程序代写代做代考 scheme concurrency algorithm PowerPoint Presentation

PowerPoint Presentation

Lecture 4: Principles of Parallel
Algorithm Design (part 4)

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Mapping Technique for Load Balancing

• Sources of overheads:
– Inter-process interaction
– Idling

• Goals to achieve:
– To reduce interaction time
– To reduce total amount of time some processes being

idle
– Remark: these two goals often conflict

• Classes of mapping:
– Static
– Dynamic

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Schemes for Static Mapping

• Mapping Based on Data Partitioning

• Task Graph Partitioning

• Hybrid Strategies

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Mapping Based on Data Partitioning

• By owner-computes rule, mapping the relevant
data onto processes is equivalent to mapping
tasks onto processes

• Array or Matrices
– Block distributions

– Cyclic and block cyclic distributions

• Irregular Data
– Example: data associated with unstructured mesh

– Graph partitioning

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1D Block Distribution

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Example. Distribute rows or columns of matrix to different
processes

Multi-D Block Distribution

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Example. Distribute blocks of matrix to different processes

Load-Balance for Block Distribution

Example. 𝑛 × 𝑛 dense matrix multiplication 𝐶 = 𝐴 × 𝐵
using 𝑝 processes

– Decomposition based on output data.

– Each entry of 𝐶 use the same amount of computation.

– Either 1D or 2D block distribution can be used:

• 1D distribution:
𝑛

𝑝
rows are assigned to a process

• 2D distribution: 𝑛/ 𝑝 × 𝑛/ 𝑝 size block is assigned to a process

– Multi-D distribution allows higher degree of concurrency.

– Multi-D distribution can also help to reduce interactions

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Cyclic and Block Cyclic Distributions

• If the amount of work differs for different
entries of a matrix, a block distribution can
lead to load imbalances.

• Example. Doolittle’s method of LU factorization
of dense matrix

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Doolittle’s method of LU factorization

𝐴 =

𝑎11 𝑎12 … 𝑎1𝑛
𝑎21 𝑎22 … 𝑎2𝑛
⋮ ⋮ ⋱ ⋮
𝑎𝑛1 𝑎𝑛2 … 𝑎𝑛𝑛

= 𝐿𝑈 =

1 0 … 0
𝑙21 1 … 0
⋮ ⋮ ⋱ ⋮
𝑙𝑛1 𝑙𝑛2 … 1

𝑢11 𝑢12 … 𝑢1𝑛
0 𝑢22 … 𝑢2𝑛
⋮ ⋮ ⋱ ⋮
0 0 … 𝑢𝑛𝑛

By matrix-matrix multiplication

𝑢1𝑗 = 𝑎1𝑗 , 𝑗 = 1,2, … , 𝑛 (1𝑠𝑡 row of 𝑈)

𝑙𝑗1 = 𝑎𝑗1/𝑢11, 𝑗 = 1,2, … , 𝑛 (1𝑠𝑡 column of 𝐿)

For 𝑖 = 2,3, … , 𝑛 − 1 do

𝑢𝑖𝑖 = 𝑎𝑖𝑖 − 𝑙𝑖𝑡𝑢𝑡𝑗
𝑖−1
𝑡=1

𝑢𝑖𝑗 = 𝑎𝑖𝑗 − 𝑙𝑖𝑡𝑢𝑡𝑗
𝑖−1
𝑡=1 for 𝑗 = 𝑖 + 1,… , 𝑛 (𝑖𝑡ℎ row of 𝑈)

𝑙𝑗𝑖 =
𝑎𝑗𝑖− 𝑙𝑗𝑡𝑢𝑡𝑖

𝑖−1
𝑡=1

𝑢𝑖𝑖
for 𝑗 = 𝑖 + 1, … , 𝑛 (𝑖𝑡ℎ column of 𝐿)

End
𝑢𝑛𝑛 = 𝑎𝑛𝑛 − 𝑙𝑛𝑡𝑢𝑡𝑛

𝑛−1
𝑡=1

Serial Column-Based LU

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• Remark: Matrices L and U share space with A

Work used to compute Entries of L and U

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• Block distribution of LU factorization tasks
leads to load imbalance.

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Block-Cyclic Distribution

• A variation of block distribution that can be
used to alleviate the load-imbalance.

• Steps
1. Partition an array into many more blocks than

the number of available processes

2. Assign blocks to processes in a round-robin
manner so that each process gets several non-
adjacent blocks.

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(a) The rows of the array are grouped into blocks each consisting of two rows,
resulting in eight blocks of rows. These blocks are distributed to four processes
in a wraparound fashion.

