1

High-Performance Computation for Path Problems in Graphs

Aydin BuluçJohn R. GilbertUniversity of California, Santa Barbara

SIAM Conf. on Applications of Dynamical SystemsMay 20, 2009

Support: DOE Office of Science, MIT Lincoln Labs, NSF, DARPA, SGI

2

Horizontal-vertical decomposition [Mezic et al.]

Slide courtesy of Igor Mezic group, UCSB

3

Combinatorial Scientific Computing

“I observed that most of the coefficients in our matrices were zero; i.e., the nonzeros were ‘sparse’ in the matrix, and that typically the triangular matrices associated with the forward and back solution provided by Gaussian elimination would remain sparse if pivot elements were chosen with care”

- Harry Markowitz, describing the 1950s work on portfolio theory that won the 1990 Nobel Prize for Economics

4

A few directions in CSC

• Hybrid discrete & continuous computations• Multiscale combinatorial computation• Analysis, management, and propagation of uncertainty• Economic & game-theoretic considerations• Computational biology & bioinformatics• Computational ecology• Knowledge discovery & machine learning• Relationship analysis • Web search and information retrieval• Sparse matrix methods• Geometric modeling• . . .

5

The Parallel Computing Challenge

LANL / IBM Roadrunner> 1 PFLOPS

Two Nvidia 8800 GPUs> 1 TFLOPS

Intel 80-core chip> 1 TFLOPS Parallelism is no longer optional…

… in every part of a computation.

6

Efficient sequential algorithms for graph-theoretic

problems often follow long chains of dependencies

Several parallelization strategies, but no silver bullet: Partitioning (e.g. for preconditioning PDE solvers) Pointer-jumping (e.g. for connected components) Sometimes it just depends on what the input looks like

A few simple examples . . .

The Parallel Computing Challenge

7

Sample kernel: Sort logically triangular matrix

• Used in sparse linear solvers (e.g. Matlab’s)

• Simple kernel, abstracts many other graph operations (see next)

• Sequential: linear time, simple greedy topological sort

• Parallel: no known method is efficient in both work and span: one parallel step per level; arbitrarily long dependent chains

Original matrix Permuted to unit upper triangular form

8

Bipartite matching

• Perfect matching: set of edges that hits each vertex exactly once

• Matrix permutation to place nonzeros (or heavy elements) on diagonal

• Efficient sequential algorithms based on augmenting paths

• No known work/span efficient parallel algorithms

1 52 3 41

5

234

A

1

5

2

3

4

1

5

2

3

4

1 52 3 44

2

531

PA

9

Strongly connected components

• Symmetric permutation to block triangular form

• Diagonal blocks are strong Hall (irreducible / strongly connected)

• Sequential: linear time by depth-first search [Tarjan]

• Parallel: divide & conquer, work and span depend on input [Fleischer, Hendrickson, Pinar]

1 52 4 7 3 61

5

247

36

PAPT G(A)

1 2

3

4 7

6

5

10

Horizontal - vertical decomposition

• Defined and studied by Mezic et al. in a dynamical systems context

• Strongly connected components, ordered by levels of DAG

• Efficient linear-time sequential algorithms

• No work/span efficient parallel algorithms known

3 54

9

7

1

8

6

2

level 1

level 2

level 3

level 45 96 7 81 2 3 4

1

5

234

9

678

11

Strong components of 1M-vertex RMAT graph

12

Dulmage-Mendelsohn decomposition

1

5

234

678

12

91011

1 52 3 4 6 7 8 9 10 111

2

5

3

4

7

6

10

8

9

12

11

1

2

3

5

4

7

6

9

8

11

10

HR

SR

VR

HC

SC

VC

13

Applications of D-M decomposition

• Strongly connected components of directed graphs

• Connected components of undirected graphs

• Permutation to block triangular form for Ax=b

• Minimum-size vertex cover of bipartite graphs

• Extracting vertex separators from edge cuts for arbitrary graphs

• Nonzero structure prediction for sparse matrix factorizations

14

Strong Hall components are independent of choice of matching

1 52 4 7 3 61

5

247

36

1 52 4 7 3 64

5

172

63

1

5

2

3

4

1

5

2

3

4

7

6

7

6

1

5

2

3

4

1

5

2

3

4

7

6

7

6

15

• By analogy to numerical linear algebra. . .

• What should the “combinatorial BLAS” look like?

The Primitives Challenge

C = A*B

y = A*x

μ = xT y

Basic Linear Algebra Subroutines (BLAS):Speed (MFlops) vs. Matrix Size (n)

16

Primitives for HPC graph programming

• Visitor-based multithreaded [Berry, Gregor, Hendrickson, Lumsdaine]

+ search templates natural for many algorithms+ relatively simple load balancing– complex thread interactions, race conditions– unclear how applicable to standard architectures

• Array-based data parallel [G, Kepner, Reinhardt, Robinson, Shah]

+ relatively simple control structure+ user-friendly interface– some algorithms hard to express naturally– load balancing not so easy

• Scan-based vectorized [Blelloch]

• We don’t know the right set of primitives yet!

