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Parallelization of Irregular Applications:
Hypergraph-based Models and Methods for
Partitioning and Load Balancing
Cevdet Aykanat
Bilkent University
Computer Engineering Department
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“Partitioning” of irregular computations
• Partitioning of computation into smaller works
• Divide the work and data for efficient parallel computation
(applicable for data-parallelism)
Partitioning of irregular computations“Good Partitioning”
• Low computational imbalance
(computational load balancing)
• Low communication overhead
– Total message volume
– Total message count (latency)
– Maximum message volume and count per processor
(Communication load minimization and balancing)
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Why graph models are not sufficient?
• Existing Graph Models– Standard graph model– Bipartite graph model– Multi-constraint / multi-objective graph partitioning– Skewed partitioning
• Flaws– Wrong cost metric for communication volume– Latency: number of messages also important– Minimize the maximum volume and/or message count
• Limitations– Standard Graph Model can only express symmetric dependencies– Symmetric = identical partitioning of input and output data– Multiple computation phases
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Preliminaries: Graph Partitioning
P1 P2
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• Partitioning constraint: maintaining balance on part weights
• Partitioning objective: minimizing cutsize
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Preliminaries: Hypergraph Partitioning
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V1 V2
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• Partitioning constraint: maintaining balance on part weights
• Partitioning objective: minimizing cutsize
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Irregular Computations – an example: Parallel Sparse Matrix Vector Multiply (y=Ax)
• Abounds in scientific computing– A concrete example is iterative methods
• SpMxV is applied repeatedly• efficient parallelization: load balance and small communication
cost
• Characterizes a wide range of applications with irregular computational dependency
• A fine-grain computation– guaranteeing efficiency will guarantee higher efficiency in
applications with coarser-grain computations
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Sparse-matrix partitioning taxonomy for parallel y=Ax
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Row-parallel y=Ax• Rows (and hence y) and x is partitioned
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10111213141516
y4
y3
y2
y1P1
P2
P3
P4
x1 x2 x3 x4
P1 P2 P3 P4
1. Expand x vector (sends/receives)
2. Compute with diagonal blocks
3. Receive x and compute with off-diagonal blocks
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Row-parallel y=Ax
Communication requirements
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y4
y3
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y1P1
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x1 x2 x3 x4
P1 P2 P3 P4
Total message volume:#nonzero column segments in off diagonal blocks (13)
Total message number :#nonzero off diagonal blocks (9)
Per processor: above two metrics confined within a column stripe
We minimize total volume and number of messages and obtain balance on per processor basis
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Column-parallel y=Ax• Columns (and hence x) and y is partitioned
1. Compute with off diagonal blocks; obtain partial y results, issue sends/receives
2. Compute with diagonal block
3. Receive partial results on y for nonzero off-diagonal blocks and add the partial results
y3
y2
y1 P1
P2
P3
P4
x1 x2 x3
P1 P2 P3 P4
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1011121314151617181920212223242526
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x4
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Row-column-parallel y=Ax• Matrix nonzeros, x, and y are partitioned
1. Expand x vector
2. Scalar multiply and add yi aijxj aikxk
3. Fold on y vector
(send and receive partial results)
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P1
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P3
P3
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P2
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P2 Send partial y9
Send x5
and x7
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Hypergraph Models for y = Ax
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Hypergraph Model for row-parallel y = Ax 1D Rowwise Partitioning: Column-net Model
• Columns are nets; rows are vertices: Connect vi to nj if aij is nonzero
• Assign yi using vi, xj using nj (permutation policy)
• Respects row coherence, disturbs column coherence
• Symmetric partitioning: aii should be nonzero
• Partitioning constraint: computational load balancing
• Partitioning objective: minimizing total communication volume
• Comm. vol. = 2(x5)+2(x12)+1(x7)+1(x11) = 6 words
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• Dual of column-net model: rows are nets; columns are vertices• Assign xj using vj, yi using ni (permutation policy)• Symmetric partitioning: nonzero diagonal entries• Respects column coherence, disturbs row coherence• Partitioning constraint: computational load balancing• Partitioning objective: minimizing total communication volume • Comm. vol. = 2(y5)+2(y12)+1(y6)+1(y10) = 6 words
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P1 P2 P3 P4 v16
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Hypergraph Model for column-parallel y = Ax 1D Columnwise Partitioning: Row-net Model
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Hypergraph Model for row-column-parallel y = Ax: 2D Fine-Grain Partitioning: Row-column-net Model
• MN matrix A with nnz nonzeros: M row nets,
N column nets
nnz vertices.
