1. High Performance Computing 1 Load-Balancing

2. 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)

3. 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.

4. High Performance Computing 1 is addressed Matrix addressing in FORTRAN

5. 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

6. 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.

7. High Performance Computing 1 Indirect addressing If addressing is indirect we may wind up jumping all over, and suffer performance hits because of it.

8. 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)

9. 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

10. High Performance Computing 1 Load-balancing nomenclature Object Edge Objects get distributed among different processes Edges represent information that need to be shared between objects

11. High Performance Computing 1 Partitioning Divides up the work 5 & 4 objects assigned to processes Creates “edge-cuts” Necessary communications between processes

12. 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 If edge weights are used, also need a good measure on how strongly objects are coupled to each other

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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. Partition “space” which results in a partition of the load-balancing objects Edge cuts are usually not explicitly dealt with

21. 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 Coordinate information Bounded domain

22. 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

23. High Performance Computing 1 Recursive Coordinate Bisection 1.Choose an axis for the cut 2.Find the proper location of the cut 3.Group objects together according to location relative to cut 4.If more partitions are needed, go to step 1

24. High Performance Computing 1 Recursive Inertial Bisection 1.Choose a direction for the cut 2.Find the proper location of the cut 3.Group objects together according to location relative to cut 4.If more partitions are needed, go to step 1

25. High Performance Computing 1 Space Filling Curves A Space Filling Curve is a 1-dimensional curve which passes through every point in an n-dimensional domain

26. High Performance Computing 1 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: 1.Sort objects by their location on the SFC 2.Calculate cuts along the SFC

27. High Performance Computing 1 Octree Partitioning/Refinement- Tree Partitioning 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

28. High Performance Computing 1 Comparisons of RCB, RIB, and SFC 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

29. High Performance Computing 1 Load-Balancing Libraries There are many load-balancing libraries downloadable from the web –Mostly graph partitioning libraries Static: Chaco, Metis, Party, Scotch Dynamic: ParMetis, DRAMA, Jostle, Zoltan Zoltan (www.cs.sandia.gov/Zoltan) –Dynamic load-balancing library with: SFC, RCB, RIB, Octree, ParMetis, Jostle –Same interface to all load-balancing algorithms

30. High Performance Computing 1 Methods to Avoid Communication Avoiding load-balancing –Load-balancing not needed every time the workload and/or edge connectivity changes Ghost cells Predictive load-balancing

31. High Performance Computing 1 Accessing Information on Other Processors Need communication between processors Use ‘ghost’ cells – need to maintain consistency of data in ghost cells

32. High Performance Computing 1 Ghost Cells 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

33. High Performance Computing 1 Predictive Load-Balancing 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

34. High Performance Computing 1 Predictive Load-Balancing Refine then load-balance – 4 objects migrated Predictive load-balance then refine – 1 object migrated