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### Load-Balancing

High Performance Computing 1

Load-Balancing

- What is load-balancing?
- Dividing up the total work between processes when running codes on a parallel machine

- Load-balancing constraints
- Minimize interprocess communication

- Also called:
- partitioning, mesh partitioning, (domain decomposition)

High Performance Computing 1

Know your data and memory

- Memory is organized by banks. Between access to any bank, there is a latency period.
- Matrix entries are stored column-wise in FORTRAN.

High Performance Computing 1

Addressing Memory

- For illustration purposes, lets imagine 8 banks [128 or 256 common on chips today], with bank busy time (bbt) of 8 cycles between accesses. Thus we have:
data a13 a23 a33 a43 a14 a24 a34 a44

data a11 a21 a31 a41 a12 a22 a32 a42

bank 1 2 3 4 5 6 7 8

High Performance Computing 1

Addressing Memory

- If we access data column-wise, we proceed through each bank in order. By the time we call a13, we (just) avoid bbt.
- On the other hand, if we access data row-wise, we get a11 in bank 1, a12 in bank 5, a13 in bank 1 again - so instead of access on clock cycle 3, we have to wait until cycle 9. Then we get a14 in bank 5 again on cycle 10, etc.

High Performance Computing 1

Indirect addressing

- If addressing is indirect we may wind up jumping all over, and suffer performance hits because of it.

High Performance Computing 1

Shared Memory

- Bank conflicts depend on granularity of memory
- If N memory refs per cycle, p processors, memory with b cycles bbt, need p*N*b memory banks to see uninterrupted access of data
- With B banks, granularity is
g = B/(p*N*b)

High Performance Computing 1

Moral

- Separate selection of data from its processing
- Each subtask requires its own data structure. Be prepared to change structures between tasks

High Performance Computing 1

Load-balancing nomenclature

Objects get distributed among different processes

Edges represent information that need to be shared between objects

Object

Edge

High Performance Computing 1

Partitioning

- Divides up the work
- 5 & 4 objects assigned to processes

- Creates “edge-cuts”
- Necessary communications between processes

High Performance Computing 1

Work/Edge Weights

- Need a good measure of what the expected work may be
- Molecular dynamics:
- number of molecules
- regions

- FEM/finite difference/finite volume, etc:
- Degrees of freedom
- Cells/elements

- Molecular dynamics:
- If edge weights are used, also need a good measure on how strongly objects are coupled to each other

High Performance Computing 1

Static/Dynamic Load-Balancing

- Static load-balancing
- Done as a “preprocessing” step before the actual calculation
- If the objects and edges don’t change very much or at all, can do static load-balancing

- Dynamic load-balancing
- Done during the calculation
- Significant changes in the objects and/or edges

High Performance Computing 1

Dynamic Load-Balancing Example

- h-adapted mesh
- Workload is changing as the computation proceeds
- Calculate a new partition
- Need to migrate the elements to their assigned process

High Performance Computing 1

Static vs. Dynamic Load Balancing

- Static partitioning insufficient for many applications
- Adaptive mesh refinement
- Multi-phase/Multi-physics computations
- Particle simulations
- Crash simulations
- Parallel mesh generation
- Heterogeneous computers

- Need dynamic load balancing

High Performance Computing 1

Dynamic Load-Balancing Constraints

- Minimize load-balancing time
- Memory constraints

- Minimize data migration -- incremental partitions
- Small changes in the computation should result in small changes in the partitioning
- Calculating new partition and data migration should take less time than the amount of time saved by performing computations on new grid

- Done in parallel

High Performance Computing 1

Methods of Load-Balancing

- Geometric
- Based on geometric location
- Faster load-balancing time with medium quality results

- Graph-based
- Create a graph to represent the objects and their connections
- Slower load-balancing time but high quality results

- Incremental methods
- Use graph representation and “shuffle” around objects

High Performance Computing 1

Choosing a Load-Balancing Algorithm/Method

No algorithm/method is appropriate for all applications!