• Vertex aij is connected to
nets ri and cj
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2D Fine-Grain Partitioning
• One vertex for each nonzero
• Disturbs both row and column coherence
• Partitioning constraint: computational load balancing
• Partitioning objective: minimizing total communication volume
• Symmetric partitioning: nonzero diagonal entries
r5
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con-1 =2
con-1=1
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Hypergraph Methods for y = Ax
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Hypergraph Methods for y = Ax
2D Jagged 2D Checkerboard
• √K–by–√K virtual processor mesh
• 2-phase method: Each phase models either expand or fold communication
• Column-net and Row-net models are used
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HP Methods for y = Ax: 2D Jagged Partitioning
• Phase 1: √K-way rowwise partitioning using column-net model
• Respects row coherence at processor-row level
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• Phase 2: √K independent √K-way columnwise partitionings using row-net model
• Symmetric partitioning
• Column coherence for cols 2 and 5 are disturbed => P3 sends messages to both P1 and P2
HP Methods for y = Ax: 2D Jagged Partitioning
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• Row coherence => fold comms confined to procs in the same row => max mssg count per proccessor = √K-1 in fold phase
• No column coherence => max mssg count per proc = K - √K in expand phase
HP Methods for y = Ax: 2D Jagged Partitioning
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HP Methods for y = Ax: 2D Checkerboard Partitioning
• Phase 1: √K-way rowwise partitioning using column-net model
• Respects row coherence at processor-row level
• Same as phase 1 of jagged partitioning
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• Phase 2: One √K-way √K-constraint columnwise partitioning using row-net model
• Respects column coherence at processor column level
• Symmetric partitioning
HP Methods for y = Ax: 2D Checkerboard Partitioning
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• Row coherence => fold comms confined to procs in the same row => max mssg count per proccessor = √K-1 in fold phase
• Column coherence => expand comms confined to procs in the same column => max mssg count per proccessor = √K-1 in fold phase
HP Methods for y = Ax: 2D Checkerboard Partitioning
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• Respects either row or column coherence at proc level• Single communication phase (either expand or fold)• Max mssg count per processor = K – 1 • Fast partitioning time• Determining the partitioning dimension
– Dense columns => rowwise partitioning– Dense rows => columnwise partitioning– Partition both rowwise and columnwise and choose the best
Comparison of Partitioning Schemes:1D Partitioning
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• Do not respect row or column coherencies at processor level• Two communication phases (both expand and fold) • Better in load balancing• 2D Fine Grain:
– Disturbs both row and column coherencies at processor-row and processor-col level– Significant reductions in total volume wrt 1D and other 2D partitionings – Worse in total number of messages– Slow partitioning time
• 2D Checkerboard– Respect both row and column coherencies at processor-row and processor-col level– Restricts max mssg count per processor to 2(√K –1)– Good for architectures with high mssg latency – As fast as 1D partitioning schemes
• 2D Jagged:– Respect either row or column coherencies at processor-row or processor-col level– Restricts max mssg count per processor to K – 2√K –1 – Better communication volume and load balance wrt checkerboard– As fast as 1D partitioning schemes
Comparison of Partitioning Schemes:2D Partitioning
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2-Phase Approach for Minimizing Multiple
Communication Hypergraph ModelCommunication-Cost Metrics:
• Communication Cost Metrics-- Total message volume
-- Total message count
-- Maximum message volume and count per processor
• 2-Phase Approach: 1D unsymmetric partitioning
-- Phase 1: Rowwise (colwise) partitioning using col-net (row-net) model
– minimize total mssg volume
– maintain computational load balance
-- Phase 2: Refine rowwise (colwise) partition using communication hypergraph model– encapsulate the remaining communication-cost metrics
– try to attain total mssg volume bound obtained in Phase 1
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• Construct communication matrix Cx– Perform rowwise compression
• Compress row stripe Rk of A into a single row rk of Cx – Sparsity pattern of rk = Union of sparsity patterns of all rows in Rk
– Discard internal columns of Rk
– Nonzeros in rk: Subset of x-vector entries needed by processor Pk.– Nonzeros in column cj: The set of processors that need xj
• Construct row-net model H of comm matrix Cx
– Vertices (columns of Cx) represent expand communication tasks
– Nets (rows of Cx) represent processors – Partitioning constraint: balancing send volume loads of processors– Partitioning objective: minimizing total number of messages
Communication Hypergraph Model:Phase 2 for 1D rowwise partitioning
