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CS 267 Applications of Parallel

ComputersLecture 19Graph Partitioning Part

II

- Kathy Yelick
- http//www-inst.eecs.berkeley.edu/cs267

Recap of Last Lecture

- Partitioning with nodal coordinates
- Inertial method
- Projection onto a sphere
- Algorithms are efficient
- Rely on graphs having nodes connected (mostly) to

nearest neighbors in space - Partitioning without nodal coordinates
- Breadth-First Search simple, but not great

partition - Kernighan-Lin good corrector given reasonable

partition - Spectral Method good partitions, but slow
- Today
- Spectral methods revisited
- Multilevel methods

Basic Definitions

- Definition The Laplacian matrix L(G) of a graph

G(N,E) is an N by N symmetric matrix, with

one row and column for each node. It is defined

by - L(G) (i,i) degree of node I (number of incident

edges) - L(G) (i,j) -1 if i ! j and there is an edge

(i,j) - L(G) (i,j) 0 otherwise

2 -1 -1 0 0 -1 2 -1 0 0 -1 -1 4

-1 -1 0 0 -1 2 -1 0 0 -1 -1 2

1

4

G

L(G)

5

2

3

Properties of Laplacian Matrix

- Theorem 1 Given G, L(G) has the following

properties (proof on web page) - L(G) is symmetric.
- This means the eigenvalues of L(G) are real and

its eigenvectors are real and orthogonal. - Rows of L sum to zero
- Let e 1,,1T, i.e. the column vector of all

ones. Then L(G)e0. - The eigenvalues of L(G) are nonnegative
- 0 l1 lt l2 lt lt ln
- The number of connected components of G is equal

to the number of li equal to 0. - Definition l2(L(G)) is the algebraic

connectivity of G - The magnitude of l2 measures connectivity
- In particular, l2 ! 0 if and only if G is

connected.

Spectral Bisection Algorithm

- Spectral Bisection Algorithm
- Compute eigenvector v2 corresponding to l2(L(G))
- For each node n of G
- if v2(n) lt 0 put node n in partition N-
- else put node n in partition N
- Why does this make sense?
- Recall l2(L(G)) is the algebraic connectivity of

G - Theorem (Fiedler) Let G1(N,E1) be a subgraph of

G(N,E), so that G1 is less connected than G.

Then l2(L(G)) lt l2(L(G)) , i.e. the algebraic

connectivity of G1 is less than or equal to the

algebraic connectivity of G. (proof on web page)

Motivation for Spectral Bisection

- Vibrating string has modes of vibration, or

harmonics - Modes computable as follows
- Model string as masses connected by springs (a 1D

mesh) - Write down Fma for coupled system, get matrix A
- Eigenvalues and eigenvectors of A are frequencies

and shapes of modes - Label nodes by whether mode - or to get N- and

N - Same idea for other graphs (eg planar graph

trampoline)

Eigenvectors of L(1D mesh)

Eigenvector 1 (all ones)

Eigenvector 2

Eigenvector 3

2nd eigenvector of L(planar mesh)

Computing v2 and l2 of L(G) using Lanczos

- Given any n-by-n symmetric matrix A (such as

L(G)) Lanczos computes a k-by-k approximation

T by doing k matrix-vector products, k ltlt n - Approximate As eigenvalues/vectors using Ts

Choose an arbitrary starting vector r b(0)

r j0 repeat jj1 q(j) r/b(j-1)

scale a vector r Aq(j)

matrix vector multiplication,

the most expensive step r r -

b(j-1)v(j-1) saxpy, or scalarvector

vector a(j) v(j)T r dot

product r r - a(j)v(j)

saxpy b(j) r

compute vector norm until convergence

details omitted

T a(1) b(1) b(1) a(2) b(2)

b(2) a(3) b(3)

b(k-2) a(k-1) b(k-1)

b(k-1) a(k)