- Graph load-balancing algorithms for:
- Static load-balancing
- Computations where computation to load-balancing time ratio is high
- Implicit schemes with a linear and non-linear solution scheme

High Performance Computing 1

Choosing a Load-Balancing Algorithm/Method

- Geometric load-balancing algorithms for:
- Computations where computation to load-balancing time ratio is low
- For explicit time stepping calculations with many time steps and varying workload (MD, FEM crash simulations, etc.)
- Problems with many load-balancing objects

- Computations where computation to load-balancing time ratio is low

High Performance Computing 1

Geometric Load-Balancing

- Based on the objects’ coordinates
- Want a unique coordinate associated with an object
- Node coordinates, element centroid, molecule coordinate/centroid, etc.

- Want a unique coordinate associated with an object
- Partition “space” which results in a partition of the load-balancing objects
- Edge cuts are usually not explicitly dealt with

High Performance Computing 1

Geometric Load-Balancing Assumptions

- Objects that are close will likely need to share information
- Want compact partitions
- High volume to surface area or high area to perimeter length ratios

- Want compact partitions
- Coordinate information
- Bounded domain

High Performance Computing 1

Geometric Load-Balancing Algorithms

- Recursive Coordinate Bisection (RCB)
- Berger & Bokhari

- Recursive Inertial Bisection (RIB)
- Taylor & Nour-Omid

- Space Filling Curves (SFC)
- Warren & Salmon, Ou, Ranka, & Fox, Baden & Pilkington

- Octree Partitioning/Refinement-tree Partitioning
- Loy & Flaherty, Mitchell

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Recursive Coordinate Bisection

- Choose an axis for the cut
- Find the proper location of the cut
- Group objects together according to location relative to cut
- If more partitions are needed, go to step 1

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Recursive Inertial Bisection

- Choose a direction for the cut
- Find the proper location of the cut
- Group objects together according to location relative to cut
- If more partitions are needed, go to step 1

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Space Filling Curves

A Space Filling Curve is a 1-dimensional curve which passes through every point in an n-dimensional domain

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Load-Balancing with Space Filling Curves

- The SFC gives a 1-dimensional ordering of objects located in an n-dimensional domain
- Easier to work with objects in 1 dimension than in n dimensions

- Algorithm:
- Sort objects by their location on the SFC
- Calculate cuts along the SFC

High Performance Computing 1

Tree based algorithms for applications with multiple levels of data, simulation accuracy, etc.

Tree is usually built from specific computational schemes

Tightly coupled with the simulation

Octree Partitioning/Refinement-Tree PartitioningHigh Performance Computing 1

Comparisons of RCB, RIB, and SFC of data, simulation accuracy, etc.

- RCB and RIB usually give slightly better partitions than SFC
- SFC is usually a little faster
- SFC is a little better for incremental partitions
- RIB can be real unstable for incremental partitions

High Performance Computing 1

Load-Balancing Libraries of data, simulation accuracy, etc.

- There are many load-balancing libraries downloadable from the web
- Mostly graph partitioning libraries
- Static: Chaco, Metis, Party, Scotch
- Dynamic: ParMetis, DRAMA, Jostle, Zoltan

- Mostly graph partitioning libraries
- Zoltan (www.cs.sandia.gov/Zoltan)
- Dynamic load-balancing library with:
- SFC, RCB, RIB, Octree, ParMetis, Jostle

- Same interface to all load-balancing algorithms

- Dynamic load-balancing library with:

High Performance Computing 1

Methods to Avoid Communication of data, simulation accuracy, etc.

- Avoiding load-balancing
- Load-balancing not needed every time the workload and/or edge connectivity changes

- Ghost cells
- Predictive load-balancing

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Accessing Information on Other Processors of data, simulation accuracy, etc.

- Need communication between processors
- Use ‘ghost’ cells – need to maintain consistency of data in ghost cells

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Ghost Cells of data, simulation accuracy, etc.

- Copies of cells assigned to other processors
- Make needed information available
- No solution values are computed at the ghost cells
- Ghost cell information needs to be updated whenever necessary
- Ghost cells need to be calculated dynamically because of changing mesh and dynamic load-balancing

High Performance Computing 1

Predictive Load-Balancing of data, simulation accuracy, etc.

- Predict the workload and/or edge connectivity and load-balance with that information
- Assumes that you can predict the workload and/or edge connectivity

- Still need to perform communication but reduces data migration

High Performance Computing 1

Predictive Load-Balancing of data, simulation accuracy, etc.

- Refine then load-balance – 4 objects migrated
- Predictive load-balance then refine – 1 object migrated

High Performance Computing 1

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