But, we want a single task per processor

So, each processor will run the sequential algorithm and find its local subtotal before communicating to the other tasks ...

Analysis of Reduction Algorithm the Boundary Value Problem

- Assume n integers are divided evenly among the p tasks, no task will handle more than n/p integers.
- The time needed to perform concurrently their subtasks is
(n/p - 1) where is the time to perform the binary operation.

- We already know the reduction can be performed in log p communication steps.
- The receiving processor must wait for the message to arrive and add its value to the received value. So each reduction step requires + time.
- Combining all of these, the overall execution time is
(n/p - 1) + log p ( + )

- What would happen on a SIMD with p = n?

Asynchronous Communication Costs & Performance Metrics

References the Boundary Value Problem

- Michael Quinn, Parallel Programming in C with MPI and OpenMP, McGraw Hill, 2004.
- Chapters 7 plus algorithm examples.
- Textbook Slides for Chapter 7 on isoefficiencyds

- Ananth Grama, Anshul Gupta, George Karypis, and Vipin Kumar, Introduction to Parallel Computing, 2nd Edition, Addison Wesley, 2003.
- Particularly, Section 2.5, pgs 53-63.
- Plan to make available online

- Chapter 5 on performance evaluation used lightly
- Authors’ Slides for Section 2.5

- Particularly, Section 2.5, pgs 53-63.

Message Passing Terminology the Boundary Value Problem

- The time to communicate a message between two nodes in a network is the sum of the following:
- The time to prepare a message for transmission.
- The time taken by the message to transverse the network to its destination.

- Link: Connection between two nodes.
- A switch enables packets to be routed through a node to other nodes without disturbing the processor.
- The links can be assumed to be bi-directional.

- Bandwidth: The number of words or bits that can be transmitted in unit time (i.e., bits per second)

Communications Cost Parameters the Boundary Value Problem

- The principal parameters that determine the communication cost are the following:
- Startup time: ts
- Time required to handle a message at the sending and receiving nodes.
- Includes the time to prepare a message and the time to execute the routing algorithm.

- Per-hop time: th
- Time taken by the header of a node to reach the next directly connected node in its path.
- Also called the node latency.)

- Per-word transfer time: tw
- If the channel bandwidth is r words per second, then each word takes tw= 1/r to traverse the link

- Startup time: ts

Store-and-Forward Routing the Boundary Value Problem

- A message traversing multiple hops is completely received at an intermediate hop before being forwarded to the next hop.
- The total communication cost for a message of size m words to traverse l communication links is
- In most platforms, th is small and the above expression can be approximated by

Packet Routing the Boundary Value Problem

- Store-and-forward makes poor use of communication resources.
- Packet routing breaks messages into packets and pipelines them through the network.
- Since packets may take different paths, each packet must carry routing information, error checking, sequencing, and other related header information.
- The total communication time for packet routing is approximated by:
- The factor tw accounts for overheads in packet headers.

Cut-Through Routing the Boundary Value Problem

- Takes the concept of packet routing to an extreme by further dividing messages into basic units called flits.
- Since flits are typically small, the header information must be minimized.
- This is done by forcing all flits to take the same path, in sequence.
- A tracer message first programs all intermediate routers. All flits then take the same route.
- Error checks are performed on the entire message, as opposed to flits.
- No sequence numbers are needed.

Cut-Through Routing the Boundary Value Problem

- The total communication time for cut-through routing is approximated by:
- This is identical to packet routing, however, tw is typically much smaller.

Routing Techniques the Boundary Value Problem

Passing a message from node P0 to P3 (a) through a store-and-forward communication network; (b) and (c) extending the concept to cut-through routing. The shaded regions represent the time that the message is in transit. The startup time associated with this message transfer is assumed to be zero.