17

Array-based graph algorithms study [Kepner, Fineman, Kahn, Robinson]

18

Multiple-source breadth-first search

X

1 2

3

4 7

6

5

AT

19

XAT ATX

1 2

3

4 7

6

5

Multiple-source breadth-first search

20

• Sparse array representation => space efficient

• Sparse matrix-matrix multiplication => work efficient

• Span & load balance depend on matrix-mult implementation

XAT ATX

1 2

3

4 7

6

5

Multiple-source breadth-first search

21

Matrices over semirings

• Matrix multiplication C = AB (or matrix/vector):

Ci,j = Ai,1B1,j + Ai,2B2,j + · · · + Ai,nBn,j

• Replace scalar operations and + by

: associative, distributes over , identity 1

: associative, commutative, identity 0 annihilates under

• Then Ci,j = Ai,1B1,j Ai,2B2,j · · · Ai,nBn,j

• Examples: (,+) ; (and,or) ; (+,min) ; . . .

• Same data reference pattern and control flow

22

• Shortest path calculations (APSP)

• Betweenness centrality

• BFS from multiple source vertices

• Subgraph / submatrix indexing

• Graph contraction

• Cycle detection

• Multigrid interpolation & restriction

• Colored intersection searching

• Applying constraints in finite element modeling

• Context-free parsing

SpGEMM: Sparse Matrix x Sparse Matrix [Buluc, G]

23

Distributed-memory parallel sparse matrix multiplication

j

* =i

kk

Cij

Cij += Aik * Bkj

2D block layout Outer product formulation Sequential “hypersparse” kernel

• Asynchronous MPI-2 implementation

• Experiments: TACC Lonestar cluster

• Good scaling to 256 processors

Time vs Number of cores -- 1M-vertex RMAT

24

• Directed graph with “costs” on edges• Find least-cost paths between all reachable vertex pairs• Several classical algorithms with

– Work ~ matrix multiplication– Span ~ log2 n

• Case study of implementation on multicore architecture:

– graphics processing unit (GPU)

All-Pairs Shortest Paths

25

GPU characteristics

• Powerful: two Nvidia 8800s > 1 TFLOPS• Inexpensive: $500 each

• Difficult programming model: One instruction stream drives 8 arithmetic units

• Performance is counterintuitive and fragile:Memory access pattern has subtle effects on cost

• Extremely easy to underutilize the device:Doing it wrong easily costs 100x in time

t1

t3t2

t4

t6t5

t7

t9t8

t10

t12t11

t13t14

t16t15

But:

26

Recursive All-Pairs Shortest Paths

A B

C DA

BD

C

A = A*; % recursive callB = AB; C = CA; D = D + CB;D = D*; % recursive callB = BD; C = DC;A = A + BC;

+ is “min”, × is “add”

Based on R-Kleene algorithm

Well suited for GPU architecture:

• Fast matrix-multiply kernel

• In-place computation => low memory bandwidth

• Few, large MatMul calls => low GPU dispatch overhead

• Recursion stack on host CPU,

not on multicore GPU

• Careful tuning of GPU code

27

Execution of Recursive APSP

28

APSP: Experiments and observations

128-core Nvidia 8800 Speedup relative to. . . 1-core CPU: 120x – 480x 16-core CPU: 17x – 45xIterative, 128-core GPU: 40x – 680x MSSSP, 128-core GPU: ~3x

Conclusions:• High performance is achievable but not simple• Carefully chosen and optimized primitives will be key

Time vs. Matrix Dimension

29

H-V decomposition

A span-efficient, but not work-efficient, method for H-V

decomposition uses APSP to determine reachability…

3 54

9

7

1

8

6

2

level 1

level 2

level 3

level 45 96 7 81 2 3 4

1

5

234

9

678

30

Reachability: Transitive closure

3 54

9

7

1

8

6

2

level 1

level 2

level 3

level 45 96 7 81 2 3 4

1

5

234

9

678

• APSP => transitive closure of adjacency matrix

• Strong components identified by symmetric nonzeros

31

H-V structure: Acyclic condensation

1 82 3 6 91

8

2

3

6

9

345

9

67

1

8

2

level 1

level 2

level 3

level 4

• Acyclic condensation is a sparse matrix-matrix product

• Levels identified by “APSP” for longest paths

• Practically speaking, a parallel method would compromise between work and span efficiency

32

Remarks

• Combinatorial algorithms are pervasive in scientific computing and will become more so.

• Path computations on graphs are powerful tools, but efficiency is a challenge on parallel architectures.

• Carefully chosen and implemented primitive operations are key.

• Lots of exciting opportunities for research!