1

Challenges in Combinatorial Scientific Computing

John R. GilbertUniversity of California, Santa Barbara

SIAM Annual MeetingJuly 10, 2009

Support: DOE Office of Science, NSF, DARPA, SGI

2

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

3

IC10: Mathematical Analysis of Hyperlinks in the World-Wide Web

IP3: Discrete Mathematics and Theoretical Computer Science

VNL: The Human Genome and Beyond

IC18: Finding Provably Near-Optimal Solutions to Discrete Optimization Problems

CP6: Computer Science: Discrete Algorithms

MS54: New Approaches for Scalable Sparse Linear System Solution

MS71: Special Session on IR Algorithms and Software

MS73: Reverse Engineering Gene Networks

MS86/110: Combinatorial Algorithms in Scientific Computing

2002 SIAM Annual Meeting

4

IC 7: Combinatorics Inspired by Biology

IC 12: Parallel Network Analysis

IC 14: On the Complexity of Game, Market, and Network Equilibria

MS1/8: Adaptive Algebraic Multigrid Methods

MS 4/11/26 : Mathematical Challenges in Cyber Security

MS 36/43/55 : High Performance Computing on Massive Real-World Graphs

MS 47: Optimization and Graph Algorithms

MS 79: Enumerative and Geometric Combinatorics

MS 71: Modeling of Large-Scale Metabolic Networks

MS 75: Combinatorial Scientific Computing

2007 SIAM Annual Meeting

5

An analogy?

As the “middleware” of scientific computing, linear algebra has supplied or enabled:

• Mathematical tools

• “Impedance match” to computer operations

• High-level primitives

• High-quality software libraries

• Ways to extract performance from computer architecture

• Interactive environments

Computers

Continuousphysical modeling

Linear algebra

6

An analogy?

Computers

Continuousphysical modeling

Linear algebra

Discretestructure analysis

Graph theory

Computers

7

An analogy? Well, we’re not there yet ….

Discretestructure analysis

Graph theory

Computers

• Mathematical tools

• “Impedance match” to computer operations

• High-level primitives

• High-quality software libs

• Ways to extract performance from computer architecture

• Interactive environments

8

Five Challenges in

Combinatorial Scientific Computing

9

• How do we know we’re solving the right problem?

– Harder in graph analytics than in numerical linear algebra

– I’ll pretend this is a math modeling question, not a CSC question

• How will applications keep us honest in the future?

– Tony Chan, 1991: T(A, B) = elapsed time before field A would notice the disappearance of field B

– minA( T(A, CSC) ) ?

Challenges I’m not going to talk about

10

#1: The Architecture & Algorithms 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.

11

High-Performance Architecture

Most high-performance computer designs allocate resources to optimize Gaussian elimination on large, dense matrices.

Originally, because linear algebra is the middleware of scientific computing.

Nowadays, largely for bragging rights.

= xP A L U

12

The Memory Wall Blues

Most of memory is hundreds or thousands of cycles away from the processor that wants it.

You can buy more bandwidth, but you can’t buy less latency. (Speed of light, for one thing.)

13

The Memory Wall Blues

Most of memory is hundreds or thousands of cycles away from the processor that wants it.

You can buy more bandwidth, but you can’t buy less latency. (Speed of light, for one thing.)

You can hide latency with either locality or (parallelism + bandwidth).

You can hide lack of bandwidth with locality, but not with parallelism.

14

The Memory Wall Blues

Most of memory is hundreds or thousands of cycles away from the processor that wants it.

You can buy more bandwidth, but you can’t buy less latency. (Speed of light, for one thing.)

You can hide latency with either locality or (parallelism + bandwidth).

You can hide lack of bandwidth with locality, but not with parallelism.

Most emerging graph problems have lousy locality.

Thus the algorithms need even more parallelism!

15

Architectural impact on algorithms

-1

0

1

2

3

4

5

6

0 1 2 3 4 5

log Problem Size

log

cycl

es/fl

opT = N4.7

Naïve algorithm is O(N5) time under UMH model.BLAS-3 DGEMM and recursive blocked algorithms are O(N3).

Size 2000 took 5 days

12000 would take1095 years

Slide from Larry Carter

Naïve 3-loop matrix multiply [Alpern et al., 1992]:

16

Efficient sequential algorithms for graph-theoretic

problems often follow long chains of dependencies

Computation per edge traversal is often small

Little locality or opportunity for reuse

A big opportunity exists for architecture to influence combinatorial algorithms.

Maybe even vice versa.

The parallel computing challenge

17

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

2

4

7

3

6

PAPT G(A)

1 2

3

4 7

6

5

18

Strong components of 1M-vertex RMAT graph

19

One architectural approach: Cray MTA / XMT

• Hide latency by massive multithreading

• Per-tick context switching

• Uniform (sort of) memory access time

• But the economic case for the machine is less than completely clear.

20

#2: The Data Size Challenge

“Can we understand anything interesting about our data when we do not even have time to read all of it?”

