1. Virtues of Good (Parallel) Software
Concurrency
Able to exploit concurrencies in algorithm/problem/hardware
Scalability
Resilient to increasing processor count
Locality
More frequent access to local data than to remote data
Modularity
Employ abstraction and modular design

2. Two Basic Requirements for Parallel Program
Safety: Produce correct results
Result computed on P processors and on 1 processor must be IDENTICAL.
Livelihood: Able to proceed and finish; free of deadlock.

3. Sources of Overhead Execution time
The time that elapses from when the first processor starts executing on the problem to when the last processor completes execution
Execution time = computation time + communication time + idle time
Communication / interprocess interaction: usually main source of overhead
T_comm = t_s + t_w*L
Minimize the volume and frequency of communications; overlap computation/communication
Idling: lack of computation or lack of data
Load imbalance
Synchronization
Presence of serial components
Wait on remote data
Replicated Computation
Communicate or replicate

4. Speedup & Efficiency
Relative speed-up: the factor by which the execution time is reduced on multiple processors
S(p) = T_1/T_p
T_1 is the execution time on one processor
T_p is execution time on p processors
Absolute speed-up: where is T_1 is the uniprocessor time for best-known (sequential) algorithm
S(p) <= p
Embarrassingly parallel (EP): no communication among cpus.
Superlinear speedup: exists in reality
Efficiency: the fraction of time that processors spend doing useful work.
E = S/p = T_1/(p*T_p)
Parallel cost: p*T_p
Parallel overhead: T_o = p*T_p – T_1

5. Amdahl’s Law This is for a fixed problem size
Alpha – fraction of operations in serial code that can be parallelized
P – number of processors
This is for a fixed problem size
T_p = alpha*T_1/p + (1-alpha)*T_1
S 1/(1-alpha) as Pinfinity
Alpha = 90%, S10
Alpha =99%, S100
Alpha = 99.9%, S1000
“Mental block”

6. Gustafson’s Law
Alpha – fraction of time spent on parallel operations in the parallel program
This is for a scaled problem size; or constant run time.
T_1 = (1-alpha)*T_p + p*alpha*T_p
As problem size increases, fraction of parallel operations increases

7. Iso-Efficiency Function
For fixed problem size N, as P increases, increase in speedup S slows down or levels off, efficiency E decreases
For fixed P, as the N increases, S increases, efficiency E increases
As P increases, can increase the problem size N such that the efficiency is kept constant
This N(p) for fixed efficiency is called iso-efficiency function
Rate of increase in N(p), dN/dp, measures the scalability of a parallel program
Smaller rate of increase  more scalable

8. Parallel Program Design
PCAM Model
(I. Foster)
Concurrency, scalability
Locality, performance-related issues

9. Partitioning
Decompose the computation to be performed and the data operated on by this computation into small tasks
Purpose: expose opportunities of parallel execution
Ignore practical issues such as number of processors in target machine etc
Avoid replicating computation and data
Focus: Define a large number of small tasks in order to yield a fine-grained decomposition of the problem
Fine grained decomposition provides the greatest flexibility in terms of potential parallel algorithms
Maximize concurrency

10. Partitioning
Good partition: divides both the computation associated with a problem and the data this computation operates on
Domain/Data decomposition: first focus on data
Partition the data associated with the problem
Associate computations with partitioned data
Functional decomposition: first focus on computation
Decompose computations to be performed
Deal with data decomposed computations work on

11. Domain Decomposition
Decompose the data first, and then associated computations
“owner computes”
Outcome: tasks comprising some data and a set of operations on that data
Some operation may require data from several tasks  communication
Data can be input data, output data, intermediate data, or all of them.
Rule of thumb: focus first on largest data structure or the data structure accessed most frequently
Mesh-based problems:
Structured mesh: 1D, 2D, 3D decompositions
Unstructured mesh: graph partitioning tools such as METIS
Favor the most aggressive decomposition possible at this stage

12. Functional Decomposition
Focus first on computation to be performed;
Divide computations into disjoint tasks
Then consider the data associated with each sub-task
Data requirements may be disjoint  done
Data may overlap significantly, communications; May just as well try domain decomposition
Provide an alternative way of thinking about problem; Hybrid decomposition maybe best
E.g. multi-physics simulations, overall functional decomposition, each component domain decomposition

