Load Balancing

• How?– Partition the computation into units of work (tasks or jobs)– Assign tasks to different processors

• Load Balancing Categories– Static (load assigned before application runs)– Dynamic (load assigned as applications run)

o Centralized (Tasks assigned by the master or root process)o De-centralized (Tasks reassigned among slaves)

– Semi-dynamic (application periodically suspended and load balanced)

• Load Balancing Algorithms are:– Adaptive if they adapt to system load levels using thresholds– Stable if load balancing traffic is independent of load levels– Symmetric if both senders and receivers initiate action– Effective if load balancing overhead is minimal

Definition: A load is balanced if no processes are idle

Note: Load balancing is an NP-Complete problem

Improving the Load Balance

By realigning processing work, we improve speed-up

Static Load Balancing

• Round Robin – Tasks assigned sequentially to

processors– If tasks > processors, the

allocation wraps around

• Randomized: Tasks are assigned randomly to processors

• Partitioning – Tasks are represented by a graph– Recursive Bisection– Simulated Annealing– Genetic Algorithms– Multi-level Contraction and

Refinement

• Advantages– Simple to implement– Minimal run time

overhead• Disadvantages

– Predicting execution times is often not knowable

– Affect of communication dynamics is often ignored

– The number of iterations required by processors to converge on a solution is often indeterminate

Done prior to executing the parallel application

Note: The Random algorithm is a popular benchmark for comparison

• The Nodes represent tasks• The Edges represent communication cost• The Node values represent processing cost• A second node value could represent reassignment

cost

A Load Balancing Partitioning Graph

Simulated Annealing• Overview

– A heuristic used to address many NP-Complete problems, which is more efficient than exhaustively searching graphs

– Metallurgy: heating and cooling causes some atoms to temporarily randomly move through higher energy states; settling in a more stable state

• AlgorithmWHILE balance is not good enough

FOR each node in the graph

Compute its energy (computation/communication requirements)

IF probabilistic calculation indicates that node should move

Pick a set of neighbors assigned to other processors

FOR each neighbor in the set

Compute impact of reassigning the node

Move the node to the best neighbor partition

Simplifying the Problem

Recursive Bisection

• Overview: Dividing the graph in two is an easier problem than tackling the entire problem

• AlgorithmIF graph is too large

– Split the graph in two, minimizing computation and communication requirements

– Color each side

– Recursively continue splitting

Multi-level Approaches

• Overview: Similar to Recursive Bisection, but refines the approach

• Contraction phase shrinks the graph, Refinement phase restores it to its original size

• Algorithm– WHILE graph is too large

• Find bisection point and contract the graph

– WHILE graph is smaller contracted, refine the graph, and reassign nodes if improvements are possible

Dynamic Load Balancing

• Centralized– A single process hands out tasks– Processes ask for more work when their processing completes– Double buffering (ask for more while still working) can be effective

• Decentralized– Processes detect that their work load is low– Processes sense an overload condition

• When new tasks are spawned during execution• When a sudden increase in task load occurs

– Questions• Which neighbors should participate in the rebalancing?• How should the adaptive thresholds be set?• What are the communications needed to balance?• How often should balancing occur?

Done as a parallel application executes

Centralized Load Balancing

Master Processor: Maintains the work pool (queue, heap, etc.)While ( task=Remove()) != null)

Receive(pi, request_msg)

Send(pi, task)While(more processes)

Receive(pi, request_msg)

Send(pi, termination_msg)

Slave Processor: Perform task and then ask for anothertask = Receive(pmaster, message)While (task!=terminate) Process task

Send(pmaster, request_msg)

task = Receive(pmaster, message)

Work Pool, Processer Farm, or Replicated Worker Algorithm

Slaves

Master

In this case, the slaves do not spawn new tasksHow would the pseudo code change if they did?

Centralized Termination

Necessary Requirements– The task queue is empty– Every process is waiting for a new task

Master ProcessorWHILE (true) Receive(pi, msg) IF msg contains a new task Add the new task to the task queue ELSE Add pi to wait queue and waitCount++

IF waitCount>0 and task queue not empty Remove pi & task respectively from wait & task queue Send(task, pi) and waitCount—-

IF waitCount==P THEN send termination messages & exit

How do we terminate when slave processes spawn new tasks?

Decentralized Load Balancing

• There is no Master Processor• Each Processor maintains a work queue• Processors interact with neighbors to request and distribute

tasks

(Worker processes interact among themselves)

Decentralized Mechanisms

• Receiver Initiated– Process requests tasks when it is about to go idle– Effective when the load is heavy– Unstable when the load is light

(A request frequency threshold is necessary)

• Sender Initiated– Process with a heavy load distributes the excess– Effective when the load is heavy– Can cause thrashing when loads are heavy

(synchronizing system load with neighbors is necessary)

Balancing is among a subset of the total running processes

ApplicationBalancingAlgorithm

Task Queue

Process Selection

• Global or Local?– Global involves all of the processors of the network

• May require expensive global synchronization• May be difficult if the load dynamic is rapidly changing

– Local involves only neighbor processes• Overall load may not be balanced• Easier to manage and less overhead than the global approach

• Neighbor selection algorithms– Random: randomly choose another process

• Easy to implement and studies show reasonable results– Round Robin: Select among neighbors using modular arithmetic

• Easy to implement. Results similar to random selection– Adaptive Contracting: Issue bids to neighbors; best bid wins

• Handshake between neighbors needed• It is possible to synchronize loads

Choosing Thresholds• How do we estimate system load?