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Communication Hypergraph Model
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Rowwise partition obtained in phase one Communication matrix
Communication hypergraphColwise permutation induced by comm HP
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• Communication Cost Metrics– Total message volume and count
– Maximum message volume and count per processor
• Balancing on external nets of parts => minimizing max mssg count per processors
Communication hypergraph model showing incoming and outgoing messages of processor Pk
Communication Hypergraph Model
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• Construct communication matrix Cy– Perform columnwise compression
• Compress kth column stripe of A into a single column ck of Cy – Sparsity pattern of ck = Union of sparsity patterns of all columns in kth
column stripe
– Discard internal rows of kth column stripe
– Nonzeros in ck: Subset of y-vector entries needed by processor Pk.– Nonzeros in row rj: The set of processors that need yj
• Construct column-net model of comm matrix Cy– Vertices (rows of Cy) represent fold communication tasks
– Nets (cols of Cy) represent processors – Partitioning constraint: balancing send volume loads of processors– Partitioning objective: minimizing total number of messages
Communication Hypergraph Model:Phase 2 for 1D columnwise partitioning
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Communication Hypergraph ModelColwise partition obtained in Phase-1 Communication matrix Communication hypergraph
Colwise permutation induced by comm HP
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Communication Hypergraph Modelfor 2D Fine-Grain Partitioning
• Phase 1: Obtain a fine-grain partition on nonzero elements• Phase 2: Obtain 2 communication matrices:
– Cx: rowwise compression– Cy: columnwise compression
• Phase 2: Construct two communication hypergraphs– Hx: column-net model of Cx– Hy: row-net model of Cy
• Phase 2 for unsymmetric partitioning– Partition Hx and Hy independently– Partitioning constraints: balancing expand and fold volume loads of
processors– Partitioning objectives: minimizing total number of messages in the expand
and fold phases
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Communication Matrix Cx
Communication Matrix Cy
Rowwise compression
Columnwise compression
Fine-grain partitioned matrix obtained in Phase-1 Cx represents
which processor needs which x vector entry
Cy represents which processor contributes to which y entry
Communication Hypergraph Modelfor 2D Fine-Grain Partitioning
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Phase-2 for unsymmetric partitioning
• Partition communication hypergraphs Hx and Hy
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P3
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m1
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Cut= 1+ 2+ 1+1= 5 Cut=5
HxHy
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Number of messages regarding x
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x1 and x2
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x5 and x7
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x10
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Number of messages regarding y
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partial y4
P3 P4
P2 P3
P4 P2
P3 P1
partial y4
partial y5
partial y7
partial y9
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Communication Hypergraph Modelfor 2D Fine-Grain Partitioning
• Phase-2 for symmetric partitioning– Combine two hypergraphs Hx and Hy into a new one H by merging
corresponding vertices
– Both net nk and mk represent processor k
• Cut net nk : processor k sending message(s) in expand phase
• Cut net mk : processor k receiving message(s) in fold phase
– Fix both net nk and mk to part Vk
• H contains K fixed vertices
– Partition augmented hypergraph H• partitioning objective: minimizing total number of messages
• 2 partitioning constraints: balancing expand and fold volume loads of processors
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Phase 2 for symmetric partitioning
• Partition augmented hypergraph H.
A portion of augmented hypergraph H
• Vertex v4 does not exist in Hx whereas it exists in Hy.
– All nonzeros of column 4 were assigned to processor 2 in fine-grain partitioning.
• Vertex v7 exists in both Hx and Hy.
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SpMxV Context: Iterative Methods
• Used for solving linear systems Axb– Usually A is sparse
• Involves– Linear vector operations
• x = xy xi = xi yi
– Inner products = x,y = sum of xi yi
– Sparse matrix-vector multiplies (SpMxV)• y = Ax yi = Ai,x
• y = ATx yi = ATi,x
– Assuming 1D rowwise partitioning of A
while not converged do computations check convergence
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Parallelizing Iterative Methods
• Avoid communicating vector entries for linear vector operations and inner products
• Nothing to do for inner products– Regular communication – Low communication volume
• partial sum values communicated
• Efficiently parallelize the SpMxV operations – in the light of previous discussion, are we done?– Preconditioning?