- Ronitt Rubinfeld, 2002

21

Approximating min spanning tree weight in sublinear time [Chazelle, Rubinfeld, Trevisan]

• Key subroutine: estimate number of connected components of a graph, in time depending on

relative error but not on size of graph

• Idea: for each vertex v define

f(v) = 1/(size of component containing v)

• Then Σv f(v) = number of connected components

• Estimate Σv f(v) by breadth-first search from a few vertices

22

Features of (many) large graph applications

• “Feasible” means O(n), or even less.

• You can’t scan all the data.– you want to poke at it in various ways

– maybe interactively.

• Multiple simultaneous queries to the same graph– maybe to differently filtered subgraphs

– throughput and response time are both important.

• Benchmark data sets are a big challenge!

• Think of data not as a finite object but as a statistical sample of an infinite process . . .

• Multiple simultaneous queries to same graph– Graph may be fixed, or slowly changing

– Throughput and response time both important

23

#3: The Uncertainty Challenge

“You could not step twice into the same river; for other waters are ever flowing on to you.”

- Heraclitus

24

• Statistical perspectives; robustness to fluctuations; streaming, noisy data; dynamic structure; analysis of sensitivity and propagation of uncertainty

• K-betweenness centrality for robustness [Bader]

• Stochastic centrality for infection vulnerability analysis [Vullikanti]

• Dynamic community updating [Bhowmick]

• Detection theory [Kepner]: “You should never worry about

implementation; you should worry about whether you can correctly model the background and foreground.”

Change and uncertainty

25

Horizontal - vertical decomposition[Mezic et al.]

• Strongly connected components, ordered by levels of DAG

• Linear-time sequentially; no work/span efficient parallel algorithms known

• … but we don’t want an exact algorithm anyway; edges are probabilities, and the application is a kind of statistical model reduction …

3 54

9

7

1

8

6

2

level 1

level 2

level 3

level 45 96 7 81 2 3 4

1

5

2

3

4

9

6

7

8

26

Approach:

1. Decompose networks

2. Propagate uncertainty through components

3. Iteratively aggregate component uncertainty

Spectral graph decomposition technique combined with dynamical Spectral graph decomposition technique combined with dynamical systems analysis leads to deconstruction of a possibly unknown network systems analysis leads to deconstruction of a possibly unknown network into inputs, outputs, forward and feedback loops and allows identification into inputs, outputs, forward and feedback loops and allows identification of a minimal functional unitof a minimal functional unit (MFU)(MFU) of a systemof a system..

(node 4 and several connections pruned, with no loss of performance)H-V decomposition

Output, execution

Feedback loops

Trim the network,preserve dynamics!

Input, initiator

Forward, production unit

Additional functional requirements

Minimal functional units: sensitive edges (leading to lack of production)

easily identifiable

Allows identification of roles ofdifferent feedback loops

Level of outputFor MFU

Level of output with feedback loops

Mezic group, UCSB

Model reduction and graph decomposition

27

• By analogy to numerical scientific computing. . .

• What should the combinatorial BLAS look like?

#4: The Primitives Challenge

C = A*B

y = A*x

μ = xT y

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

28

Primitives should…

• Supply a common notation to express computations

• Have broad scope but fit into a concise framework

• Allow programming at the appropriate level of

abstraction and granularity

• Scale seamlessly from desktop to supercomputer

• Hide architecture-specific details from users

29

Frameworks for graph primitives

Many possibilities; none completely satisfactory; relatively

little work on common frameworks or interoperability.

• Visitor-based, distributed-memory: PBGL

• Visitor-based, multithreaded: MTGL

• Heterogeneous, tuned kernels: SNAP

• Sparse array-based: Matlab dialects, CBLAS

• Scan-based vectorized: NESL

• Map-reduce: lots of visibility, utility unclear

30

Distributed-memory parallel sparse matrix-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

31

Recursive All-Pairs Shortest Paths

A B

C DA

B

DC

A = A*; % recursive call

B = AB; C = CA;

D = D + CB;

D = D*; % recursive call

B = 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

32

APSP: Experiments and observations

128-core Nvidia 8800 GPU

Speedup relative to. . .

1-core CPU: 120x – 480x

16-core CPU: 17x – 45x

Iterative, 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

33

#5: The Education Challenge

Or, the interdisciplinary challenge.

Where do you go to take courses in

Graph algorithms …

… on massive data sets …

… in the presence of uncertainty …

… analyzed on parallel computers …

… applied to a domain science?

CSC needs to be both coherent and interdisciplinary!

34

Five Challenges In CSC

1. Architecture & algorithms

2. Data size & management

3. Uncertainty

4. Primitives & software

5. Education

35

Morals

• Things are clearer if you look at them from multiple perspectives

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

• This is a great time to be doing combinatorial scientific computing!

36

Thanks …

David Bader, Jon Berry, Nadya Bliss, Aydin Buluc, Larry Carter, Alan Edelman, John Feo,

Bruce Hendrickson, Jeremy Kepner, Jure Leskovic, Kamesh Madduri, Michael Mahoney, Igor Mezic, Cleve Moler, Steve Reinhardt, Eric

Robinson, Ronitt Rubinfeld, Rob Schreiber, Viral Shah, Sivan Toledo, Anil Vullikanti