Simplified Cost Model for Communicating Messages the Boundary Value Problem

- The cost of communicating a message between two nodes l hops away using cut-through routing is given by
- In this expression, th is typically smaller than ts and tw. For this reason, the second term in the RHS does not show, particularly, when m is large.
- Furthermore, it is often not possible to control routing and placement of tasks (e.g., when using MPI).
- For these reasons, we can approximate the cost of message transfer by

Simplified Cost Model for Communicating Messages the Boundary Value Problem

- It is important to note that the original expression for communication time is valid for only uncongested networks.
- If a link takes multiple messages, the corresponding tw term must be scaled up by the number of messages.
- Different communication patterns congest different networks to varying extents.
- It is important to understand and account for this in the communication time accordingly.

Cost Models for the Boundary Value ProblemShared Address Space Machines

- While the basic messaging cost applies to these machines as well, a number of other factors make accurate cost modeling more difficult.
- Memory layout is typically determined by the system.
- Smaller cache sizes can result in cache thrashing.
- Overheads associated with ‘invalidate and update’ operations are difficult to quantify.
- Spatial locality is difficult to model.
- Pre-fetching can play a role in reducing the overhead associated with data access.
- False sharing and contention are difficult to model.

Including the

Isoefficiency Metric & Scalability

Performance Metrics Revisited the Boundary Value Problem

- Performance metrics were discussed in the first set of slides (Introduction & General Concepts).
- At that time, no restrictions were made as to whether these metrics were for synchronous or asynchronous models.
- The definitions of the metrics introduced there are the same for both synchronous & asynchronous models.
- However, there is a difference in the communication cost and how it is measured:
- A basic communication step in a synchronous model is treated the same as a basic computation step and charged a cost of O(1).
- For data parallel algorithms on asychronous models, data movements costs may be essentially the same as above.
- However, for the asynchronous communications covered here, asynchronous communication cost estimates should be used.

Performance Metrics and the Boundary Value ProblemAsynchronous Communication

- Running Time (or Execution Time): tp
- Time elapsed between when the first processor starts executing until the last processor terminates.
- While this definition is the same as the one given earlier, the communication is calculated separately and tp = tcomp + tcomm.

- Speedup: (n,p)
- As before, (n,p) = ts/tp, where ts is the fastest known sequential time for an algorithm.

- Total Parallel Overhead
- T0(n,p) = ptp – ts = cost – ts
- Note that ts time units are needed to do useful work and the remainder is overhead caused by parallelism.

Notation needed for the the Boundary Value ProblemIsoefficiency Relation Slides

- n data size
- p number of processors
- T(n,p) Execution time, using p processors
- (n,p) speedup
- (n) Inherently sequential computations
- (n) Potentially parallel computations
- (n,p) Communication operations
- (n,p) Efficiency
- Note: If (n) occurs, it is a misprint; Replace it with (n)

The Isoefficiency Metric the Boundary Value Problem

- Parallel system – a parallel program executing on a parallel computer
- Scalability of a parallel system - a measure of its ability to increase performance as number of processors increases
- A scalable system maintains efficiency as processors are added
- Isoefficiency - a way to measure scalability

Isoefficiency Concepts the Boundary Value Problem

- T0(n,p) is the total time spent by processes doing work not done by sequential algorithm.
- T0(n,p) = (p-1)(n) + p(n,p)
- We want the algorithm to maintain a constant level of efficiency as the data size n increases, so (n,p) is required to be a constant.
- Recall that T(n,1) represents the sequential execution time.

The Isoefficiency Relation the Boundary Value Problem

Suppose a parallel system exhibits efficiency (n,p). Define

In order to maintain the same level of efficiency as the number of processors increases, n must be increased so that the following inequality is satisfied.

Isoefficiency Derivation Steps the Boundary Value Problem

- Begin with speedup formula
- Compute total amount of overhead
- Assume efficiency remains constant
- Determine relation between sequential execution time and overhead

Deriving Isoefficiency Relation the Boundary Value Problem

Determine overhead

Substitute overhead into speedup equation

Substitute T(n,1) = (n) + (n). Assume efficiency is constant.