13. Partitioning: Questions to Ask
Does your partition define more tasks (an order of magnitude more?) than the number of processors of the target machine?
No  reduced flexibility in subsequent stages
Does your partition avoid redundant computation and storage requirements?
No  may not be scalable to large problems
Are tasks of comparable size?
No  hard to allocate to cpus with equal amount of work  load imbalance
Does the number of tasks scale with problem size?
Ideal: increased problem size  increase in number of tasks
No  may not be able to solve larger problems with more processors
Have you identified alternative partitions?
Maximize flexibility; try both domain and functional decompositions

14. Communication Purpose: Determine the interaction among tasks
Distribute communication operations among many tasks
Organize communication operations in a way that permits concurrent execution
4 categories of communications:
Local/global communications:
Local: each task communicates with a small set of other tasks (neighbors)
Global: communicate with many or all other tasks

15. Communication Structured/un-structured communication
Structured: A task and neighbors form a regular structure, grid or tree
Un-structured: communication represented by arbitrary graphs
Static/dynamic communication:
Static: identity of communication partners does not change over time
Dynamic: identity of partners determined by data computed at runtime and highly variable
Synchronous/asynchronous communication
Synchronous: requires coordination between communication partners
Asynchronous: without cooperation

16. Task Dependency Graph
Task dependencies: one task cannot start until some other task(s) finishes.
E.g. the output of one task is the input to another task
Represented by the task dependency graph:
Directed acyclic
Nodes: tasks (task size as the weight of node)
Directed edges: dependencies among tasks

17. Task Dependency Graph
Degree of concurrency: number of tasks that can run concurrently
Maximum degree of concurrency: the maximum number of tasks that can be executed simultaneously at any given time
Average degree of concurrency: the average number of tasks that can run concurrently over the duration of program
Critical path: The longest vertex-weighted directed path between any pair of start and finish nodes
Critical path length: sum of vertex weights along the critical path
Average degree of concurrency = total amount of work / critical path length

18. Task Interaction Graph
Even independent tasks may need to interact, e.g. sharing data
Interaction graph: captures interaction patterns among tasks
Nodes: tasks
Edges: communications / interactions
Usually contains task dependency graph as sub-graph
Example interaction graph

19. Communication: Questions to Ask
Do all tasks perform the same number of communication operations?
Unbalanced communication  poor scalability
Distribute communications equitably
Does each task communicate only with a small number of neighbors?
May need to re-formulate global communication in terms of local communication structures
Can communications proceed concurrently?
Can computations associated with different tasks proceed concurrently?
No  may need to re-order computations / communications

20. Agglomeration
Improve performance: Combine tasks to reduce the task interaction strength, increase locality, increase the computation and communication granularity. Also determine if it is worthwhile to replicate data/computation
Dependent tasks will be combined
Independent tasks may also be agglomerated to increase granularity
Goals: reduce communication cost, retain flexibility w.r.t. scalability and mapping decisions

21. Increasing Granularity
Coarse-grain usually performs better:
Send less data (reduce volume of communication)
Use fewer messages when sending same amount of data (reducing frequency of communications)
Surface-to-volume effects:
Communication cost usually proportional to surface area of domain
Computation cost usually proportional to volume of domain
As task size increases, amount of communication per unit computation decreases
High-D decomposition usually more efficient than low-D decompositions, due to reduced surface area for a given volume.
Replicate computation:
May trade off replicated computation for reduced communication or execution time.

22. Agglomeration: Questions to Ask
Has agglomeration reduced communication costs by increasing locality?
If computation is replicated, have you verified that the benefits of replication out-weigh its costs for a range of problem size and processor counts?
If data is replicated, have you verified that it does not comprise scalability
Do the tasks have similar computation and communication costs after agglomeration?
Load balance
Does the number of tasks still scale with problem size?

23. Mapping Map tasks to processors or processes.
If the number of tasks is larger than the number of processors, may need to place more than one task on a single processor
Goal: minimize total execution time
Place tasks that execute concurrently on different processors
Place tasks that communicate frequently on the same processor
In general case, no computationally tractable algorithm for the mapping problem, NP-complete.
If SPMD-style, one task per processor

24. Parallel Algorithm Models
Data parallel model: processors perform similar operations on different data
Work/task pool model (replicated workers):
Pool of tasks, a number of processors
A processor can remove a task from pool and work on it
A processor may generate a new task during computation and add it to the pool
Master-slave/manager-worker model: master processors generate work and allocate it to worker processors
Pipeline/producer-consumer model: a stream of data passes through a succession of processors, each perform some task on it.
Hybrid model: combination of two or more models