Spectral Bisection Summary

- Laplacian matrix represents graph connectivity
- Second eigenvector gives a graph bisection
- Roughly equal weights in two parts
- Weak connection in the graph will be separator
- Implementation via the Lanczos Algorithm
- To optimize sparse-matrix-vector multiply, we

graph partition - To graph partition, we find an eigenvector of a

matrix associated with the graph - To find an eigenvector, we do sparse-matrix

vector multiply - Have we made progress?
- The first matrix-vector multiplies are slow, but

use them to learn how to make the rest faster

Introduction to Multilevel Partitioning

- If we want to partition G(N,E), but it is too big

to do efficiently, what can we do? - 1) Replace G(N,E) by a coarse approximation

Gc(Nc,Ec), and partition Gc instead - 2) Use partition of Gc to get a rough

partitioning of G, and then iteratively improve

it - What if Gc still too big?
- Apply same idea recursively

Multilevel Partitioning - High Level Algorithm

(N,N- ) Multilevel_Partition( N, E )

recursive partitioning routine

returns N and N- where N N U N-

if N is small (1) Partition G

(N,E) directly to get N N U N-

Return (N, N- ) else (2)

Coarsen G to get an approximation Gc

(Nc, Ec) (3) (Nc , Nc- )

Multilevel_Partition( Nc, Ec ) (4)

Expand (Nc , Nc- ) to a partition (N , N- ) of

N (5) Improve the partition ( N ,

N- ) Return ( N , N- )

endif

(5)

V - cycle

(2,3)

(4)

How do we Coarsen? Expand? Improve?

(5)

(2,3)

(4)

(5)

(2,3)

(4)

(1)

Multilevel Kernighan-Lin

- Coarsen graph and expand partition using maximal

matchings - Improve partition using Kernighan-Lin

Maximal Matching

- Definition A matching of a graph G(N,E) is a

subset Em of E such that no two edges in Em share

an endpoint - Definition A maximal matching of a graph G(N,E)

is a matching Em to which no more edges can be

added and remain a matching - A simple greedy algorithm computes a maximal

matching

let Em be empty mark all nodes in N as

unmatched for i 1 to N visit the nodes

in any order if i has not been matched

mark i as matched if there is

an edge e(i,j) where j is also unmatched,

add e to Em mark j

as matched endif endif endfor

Maximal Matching Example

Coarsening using a maximal matching

1) Construct a maximal matching Em of G(N,E) for

all edges e(j,k) in Em 2) collapse

matches nodes into a single one Put node

n(e) in Nc W(n(e)) W(j) W(k) gray

statements update node/edge weights for all nodes

n in N not incident on an edge in Em 3) add

unmatched nodes Put n in Nc do not

change W(n) Now each node r in N is inside a

unique node n(r) in Nc 4) Connect two nodes in

Nc if nodes inside them are connected in E for

all edges e(j,k) in Em for each other

edge e(j,r) in E incident on j Put

edge ee (n(e),n(r)) in Ec W(ee)

W(e) for each other edge e(r,k) in E

incident on k Put edge ee

(n(r),n(e)) in Ec W(ee) W(e) If

there are multiple edges connecting two nodes in

Nc, collapse them, adding edge weights

Example of Coarsening

Expanding a partition of Gc to a partition of G

Multilevel Spectral Bisection

- Coarsen graph and expand partition using maximal

independent sets - Improve partition using Rayleigh Quotient

Iteration

Maximal Independent Sets

- Definition An independent set of a graph G(N,E)

is a subset Ni of N such that no two nodes in Ni

are connected by an edge - Definition A maximal independent set of a graph

G(N,E) is an independent set Ni to which no more

nodes can be added and remain an independent set - A simple greedy algorithm computes a maximal

independent set

let Ni be empty for k 1 to N visit the

nodes in any order if node k is not

adjacent to any node already in Ni add

k to Ni endif endfor

Coarsening using Maximal Independent Sets

Build domains D(k) around each node k in Ni

to get nodes in Nc Add an edge to Ec whenever

it would connect two such domains Ec empty

set for all nodes k in Ni D(k) ( k,

empty set ) first set contains nodes

in D(k), second set contains edges in D(k) unmark

all edges in E repeat choose an unmarked

edge e (k,j) from E if exactly one of k

and j (say k) is in some D(m) mark e

add j and e to D(m) else if k and j

are in two different D(m)s (say D(mi) and

D(mj)) mark e add edge (mk,

mj) to Ec else if both k and j are in the

same D(m) mark e add e to

D(m) else leave e unmarked

endif until no unmarked edges

Example of Coarsening

- encloses domain Dk node of Nc

Expanding a partition of Gc to a partition of G

- Need to convert an eigenvector vc of L(Gc) to an

approximate eigenvector v of L(G) - Use interpolation

For each node j in N if j is also a node in

Nc, then v(j) vc(j) use same

eigenvector component else v(j)

average of vc(k) for all neighbors k of j in

Nc end if endif

Example 1D mesh of 9 nodes

Improve eigenvector Rayleigh Quotient Iteration

j 0 pick starting vector v(0) from

expanding vc repeat jj1 r(j)

vT(j-1) L(G) v(j-1) r(j)