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Preconditioning
• Iterative methods may converge slowly, or diverge
• Transform Axb to another system that is easier to solve
• Preconditioner is a matrix that helps in obtaining desired transformation
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Preconditioning
• We consider parallelization of iterative methods that use approximate inverse preconditioners
• Approximate inverse is a matrix M such that AMI
• Instead of solving Axb, use right preconditioning and solve
AMyb and then set
x = My• Preconditioning cliché: “Preconditioning is art rather
than science”
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Preconditioned Iterative Methods• Additional SpMxV operations with M
– never form matrix AM; perform successive SpMxVs
• Parallelizing a full step in these methods requires efficient SpMxV operations with A and M– partition A and M
• What have been done– a bipartite graph model with limited usage
• A blend of dependencies and interactions among matrices and vectors– partition A and M simultaneously
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Preconditioned Iterative Methods
• Partition A and M simultaneously
• Figure out partitioning requirements through analyzing linear vector operations and inner products
– Reminder: never communicate vector entries for these operations
• Different methods have different partitioning requirements
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Preconditioned BiCG-STAB
tsr
spxx
ttst
sAt
Mss
vrs
pAv
Mpp
vprp
ii
ii
iii
i
ii
i
i
i
ii
ii
ii
1
1
11
11
1
,,
ˆ
ˆ
ˆ
ˆ
p, r, v should be partitioned conformably
s should be with r and v
t should be with s
x should be with p and s
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Preconditioned BiCG-STAB p, r, v, s, t, and, x should be partitioned conformably
• What remains?
sAt
Mss
pAv
Mppi
i
ˆ
ˆ
ˆ
ˆ
Columns of M and rows of A should be conformal
should be conformal
p
s
Rows of M and columns of A should be conformal
PAQT QMPT
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Partitioning Requirements
• “and” means there is a synchronization
point between successive SpMxV’s– Load balance each SpMxV individually
BiCG-STAB PAQTQMPTPAMPT
TFQMR PAPT and PM1M2PT
GMRES PAPT and PMPT
CGNE PAQ and PMPT
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Model for simultaneous partitioning
• We use the previously proposed models– define operators to build composite models
ci(A)
ri(M)
ri(A)
| ri(A) |
ci(M)
| ci(M) |
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Combining Hypergraph Models× Net amalgamation:
× newer combine nets of individual hypergraphs
• Vertex amalgamation: – combine vertices of individual hypergraphs, and connect the
composite vertex to the nets of the individual vertices
• Vertex weighting: – define multiple weights; individual vertex weights are not added up
• Vertex insertion: – create a new vertex, di, to be connected to nets ni of individual
hypergraphs
• Pin addition: – connect a specific vertex to a net
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Combining Guideline
1. Determine partitioning requirements
1. Decide on partitioning dimension for each matrix
• generate column-net model for the matrices to be partitioned rowwise
• generate row-net model for the matrices to be partitioned columnwise
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Combining Guideline1. Apply vertex operations
i. to impose identical partition on two vertices amalgamate them
ii. if the application of matrices are interleaved with synchronization apply vertex weighting
2. Apply net operationsi. if a net ought to be permuted with a specific vertex
then establish a policy for that net. If it is not connected then apply pin addition
ii. if two nets ought to be permuted together independent of the existing vertices then apply vertex insertion
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Combining Example
• BiCG-STAB requires PAMPT
– Reminder: rows of A and columns of M;
columns of A and rows of M
• A rowwise and M columnwise
ci(A)
ri(M)
ri(A)
| ri(A) |
ci(M)
| ci(M) |
ci(A)
ri(M)
ri(A)ci(M)
| ri(A) | + | ci(M) |
1
2 3i
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Combining Example
ci(A)
ri(M)
diri(A)ci(M)
| ri(A) | + | ci(M) |, 0
0, ?