Isoefficiency Relation

Isoefficiency the Boundary Value ProblemRelation Usage

- Used to determine the range of processors for which a given level of efficiency can be maintained
- The way to maintain a given efficiency is to increase the problem size when the number of processors increase.
- The maximum problem size we can solve is limited by the amount of memory available
- The memory size is a constant multiple of the number of processors for most parallel systems

The Scalability Function the Boundary Value Problem

- Suppose the isoefficiency relation reduces to n f(p)
- Let M(n) denote memory required for problem of size n
- M(f(p))/p shows how memory usage per processor must increase to maintain same efficiency
- We call M(f(p))/p the scalability function[i.e., scale(p) = M(f(p))/p) ]

Meaning of Scalability Function the Boundary Value Problem

- To maintain efficiency when increasing p, we must increase n
- Maximum problem size is limited by available memory, which increases linearly with p
- Scalability function shows how memory usage per processor must grow to maintain efficiency
- If the scalability function is a constant this means the parallel system is perfectly scalable

Interpreting Scalability Function the Boundary Value Problem

Cplogp

Cannot maintain

efficiency

Cp

Memory Size

Memory needed per processor

Can maintain

efficiency

Clogp

C

Number of processors

Example 1: Reduction the Boundary Value Problem

- Sequential algorithm complexityT(n,1) = (n)
- Parallel algorithm
- Computational complexity = (n/p)
- Communication complexity = (log p)

- Parallel overheadT0(n,p) = (p log p)

Reduction (continued) the Boundary Value Problem

- Isoefficiency relation: n C p log p
- We ask: To maintain same level of efficiency, how must n increase when p increases?
- Since M(n) = n,
- The system has good scalability

Example 2: Floyd’s Algorithm the Boundary Value Problem(Chapter 6 in Quinn Textbook)

- Sequential time complexity: (n3)
- Parallel computation time: (n3/p)
- Parallel communication time: (n2log p)
- Parallel overhead: T0(n,p) = (pn2log p)

Floyd’s Algorithm (continued) the Boundary Value Problem

- Isoefficiency relationn3 C(p n3 log p) n C p log p
- M(n) = n2
- The parallel system has poor scalability

Example 3: Finite Difference the Boundary Value Problem

- See Figure 7.5
- Sequential time complexity per iteration: (n2)
- Parallel communication complexity per iteration: (n/p)
- Parallel overhead: (n p)

Finite Difference (continued) the Boundary Value Problem

- Isoefficiency relationn2 Cnp n C p
- M(n) = n2
- This algorithm is perfectly scalable

Outline for Revision the Boundary Value Problem

- Task-Channel (Computational) Model Basics
- ---------Revised to here-----------------------------
- Comments following this outline give general ideas

- 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

Outline (cont.) the Boundary Value Problem

- 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

- BSP (Computational) Model
- Slides posted separately on course website

Proposed Revision Comments the Boundary Value Problem

- Change title of “strictly asynchronous models” to “multitasking models”.
- This has been partly accomplished, but most interior slides still use the old terminology.

- The slides for the “Multi-Tasking Models” is in a “second draft” stage.
- An initial set of slides that partially covered this material was first created in Spring 2005, when this “Parallel Algorithms and Models” material was last ta\ught.

- The current set of slides needs to be revised to improve the integration of the material covered.
- Some topics are partially covered in two places, such as data decomposition

- Since coverage of other models start with the definition of the model, the “Multi-Tasking Models” material should also start with a model definition.
- The Task/Channel model seems to be the better model to use for this material, with the BSP mentioned afterwards as another “strictly asynchronous” model.

Proposed Revision Comments (cont) the Boundary Value Problem

- The material covered from Quinn’s book and Grama et.al. need to be better integrated.
- Quinn’s presentation is overly simplistic and does not cover all issues that need to be covered.
- Some items (e.g., data decomposition) are essentially covered twice.
- The Foster-Quinn assignment of tasks to processors could be covered towards the end of material as one possible mapping.

- “Asynchronous Communication and Performance Evaluation” relocation
- Probably put isoefficiency material with material in first chapter on analysis of algorithms, as it makes sense for earlier models as well.

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