(b) The matrix is blocked into 16 blocks each of size 4×4, and it is mapped onto a
2×2 grid of processes in a wraparound fashion.

• Cyclic distribution: when the block size =1

Graph Partitioning

• Assign equal number of nodes (or cells) to each process

• Minimize edge count of the graph partition

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Random Partitioning Partitioning for Minimizing Edge-Count

Mappings Based on Task Partitioning

• Mapping based on task partitioning can be used
when computation is naturally expressed in the
form of a static task-dependency graph with
known sizes.

• Finding optimal mapping minimizing idle time and
minimizing interaction time is NP-complete

• Heuristic solutions exist for many structured
graphs

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Mapping a Binary Tree Task-Dependency Graph

• Finding min.

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• Mapping the tree graph onto 8 processes
• Mapping minimizes the interaction overhead by mapping independent

tasks onto the same process (i.e., process 0) and others on processes
only one communication link away from each other

• Idling exists. This is inherent in the graph

Mapping a Sparse Graph

Example. Sparse matrix-vector multiplication using 3
processes

• Arrow distribution

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• Partitioning task interaction graph to reduce
interaction overhead

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Schemes for Dynamic Mapping

• When static mapping results in highly imbalanced
distribution of work among processes or when
task-dependency graph is dynamic, use dynamic
mapping

• Primary goal is to balance load – dynamic load
balancing
– Example: Dynamic load balancing for AMR

• Types
– Centralized

– Distributed

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Centralized Dynamic Mapping

• Processes
– Master: mange a group of available tasks
– Slave: depend on master to obtain work

• Idea
– When a slave process has no work, it takes a portion of available

work from master
– When a new task is generated, it is added to the pool of tasks in

the master process

• Potential problem
– When many processes are used, mast process may become

bottleneck

• Solution
– Chunk scheduling: every time a process runs out of work it gets

a group of tasks.

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Distributed Dynamic Mapping

• All processes are peers. Tasks are distributed
among processes which exchange tasks at run
time to balance work

• Each process can send or receive work from other
processes
– How are sending and receiving processes paired

together

– Is the work transfer initiated by the sender or the
receiver?

– How much work is transferred?

– When is the work transfer performed?

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Techniques to Minimize Interaction Overheads

• Maximize data locality

– Maximize the reuse of recently accessed data

– Minimize volume of data-exchange

• Use high dimensional distribution. Example: 2D block
distribution for matrix multiplication

– Minimize frequency of interactions

• Reconstruct algorithm such that shared data are accessed
and used in large pieces.

• Combine messages between the same source-destination
pair

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Techniques to Minimize Interaction Overheads

• Minimize contention and hot spots
– Contention occur when multi-tasks try to access the same resources

concurrently: multiple processes sending message to the same
process; multiple simultaneous accesses to the same memory block

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• Using 𝐶𝑖,𝑗 = 𝐴𝑖,𝑘𝐵𝑘,𝑗
𝑝−1
𝑘=0

causes contention. For example, 𝐶0,0,

𝐶0,1, 𝐶0, 𝑝−1 attempt to read 𝐴0,0, at once.

• A contention-free manner is to use:

𝐶𝑖,𝑗 = 𝐴𝑖, 𝑖+𝑗+𝑘 % 𝑝𝐵 𝑖+𝑗+𝑘 % 𝑝,𝑗
𝑝−1
𝑘=0

All tasks 𝑃∗,𝑗 that work on the same row of C access block

𝐴𝑖, 𝑖+𝑗+𝑘 % 𝑝, which is different for each task.

Techniques to Minimize Interaction Overheads

• Overlap computations with interactions

– Use non-blocking communication

• Replicate data or computations

– Replicate a copy of shared data on each process if
possible, so that there is only initial interaction during
replication.

• Use collective interaction operations

• Overlap interactions with other interactions

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Parallel Algorithm Models

• Data parallel
– Each task performs similar operations on different data
– Typically statically map tasks to processes

• Task graph
– Use task dependency graph to promote locality or reduce

interactions

• Master-slave
– One or more master processes generating tasks
– Allocate tasks to slave processes
– Allocation may be static or dynamic

• Pipeline/producer-consumer
– Pass a stream of data through a sequence of processes
– Each performs some operation on it

• Hybrid
– Apply multiple models hierarchically, or apply multiple models

in sequence to different phases
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