3ii
4ii
– PAMPT: Columns of A and rows of M should be conformable
– di vertices are different
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Remarks on composite models• Operations on hypergraphs are defined to built
composite hypergraphs• Partitioning the composite hypergraphs
– balances computational loads of processors– minimizes the total communication volume
in a full step of the preconditioned iterative methods
• Can meet a broad spectrum of partitioning requirements: multi-phase, multi-physics applications
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Parallelization of a sample application (other than SpMxV)
A Remapping Model for Image-Space Parallel Volume Rendering
• Volume Rendering– mapping a set of scalar or vectoral values defined in a
3D dataset to a 2D image on the screen.
– Surface-based rendering– Direct volume rendering (DVR)
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Ray-Casting-Based DVR
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Parallel Rendering: Object-space (OS) and Image-space (IS)
Parallelization
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Screen Partitioning• Static
simplicity good load balancing high primitive replication
• Dynamic good load balancing– overhead of broadcasting pixel assignments– high primitive replication
• Adaptive good load balancing respects image-space coherency view-dependent preprocessing overhead
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Adaptive Parallel DVR Pipeline
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Screen Partitioning
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View Independent Cell Clustering
Tetrahedral dataset Graph representation
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View Independent Cell Clustering
Resulting 6 cell clusters6-way partitioned graph
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Pixel Clustering
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Cons and Pros of Cell and Pixel Clustering
Reduction in the 3D-2D interaction
Less preprocessing overhead
Estimation errors in workload calculations
Increase in data replication
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Screen Partitioning Model
A sample visualization instance 15 cell clusters and eight pixel blocks
Interaction hypergraph
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Screen Partitioning Model
3-way partition of Interaction hypergraph
Cell cluster (net)
Pixel block (vertex)
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Two-Phase Remapping ModelCell cluster (net)
Pixel block (vertex)
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Two-Phase Remapping ModelWeighted bipartite graph matching
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One-Phase Remapping ModelHypergraph Partitioning with Fixed Vertices
3-way partitioning of a remapping hypergraph
Cell cluster (net)
Pixel block (free vertex)
Processor (fixed vertex)
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Adaptive IS-Parallel DVR Algorithm
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Example Rendering
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Example Screen PartitioningsJagged partitioning Hypergraph partitioning
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Rendering Load Imbalance
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Total Volume of Communication
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Speedups
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Related works of our groupSparse matrix partitioning for parallel processing
• U.V. Çatalyürek, C. Aykanat, and B. Ucar, “On Two-Dimensional Sparse Matrix Partitioning: Models, Methods and a Recipe”, submitted to SIAM Journal on Scientific Computing.
• B. Ucar and C. Aykanat, “Revisiting Hypergraph Models for Sparse Matrix Partitioning,” SIAM Review, vol. 49(4), pp. 595–603, 2007.
• B. Uçar and C. Aykanat, “Partitioning Sparse Matrices for Parallel Preconditioned Iterative Methods,” SIAM Journal on Scientific Computing, vol. 29(4), pp. 1683–1709, 2007.
• B. Ucar and C. Aykanat, “Encapsulating Multiple Communication-Cost Metrics in Partitioning Sparse Rectangular Matrices for Parallel Matrix-Vector Multiplies," SIAM Journal on Scientific Computing, vol. 25(6), pp. 1837–1859, 2004.
• C. Aykanat, A. Pinar, and U.V. Catalyurek, “Permuting Sparse Rectangular Matrices into Block Diagonal Form," SIAM Journal on Scientific Computing, vol. 25(6), pp. 1860–1879, 2004.
• B.Ucar and C.Aykanat, "Minimizing Communication Cost in Fine-Grain Partitioning of Sparse Matrices,” Lecture Notes in Computer Science, vol. 2869, pp. 926–933, 2003.
• U.V. Çatalyürek and C. Aykanat, "Hypergraph-Partitioning-Based Decomposition for Parallel Sparse-Matrix Vector Multiplication," IEEE Transactions on Parallel and Distributed Systems, vol. 10, pp. 673–693, 1999.
• U.V. Catalyurek and C. Aykanat, "Decomposing Irregularly Sparse Matrices for Parallel Matrix-Vector Multiplication," Lecture Notes in Computer Science, vol. 1117, pp. 75–86, 1996.
• A. Pinar, C. Aykanat and M. Pinar, "Decomposing Linear Programs for Parallel Solution," Lecture Notes in Computer Science, vol. 1041, pp. 473–482, 1996.
