CS 484

• Load Balancing

Load Balancing

Goal: All processors working all the time Efficiency of 1 Distribute the load (work) to meet the goal

Two types of load balancing Static Dynamic

Load Balancing

The load balancing problem can be reduced to the bin-packing problem NP-complete

For simple cases, we can do well, but … Heterogeneity Different types of resources

ProcessorNetwork, etc.

Evaluation of load balancing

Efficiency Are the processors always working? How much processing overhead is associated with the load balance algorithm?

Communication Does load balance introduce or affect the communication pattern?

How much communication overhead is associated with the load balance algorithm?

How many edges are cut in communication graph?

Partitioning Techniques

Regular grids (-: Easy :-) striping blocking use processing power to divide load more fairly

Generalized Graphs Levelization Scattered Decomposition Recursive Bisection

Example

• consider a set of twelve independent tasks with the following set of execution times: {10, 6, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1}

• How would you distribute these tasks among 4 processors?

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Consecutive block assignment

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execution time for these twelve tasks would be 20 time units

This schedule would take only 10 time units

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Evaluation of Load Balancing

• Goal: Find a good mapping from the application graph G = (V,E) onto the processor graph H = (U,F)

• Consider Load: max number of nodes from G assigned to any single node of H

Dilation: max distance of any route of a single edge from G in H

Congestion: max number of edges from G that have to be routed via any single edge in H

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Overall Goal: Find a mapping π that minimizes all three measures -load, dilation, and congestion

Note: Today’s networks make dilation inconsequential to some extent

Levelization

Begin with a boundary Number these nodes 1

All nodes connected to a level 1 node are labeled 2, etc.Partitioning is performed determine the number of nodes per processor

count off the nodes of a level until exhausted

proceed to the next level

Levelization

Recursive Coordinate Bisection

Divide the domain based on the physical coordinates of the nodes.Pick a dimension and divide in half.RCB uses no connectivity informationlots of edges crossing boundariespartitions may be disconnected

Some new research based on graph separators overcomes some problems.

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Unbalanced Recursive Bisection

• An attempt at reducing communication costs

• Create subgrids that have better aspect ratios

• Instead of dividing the grid in half, consider unbalanced subgrids of size: 1/p and (p-1)/p 2/p and (p-2)/p Etc.

• Choose the partition size that minimizes the subgrid aspect ratio

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Unbalanced Recursive Bisection

Graph Theory Based Algorithms

Geometric algorithms are generally low quality they don’t take into account connectivity

Graph theory algorithms apply what we know about generalized graphs to the partitioning problemHopefully, they reduce the cut size

Greedy Bisection

Start with a vertex of the smallest degree least number of

edges

Mark all its neighborsMark all its neighbors neighbors, etc.The first n/p marked vertices form one subdomainApply the algorithm on the remaining

Recursive Graph Bisection

Based on graph distance rather than coordinate distance.Determine the two furthest separated nodes

Organize and partition nodes according to their distance from extremities.

Computationally expensiveCan use approximation methods.

Recursive Spectral Bisection

• Minimize the number of edges cut with the partition

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RCB 529 edges cut RGB 618 edges cut

RSB299 edges cut

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Dynamic Load Balancing

Dynamic Load BalancingLoad is statically partitioned initially

Adjust load when an imbalance is detected.

Objectives rebalance the load keep edge cut minimized (communication) avoid having too much overhead

Dynamic Load Balancing

Consider adaptive algorithmsAfter an interval of computation mesh is adjusted according to an estimate of the discretization errorcoarsened in areasrefined in others

Mesh adjustment causes load imbalance

Centralized DLB

Control of the load is centralized

Two approaches Master-worker (Task scheduling)

Tasks are kept in central location Workers ask for tasks Requires that you have lots of tasks with weak locality requirements. No major communication between workers

Load Monitor Periodically, monitor load on the processors

Adjust load to keep optimal balance

Decentralizing DLB

Generally focused on work poolTwo approaches Hierarchy

Fully distributed

Fully Distributed DLB

Lower overhead than centralized schemes.No global information Load is locally optimized Propagation is slow Load balance may not be as good as centralized load balance scheme

Three steps Flow calculation (How much to move) Mesh node selection (Which work to move)

Actual mesh node migration

Flow calculation

View as a network flow problem Add source and sink nodes Connect source to all nodes

edge value is current load Connect sink to all nodes

edge value is mean loadprocessor communication graph

Flow calculation

Many network flow algorithms more intense than necessary not parallel

Use simpler, more scalable algorithmsRandom Matchings pick random neighboring processes

exchange some load eventually you may get there

Diffusion

Each processor balances its load with all its neighbors How much work should I have? ( is weighting factor)

How much to send on an edge?

Repeat until all load is balancedsteps

Diffusion

Convergence to load balance can be slow

Can be improved with over-relaxation Monitor what is sent in each step Determine how much to send based on current imbalance and how much was sent in previous steps

Diffuses load in steps

Dimension Exchange

Rather than communicate with all neighbors each round, only communicate with one synchronous algorithm Comes from dimensions of hypercube Use edge coloring for general graphs

Exchange load with neighbor along a dimension l = (li + lj)/2

Will converge in d steps if hypercubeSome graphs may need different factor to converge faster l = li * a + lj * (1 –a)