Rayleigh Quotient of v(j-1)

good approximate eigenvalue v(j) (L(G) -

r(j)I)-1 v(j-1) expensive to do

exactly, so solve approximately using an

iteration called SYMMLQ, which uses

matrix-vector multiply (no surprise) v(j)

v(j) / v(j) normalize v(j) until

v(j) converges Convergence is very fast cubic

Example of convergence for 1D mesh

Available Implementations

- Multilevel Kernighan/Lin
- METIS (www.cs.umn.edu/metis)
- ParMETIS - parallel version
- Multilevel Spectral Bisection
- S. Barnard and H. Simon, A fast multilevel

implementation of recursive spectral bisection

, Proc. 6th SIAM Conf. On Parallel Processing,

1993 - Chaco (www.cs.sandia.gov/CRF/papers_chaco.html)
- Hybrids possible
- Ex Using Kernighan/Lin to improve a partition

from spectral bisection

Comparison of methods

- Compare only methods that use edges, not nodal

coordinates - CS267 webpage and KK95a (see below) have other

comparisons - Metrics
- Speed of partitioning
- Number of edge cuts
- Other application dependent metrics
- Summary
- No one method best
- Multi-level Kernighan/Lin fastest by far,

comparable to Spectral in the number of edge cuts - www-users.cs.umn.edu/karypis/metis/publications/m

ail.html - see publications KK95a and KK95b
- Spectral give much better cuts for some

applications - Ex image segmentation
- www.cs.berkeley.edu/jshi/Grouping/overview.html
- see Normalized Cuts and Image Segmentation

Number of edges cut for a 64-way partition

For Multilevel Kernighan/Lin, as implemented in

METIS (see KK95a)

Expected cuts for 2D mesh 6427 2111

1190 11320 3326 4620 1746

8736 2252 4674 7579

Expected cuts for 3D mesh 31805 7208

3357 67647 13215 20481 5595

47887 7856 20796 39623

of Nodes 144649 15606 4960

448695 38744 74752 10672 267241

17758 76480 201142

of Edges 1074393 45878

9462 3314611 993481 261120 209093 334931

54196 152002 1479989

Edges cut for 64-way partition

88806 2965 675

194436 55753 11388 58784

1388 17894 4365

117997

Graph 144 4ELT ADD32 AUTO BBMAT FINAN512 LHR10 MA

P1 MEMPLUS SHYY161 TORSO

Description 3D FE Mesh 2D FE Mesh 32 bit

adder 3D FE Mesh 2D Stiffness M. Lin. Prog. Chem.

Eng. Highway Net. Memory circuit Navier-Stokes 3D

FE Mesh

Expected cuts for 64-way partition of 2D mesh

of n nodes n1/2 2(n/2)1/2 4(n/4)1/2

32(n/32)1/2 17 n1/2 Expected cuts

for 64-way partition of 3D mesh of n nodes

n2/3 2(n/2)2/3 4(n/4)2/3

32(n/32)2/3 11.5 n2/3

Speed of 256-way partitioning (from KK95a)

Partitioning time in seconds

of Nodes 144649 15606 4960

448695 38744 74752 10672 267241

17758 76480 201142

of Edges 1074393 45878

9462 3314611 993481 261120 209093 334931

54196 152002 1479989

Multilevel Spectral Bisection 607.3

25.0 18.7 2214.2

474.2 311.0 142.6 850.2

117.9 130.0 1053.4

Multilevel Kernighan/ Lin 48.1

3.1 1.6 179.2 25.5

18.0 8.1 44.8 4.3

10.1 63.9

Graph 144 4ELT ADD32 AUTO BBMAT FINAN512 LHR10 MA

P1 MEMPLUS SHYY161 TORSO

Description 3D FE Mesh 2D FE Mesh 32 bit

adder 3D FE Mesh 2D Stiffness M. Lin. Prog. Chem.

Eng. Highway Net. Memory circuit Navier-Stokes 3D

FE Mesh

Kernighan/Lin much faster than Spectral Bisection!