– Synchronization averages task queue length or processes– Average number of tasks or projected execution time

• When is the load low?– When a process is about to go idle– Goal: prevent idleness, not achieve perfect balance– A low threshold constant is sufficient

• When is the load high?– When some processes have many tasks and others are idle– Goal: prevent thrashing– Synchronization among processors is necessary– An exponentially growing threshold works well

• What is the job request frequency?– Goal: minimize load balancing overhead

Gradient Algorithm

• Node Data Structures– For each neighbor

• Distance, in hops, to the nearest lightly-loaded process

– A load status flag indicating if the current processor is lightly-loaded, or normal

• Routing– Spawned jobs go to the nearest

lightly-loaded process

• Local Synchronization– Node status changes are

multicast to its neighbors

L

2

1 2

211

2 2

Maintains a global pressure grid

Symmetric Broadcast Networks (SBN)

• Characteristics– A unique SBN starts at each node– Each SBN is lg P deep– Simple operations algebraically

compute successors– Easily adapts to the hypercube

• Algorithm– Starts with a lightly loaded process– Phase 1: SBN Broadcast– Phase 2: Gather task queue lengths– Load is balanced during the load

and gather phases

1

3

4 2

7

60

Global Synchronization

Stage 0

Stage 1

Stage 2

5Stage 3

Successor 1 = (p+2s-1) %P; 1≤s≤3Successor 2 = (p-2s-1); 1≤s<3

Note: If successor 2<0 successor2 +=P

Hypercube Based SBN

Four Dimension Hypercube SBN Hypercube Spanning Tree

• Stage = number of bits set• Successors: exclusive or with zero bits to the

left, or if none with the first leftmost unset bit• Single hop required between adjacent nodes• Load balance requires 2 lg(P) communications

Find Successor Ranksfor (int rank=0; rank<P-1; rank++)

{ int stage = 0, n = rank; // Compute the number of bits set

while (n!=0) { n = n & (n-1); stage++; }

System.out.printf("Successors rank %d stage %d = ", rank, stage);

if (rank>=P/2) // No unset bits to the left

{ mask = (1<<(dimension-stage))-1;

successor = ((rank & ~mask)>>1) | rank;

System.out.println(successor);

}

else // Successors are all ranks with unset bits to the left

{ for (int bit = 1<<(dimension-1); bit>rank; bit/=2)

{ successor = rank | bit; System.out.print(successor + " "); }

System.out.println();

} } }

SBN PatternRank 0

Successors rank 0 stage 0 = 8 4 2 1

Successors rank 1 stage 1 = 9 5 3

Successors rank 2 stage 1 = 10 6

Successors rank 3 stage 2 = 11 7

Successors rank 4 stage 1 = 12

Successors rank 5 stage 2 = 13

Successors rank 6 stage 2 = 14

Successors rank 7 stage 3 = 15

Successors rank 8 stage 1 = 12

Successors rank 9 stage 2 = 13

Successors rank 10 stage 2 = 14

Successors rank 11 stage 3 = 15

Successors rank 12 stage 2 = 14

Successors rank 13 stage 3 = 15

Successors rank 14 stage 3 = 15

• For other ranks, simply exclusive or the desired source rank with the above pattern

• For example: Consider the source rank of 5– The stage 2 processors are

3^5 (6), 5^5 (0), 6^5 (3), 9^5 (12), 10^5 (15), and 12^5 (9)

– The successors to 3^5 (6) are 11^5 (14) and 7^5 (2)

Line BalancingAlgorithm

• Master or slave processors adjust pipeline

• Slave processors– Request and receives tasks if queue not full

– Pass tasks on if task request is posted

• Non blocking receives are necessary to implement this algorithm

Uses a pipeline approach

Request taskif queue not full

Receive taskfrom request

Deliver task to pi+1

pi+1 requests task

Dequeue andprocess task

pi

Note: This algorithm easily extends to a tree topology

Semi-dynamic• Pseudo codeRun algorithmTime to check balance? Suspend application IF load is balanced, resume application Re-partition the load Distribute data structures among processors Resume execution

• Partitioning– Model application execution by a partitioning graph– Partitioning is an NP-Complete problem– Goals: Balance processing and minimize communication and

relocation costs– Partitioning Heuristics

• Recursive Bisection, Simulated Annealing, Multi-level, MinEx

Partitioning Graph

P2R1

P5R3

P8R3

P4R1

P6R6

P2R1

P9R6

P4R4P7

R5

P1 P2

c4 c6

c2

c1

c7

c1c3

c8

c5c3

P1 Load = (9+4+7+2) + (4+3+1+7) = 37P2 Load = (6+2+4+8+5) + (4+3+1+7) = 40

Question: When can we move a task to improve load balance?