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Related works of our groupGraph and Hypergraph Models and Methods for other Parallel & Distributed Applications
• B.B. Cambazoğlu and C. Aykanat, “Hypergraph-Partioning-Based Remapping Models for Image-Space-Parallel Direct Volume Rendering of Unstructured Grids,” IEEE Transactions on Parallel and Distributed Systems, vol. 18(1), pp. 3–16, 2007.
• K. Kaya, B. Uçar and C. Aykanat, “Heuristics for Scheduling File-Sharing Tasks on Heterogeneous Systems with Distributed Repositories,” Journal of Parallel and Distributed Computing, vol. 67, pp. 271–285, 2007.
• B. Uçar, C. Aykanat, M. Pınar and T. Malas, “Parallel Image Restoration Using Surrogate Constraint Methods,” Journal of Parallel and Distributed Computing, vol. 67, pp. 186–204, 2007.
• C. Aykanat, B. B. Cambazoğlu, F. Findik, and T.M. Kurc, “Adaptive Decomposition and Remapping Algorithms for Object-Space-Parallel Direct Volume Rendering of Unstructured Grids,” Journal of Parallel and Distributed Computing, vol. 67, pp. 77–99, 2006.
• K. Kaya and C. Aykanat, “Iterative-Improvement-Based Heuristics for Adaptive Scheduling of Tasks Sharing Files on Heterogeneous Master-Slave Environments,” IEEE Transactions on Parallel and Distributed Systems, vol. 17(8), pp. 883–896, August 2006.
• B. Ucar, C. Aykanat, K. Kaya and M. İkinci, “Task Assignment in Heterogeneous Systems,” Journal of Parallel and Distributed Computing, vol. 66(1), pp. 32–46, 2006.
• M. Koyuturk and C. Aykanat, “Iterative-Improvement Based Declustering Heuristics for Multi-Disk Databases,” Information Systems, vol. 30, pp. 47–70, 2005.
• B.B. Cambazoglu, A. Turk and C. Aykanat, "Data-Parallel Web-Crawling Models,” Lecture Notes in Computer Science, vol. 3280, pp. 801–809, 2004.
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Related works of our groupHypergraph Models and Methods for Sequential Applications
• E. Demir and C. Aykanat, “A Link-Based Storage Scheme for Efficient Aggregate Query Processing on Clustered Road Networks,” Information Systems, accepted for publication.
• E. Demir, C. Aykanat and B. B. Cambazoglu, “Clustering Spatial Networks for Aggregate Query Processing, a Hypergarph Approach,” Information Systems, vol. 33(1), pp. 1–17, 2008.
• M. Özdal and C. Aykanat, “Hypergraph Models and Algorithms for Data-Pattern Based Clustering," Data Mining and Knowledge Discovery, vol. 9, pp. 29–57, 2004.
Software Package / Tool Development
• C. Aykanat, B.B. Cambazoglu and B. Ucar, “Multi-level Direct K-way Hypergraph Partitioning with Multiple Constraints and Fixed vertices,” Journal of Parallel and Distributed Computing, vol. 68, pp 609–625, 2008.
• B. Ucar and C. Aykanat, “A Library for Parallel Sparse Matrix Vector Multiplies”, Tech. Rept. BU-CE-0506, Dept. of Computer Eng., Bilkent Univ., 2005.
• Ümit V. Çatalyürek and Cevdet Aykanat. PaToH: A multilevel hypergraph- partitioning tool, ver. 3.0.Tech. Rept. BU-CE-9915, Dept. of Computer Eng., Bilkent Univ., 1999.
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Conclusion• Computational hypergraph models & methods for
1D & 2D sparse matrix partitioning for parallel SpMxV– All models encapsulate exact total message volume– 1D models: Good trade-off between comm overhead & partitioning time– 2D Fine-Grain: Lowest total mssg volume– 2D Jagged & Checkerboard: Restrict mssg latency
• Communication hypergraph models– Minimize mssg latency (number of messages)– Balance communication loads of processors
• Composite hypergraph models – Parallelization of preconditioned iterative methods– Partition two or more matrices together for efficient SpMxVs
• Extension and adaptation of these models & methods for the parallelization of other irregular applications
– e.g. Image-space parallelization of direct volume rendering