Coordinate-Free Partitioning Summary

- Several techniques for partitioning without

coordinates - Breadth-First Search simple, but not great

partition - Kernighan-Lin good corrector given reasonable

partition - Spectral Method good partitions, but slow
- Multilevel methods
- Used to speed up problems that are too large/slow
- Coarsen, partition, expand, improve
- Can be used with K-L and Spectral methods and

others - Speed/quality
- For load balancing of grids, multi-level K-L

probably best - For other partitioning problems (vision,

clustering, etc.) spectral may be better - Good software available

Is Graph Partitioning a Solved Problem?

- Myths of partitioning due to Bruce Hendrickson
- Edge cut communication cost
- Simple graphs are sufficient
- Edge cut is the right metric
- Existing tools solve the problem
- Key is finding the right partition
- Graph partitioning is a solved problem
- Slides and myths based on Bruce Hendricksons
- Load Balancing Myths, Fictions Legends

Myth 1 Edge Cut Communication Cost

- Myth1 The edge-cut deceit
- edge-cut communication cost
- Not quite true
- vertices on boundary is actual communication

volume - Do not communicate same node value twice
- Cost of communication depends on of messages

too (a term) - Congestion may also affect communication cost
- Why is this OK for most applications?
- Mesh-based problems match the model cost is

edge cuts - Other problems (data mining, etc.) do not

Myth 2 Simple Graphs are Sufficient

- Graphs often used to encode data dependencies
- Do X before doing Y
- Graph partitioning determines data partitioning
- Assumes graph nodes can be evaluated in parallel
- Communication on edges can also be done in

parallel - Only dependence is between sweeps over the graph
- More general graph models include
- Hypergraph nodes are computation, edges are

communication, but connected to a set (gt 2) of

nodes - Bipartite model use bipartite graph for directed

graph - Multi-object, Multi-Constraint model use when

single structure may involve multiple

computations with differing costs

Myth 3 Partition Quality is Paramount

- When structure are changing dynamically during a

simulation, need to partition dynamically - Speed may be more important than quality
- Partitioner must run fast in parallel
- Partition should be incremental
- Change minimally relative to prior one
- Must not use too much memory
- Example from Touheed, Selwood, Jimack and Bersins
- 1 M elements with adaptive refinement on SGI

Origin - Timing data for different partitioning

algorithms - Repartition time from 3.0 to 15.2 secs
- Migration time 17.8 to 37.8 secs
- Solve time 2.54 to 3.11 secs

Load Balancing in General

- In some communities, load balancing is equated

with graph partitioning - Some load balancing problems do not fit this

model - Made several assumptions about the problem
- Task costs (node weights) are known
- Communication volumes (edge weights) are known
- Dependencies are known
- For basic partitioning techniques covered in

class, the dependencies were only between

iterations - What if we have less information?

Load Balancing in General

- Spectrum of solutions
- Static - all information available before

starting - Semi-Static - some info before starting
- Dynamic - little or no info before starting
- Survey of solutions
- How each one works
- Theoretical bounds, if any
- When to use it
- Enormous and diverse literature on load balancing
- Computer Science systems
- operating systems, compilers, distributed

computing - Computer Science theory
- Operations research (IEOR)
- Application domains

Understanding Load Balancing Problems

- Load balancing problems differ in
- Tasks costs
- Do all tasks have equal costs?
- If not, when are the costs known?
- Before starting, when task created, or only when

task ends - Task dependencies
- Can all tasks be run in any order (including

parallel)? - If not, when are the dependencies known?
- Before starting, when task created, or only when

task ends - Locality
- Is it important for some tasks to be scheduled on

the same processor (or nearby) to reduce

communication cost? - When is the information about communication

between tasks known?

Task Cost Spectrum

Task Dependency Spectrum

Task Locality Spectrum (Data Dependencies)

Spectrum of Solutions

- One of the key questions is when certain

information about the load balancing problem is

known - Leads to a spectrum of solutions
- Static scheduling. All information is available

to scheduling algorithm, which runs before any

real computation starts. (offline algorithms) - Semi-static scheduling. Information may be known

at program startup, or the beginning of each

timestep, or at other well-defined points.

Offline algorithms may be used even though the

problem is dynamic. - Dynamic scheduling. Information is not known

until mid-execution. (online algorithms)

Approaches

- Static load balancing
- Semi-static load balancing
- Self-scheduling
- Distributed task queues
- Diffusion-based load balancing
- DAG scheduling
- Mixed Parallelism
- Note these are not all-inclusive, but represent

some of the problems for which good solutions

exist.