Distributed Termination• Insufficient condition for distributed termination

– Empty task queues at every process

• Sufficient condition for distributed termination requires– All local termination conditions satisfied– No messages in transit that could restart an inactive process

• Termination algorithms– Acknowledgment– Ring– Tree– Fixed energy distribution

Acknowledgement Termination• Process Receives task

– Immediately acknowledge if source is not parent– Acknowledge parent as process goes idle

• Process goes idle after it– completes processing local tasks– Sends all acknowledgments– Receives all pending acknowledgments

• Notes– The process sending an initial task that activates

another process becomes that process's parent– A process always goes inactive before its parent– If the master goes inactive, termination occurs

Active

Inactive

First task

Acknowledge first task

Pi

Pj

Single Pass Ring Termination• Pseudo codeP0 sends a token to P1 when it goes idle

Pi receives token

IF Pi is idle it passes token to Pi+1

ELSE Pi sends token to Pi+1 when it goes idle

P0 receives token

Broadcast final termination message

• Assumptions – Processes cannot reactivate after going idle

– Processes cannot pass new tasks to an idle process

P0 P1 P2 Pn

Token

Dual Pass Ring Termination

Pseudo code (Only idle processors send tokens)

WHEN P0 goes idle and has token, it sends white token to p1

IF Pi sends a task to pj where j<i

Pi becomes a black process

WHEN Pi>0 receives token and goes idle

IF Pi is a black process

Pi colors the token black, Pi becomes White

ELSE Pi sends token to p(i+1)%P unchanged in color

IF P0 receives token and is idle IF token is White, application terminates

ELSE po sends a White token to P1

Handles task sent to a process that already passed the token onKey Point: Processors pass either Black or White tokens on only if they are idle

Process: white=ready for termination, black: sent a task to Pj-x

Token: white=ready for termination, black=communication possible

Tree Termination

• If a Leaf process terminates, it sends a token to it’s parent process• Internal nodes send tokens to their parent when all of their child

processes terminate• If the root node receives the token, the application can terminate

AND

Leaf Nodes

Terminated

Fixed Energy Termination

• P0 starts with full energy– When Pi receives a task, it also receives an energy allocation– When Pi spawns tasks, it assigns them to processors with

additional energy allocations within its allocation– When a process completes it returns its energy allotment

• The application terminates when the master becomes idle• Implementation

– Problem: Integer division eventually becomes zero– Solution:

o Use two level energy allocation <generation, energy>o The generation increases each time energy value goes to zero

Energy defined by an integer or long value

Example: Shortest Path Problem

DefinitionsGraph: Collection of nodes (vertices) and edgesDirected Graph: Edge can be traversed in only one directionWeighted Graph: Edges have weights that define costShortest Path Problem: Find the path from one node to another in a weighted graph that has the smallest accumulated weights

Applications1.Shortest distance between points on a map2.Quickest travel route3.Least expensive flight path4.Network routing5.Efficient manufacturing design

Climbing a Mountain

• Weights: expended effort• Directed graph

– Effort in one direction ≠ effort in another direction

– Ex: Downhill versus uphill

A B C D E F

A 10

B 8 13 24 51

C 14

D 9

E 17

F

A B C

DE

F

10 8132451

14

917

Adjacency Matrix

C 8

D 14 X

E 9 X

F 17 X

X

B 10 X

D 13

E 24

F 51

A

B

C

D

E

F Adjacency List

Graphic Representation

Moore’s Algorithm

• Assume – w[i][j] = weight of edge (i,j)– Dist[v] = distance to vertex v– Pred[v] = predecessor to vertex v

• Pseudo codeInsert the source vertex into a queueFor each vertex, v,

dist[v]=∞ infinity, dist[0] = 0WHILE (v = dequeue() exists) FOR (j=; j<n; j++) newdist = dist[i] + w[i][j] IF (newdist < dist[j]) dist[j] = newdist pred[j] = I append(j)

Less efficient than Dijkstra but more easily parallelized

i j

diwi,j

dj

dj=min(dj,di+wi,j)

Centralized Work Pool Solution

• The Master maintains– The work pool queue of unchecked vertices– The distance array

• Every slave holds: The graph weights which is static• The Slaves

– Request a vertex– Compute new minimums– Send updated distance values and vertex to master

• The Master– Appends received vertices to its work queue– Sends new vertex and the updated distance array.

Distributed Work Pool Solution

• Data held in each processor– The graph weights– The distances to vertices stored locally– The processor assignments

• When a process receiving a distance:– If its local value is reduced

o Updates its local value of dist[v]o Send distances to adjacent vertices to appropriate processors

• Notes– The load can be very imbalanced– One of the termination algorithms is necessary