Static Load Balancing

- Static load balancing is use when all information

is available in advance - Common cases
- dense matrix algorithms, such as LU factorization

- done using blocked/cyclic layout
- blocked for locality, cyclic for load balance
- most computations on a regular mesh, e.g., FFT
- done using cyclictransposeblocked layout for 1D
- similar for higher dimensions, i.e., with

transpose - sparse-matrix-vector multiplication
- use graph partitioning
- assumes graph does not change over time (or at

least within a timestep during iterative solve)

Semi-Static Load Balance

- If domain changes slowly over time and locality

is important - use static algorithm
- do some computation (usually one or more

timesteps) allowing some load imbalance on later

steps - recompute a new load balance using static

algorithm - Often used in
- particle simulations, particle-in-cell (PIC)

methods - poor locality may be more of a problem than load

imbalance as particles move from one grid

partition to another - tree-structured computations (Barnes Hut, etc.)
- grid computations with dynamically changing grid,

which changes slowly

Self-Scheduling

- Self scheduling
- Keep a centralized pool of tasks that are

available to run - When a processor completes its current task, look

at the pool - If the computation of one task generates more,

add them to the pool - Originally used for
- Scheduling loops by compiler (really the

runtime-system) - Original paper by Tang and Yew, ICPP 1986

When is Self-Scheduling a Good Idea?

- Useful when
- A batch (or set) of tasks without dependencies
- can also be used with dependencies, but most

analysis has only been done for task sets without

dependencies - The cost of each task is unknown
- Locality is not important
- Using a shared memory multiprocessor, so a

centralized pool of tasks is fine

Variations on Self-Scheduling

- Typically, dont want to grab smallest unit of

parallel work. - Instead, choose a chunk of tasks of size K.
- If K is large, access overhead for task queue is

small - If K is small, we are likely to have even finish

times (load balance) - Four variations
- Use a fixed chunk size
- Guided self-scheduling
- Tapering
- Weighted Factoring
- Note there are more

Variation 1 Fixed Chunk Size

- Kruskal and Weiss give a technique for computing

the optimal chunk size - Requires a lot of information about the problem

characteristics - e.g., task costs, number
- Results in an off-line algorithm. Not very

useful in practice. - For use in a compiler, for example, the compiler

would have to estimate the cost of each task - All tasks must be known in advance

Variation 2 Guided Self-Scheduling

- Idea use larger chunks at the beginning to avoid

excessive overhead and smaller chunks near the

end to even out the finish times. - The chunk size Ki at the ith access to the task

pool is given by - ceiling(Ri/p)
- where Ri is the total number of tasks remaining

and - p is the number of processors
- See Polychronopolous, Guided Self-Scheduling A

Practical Scheduling Scheme for Parallel

Supercomputers, IEEE Transactions on Computers,

Dec. 1987.

Variation 3 Tapering

- Idea the chunk size, Ki is a function of not

only the remaining work, but also the task cost

variance - variance is estimated using history information
- high variance gt small chunk size should be used
- low variant gt larger chunks OK

- See S. Lucco, Adaptive Parallel Programs, PhD

Thesis, UCB, CSD-95-864, 1994. - Gives analysis (based on workload distribution)
- Also gives experimental results -- tapering

always works at least as well as GSS, although

difference is often small

Variation 4 Weighted Factoring

- Idea similar to self-scheduling, but divide task

cost by computational power of requesting node - Useful for heterogeneous systems
- Also useful for shared resource NOWs, e.g., built

using all the machines in a building - as with Tapering, historical information is used

to predict future speed - speed may depend on the other loads currently

on a given processor - See Hummel, Schmit, Uma, and Wein, SPAA 96
- includes experimental data and analysis

Distributed Task Queues

- The obvious extension of self-scheduling to

distributed memory is - a distributed task queue (or bag)
- When are these a good idea?
- Distributed memory multiprocessors
- Or, shared memory with significant

synchronization overhead - Locality is not (very) important
- Tasks that are
- known in advance, e.g., a bag of independent ones
- dependencies exist, i.e., being computed on the

fly - The costs of tasks is not known in advance

Theoretical Results

- Main result A simple randomized algorithm is

optimal with high probability - Adler et al 95 show this for independent, equal

sized tasks - throw balls into random bins
- tight bounds on load imbalance show p log p

tasks leads to good balance - Karp and Zhang 88 show this for a tree of unit

cost (equal size) tasks - parent must be done before children, tree unfolds

at runtime - children pushed to random processors
- Blumofe and Leiserson 94 show this for a fixed

task tree of variable cost tasks - their algorithm uses task pulling (stealing)

instead of pushing, which is good for locality - I.e., when a processor becomes idle, it steals

from a random processor - also have (loose) bounds on the total memory

required - Chakrabarti et al 94 show this for a dynamic

tree of variable cost tasks - works for branch and bound, I.e. tree structure

can depend on execution order - uses randomized pushing of tasks instead of

pulling, so worse locality - Open problem does task pulling provably work

well for dynamic trees?

Engineering Distributed Task Queues

- A lot of papers on engineering these systems on

various machines, and their applications - If nothing is known about task costs when created
- organize local tasks as a stack (push/pop from

top) - steal from the stack bottom (as if it were a

queue), because old tasks likely to cost more - If something is known about tasks costs and

communication costs, can be used as hints. (See

Wen, UCB PhD, 1996.) - Part of Multipol (www.cs.berkeley.edu/projects/mul

tipol) - Try to push tasks with high ratio of cost to

compute/cost to push - Ex for matmul, ratio 2n3 cost(flop) / 2n2

cost(send a word) - Goldstein, Rogers, Grunwald, and others

(independent work) have all shown - advantages of integrating into the language

framework - very lightweight thread creation
- CILK (Leicerson et al) (supertech.lcs.mit.edu/cil

k)

Diffusion-Based Load Balancing

- In the randomized schemes, the machine is treated

as fully-connected. - Diffusion-based load balancing takes topology

into account - Locality properties better than prior work
- Load balancing somewhat slower than randomized
- Cost of tasks must be known at creation time
- No dependencies between tasks

Diffusion-based load balancing

- The machine is modeled as a graph
- At each step, we compute the weight of task

remaining on each processor - This is simply the number if they are unit cost

tasks - Each processor compares its weight with its

neighbors and performs some averaging - Markov chain analysis
- See Ghosh et al, SPAA96 for a second order

diffusive load balancing algorithm - takes into account amount of work sent last time
- avoids some oscillation of first order schemes
- Note locality is still not a major concern,

although balancing with neighbors may be better

than random

DAG Scheduling

- For some problems, you have a directed acyclic

graph (DAG) of tasks - nodes represent computation (may be weighted)
- edges represent orderings and usually

communication (may also be weighted) - not that common to have the DAG in advance
- Two application domains where DAGs are known
- Digital Signal Processing computations
- Sparse direct solvers (mainly Cholesky, since it

doesnt require pivoting). More on this in

another lecture. - The basic offline strategy partition DAG to

minimize communication and keep all processors

busy - NP complete, so need approximations
- Different than graph partitioning, which was for

tasks with communication but no dependencies - See Gerasoulis and Yang, IEEE Transaction on

PDS, Jun 93.

Mixed Parallelism

- As another variation, consider a problem with 2

levels of parallelism - course-grained task parallelism
- good when many tasks, bad if few
- fine-grained data parallelism
- good when much parallelism within a task, bad if

little - Appears in
- Adaptive mesh refinement
- Discrete event simulation, e.g., circuit

simulation - Database query processing
- Sparse matrix direct solvers

Mixed Parallelism Strategies

Which Strategy to Use

Switch Parallelism A Special Case

Simple Performance Model for Data Parallelism

(No Transcript)

Values of Sigma (Problem Size for Half Peak)

Modeling Performance

- To predict performance, make assumptions about

task tree - complete tree with branching factor dgt 2
- d child tasks of parent of size N are all of

size N/c, cgt1 - work to do task of size N is O(Na), agt 1
- Example Sign function based eigenvalue routine
- d2, c4 (on average), a1.5
- Example Sparse Cholesky on 2D mesh
- d4, c4, a1.5
- Combine these assumptions with model of data

parallelism

Simulated Efficiency of Eigensolver

- Starred lines are optimal mixed parallelism
- Solid lines are data parallelism
- Dashed lines are switched parallelism

Simulated efficiency of Sparse Cholesky

- Starred lines are optimal mixed parallelism
- Solid lines are data parallelism
- Dashed lines are switched parallelism

Actual Speed of Sign Function Eigensolver

- Starred lines are optimal mixed parallelism
- Solid lines are data parallelism
- Dashed lines are switched parallelism
- Intel Paragon, built on ScaLAPACK
- Switched parallelism worthwhile!