1.
CITED BY 9
Space-Time Trade-Off Optimization for a Class of
Electronic Structure Calculations
Daniel Cociorva Gerald Baumgartner Chi-Chung Lam P. Sadayappan
J. Ramanujam Marcel Nooijen David E. Bernholdt Robert Harrison
Dept. of Computer and Information Science Department of Chemistry
The Ohio State University Princeton University
cociorva,gb,clam,saday Nooijen@Princeton.edu
@cis.ohio-state.edu Oak Ridge National Laboratory
bernholdtde@ornl.gov
Dept. of Electrical and Computer Engineering
Louisiana State University Paciﬁc Northwest National Laboratory
jxr@ece.lsu.edu Robert.Harrison@pnl.gov
ABSTRACT : 1. INTRODUCTION
The development of high-performance parallel programs for sci-
The accurate modeling of the electronic structure of atoms and
molecules is very computationally intensive. Many models of elec- entiﬁc applications is usually very time consuming. The time to de-
tronic structure, such as the Coupled Cluster approach, involve col- velop an efﬁcient parallel program for a computational model can
lections of tensor contractions. There are usually a large number be a primary limiting factor in the rate of progress of the science.
of alternative ways of implementing the tensor contractions, rep- Our long term goal is to develop a program synthesis system to fa-
resenting different trade-offs between the space required for tem- cilitate the development of high-performance parallel programs for
porary intermediates and the total number of arithmetic operations. a class of scientiﬁc computations encountered in quantum chem-
In this paper, we present an algorithm that starts with an operation- istry. The domain of our focus is electronic structure calculations,
3.
n the class ofrequired. Consider the following expression: be
operations computations considered, the ﬁnal result to ing to the fused loop to be eliminate
puted can be expressed in terms of tensor contractions, essen- quirement for the computation is to view
a smaller intermediate array and thu
y a collection of multi-dimensional summations of the product loop fusions. Loop fusion merges loop ne
ments. For the example considered
everal input arrays. Due to commutativity, associativity, and loops illustrated in Fig. 1(c). By use loop
into larger imperfectly nested of lo
ributivity, expressionmany different ways to compute the ﬁnalnested produces an intermediate array actually be
If this there are is directly translated to code (with ten can be seen that can which is co
in the number arithmetic operations nest, fusing the two loop nests allows the
lt,loops, for could differ widely total number ofof ﬂoating point
and they indices ), the
a 2-dimensional array, without chan
rations required. be Consider theif the range of each index
following expression: ing to operations.
the fused loop to be eliminated in t
required will is . a smallerFor a computation comprising of a
intermediate array and thus reduc
Instead, the same expression can be rewritten by use of associative ments.will generally be a number of fusio
For the example considered, the
and distributive laws as the following: illustrated incompatible.By useis because
tually Fig. 1(c). This of loop fus
can berequire different loops to bebe reduc
seen that can actually made th
his expression is directly translated to code (with ten nested a 2-dimensional the problem ofchanging t
ps, for indices ), the total number of arithmetic operations addressed array, without ﬁnding the
operations.
operator tree that minimized the tota
uired will be if the range of each index is . For after fusion [14, 16, 15]. of a numb
a computation comprising
ead, the same expression can be rewritten by use of associative will generally be a for manyof fusion choi
However, number of the compu
distributive laws as the the formula sequence shown in Fig. 1(a) and tually compatible. This is because differe
This corresponds to following:
can be directly translated into code as shown in Fig. 1(b). This require different loopscomponent of the NW
coupled cluster
form only requires instances where to be made the outer
operations. However, additional space addressed the problem theﬁnding the choic minimal memo
and . S=0 fusion is still tooof
S = 0
is required to store temporary arrays T1=0; T2=0;Often, the space operator tree that minimized theIn such sit large.
for b, c
for b, c, d, e, f, l executable implementation, it isspac
requirements for the temporary arrays poses a serious problem. For after fusion [14,0; T2f = 0 T1f =
total ess
T1bcdf += Bbefl Dcdel by for d, 16, 15].
f
only storing lower dimensional s
this example, abstracted from a quantum chemistry model, the ar- However, for e, l of the computation
for b, c, d, f, j, k for many
s corresponds to the indices sequence the largest, while the dfjk
ray extents along formula shown in += T1bcdf and
T2bcjk Fig. 1(a) C extents
are for a, b, c, i, j, k recomputing the befl Dcdel
T1f += B slices as needed. Th
into code as shown in Fig. 1(b). of tempo- coupled clusterwe address in this paper. We
be directly translatedare the smallest. Therefore, the sizeThis
along indices problem component of the NWChem
for j, k
Sabij += T2bcjk Aacik instances whereconceptT1f aCfusion graph
T2f the minimal dfjk
m only requires would dominate theHowever, additional space
rary array operations. total memory requirement. proposed jk += k of memory req
quired to store temporary arrays and (b) Direct implementation
. Often, the space fusion isfor a, i, j, In such situation
still too large.
Sabij += T2fjk A
(a) Formula sequence (unfused code)
uirements for the temporary arrays poses a serious problem. For executable implementation, acik essential
it is
(c) Memory-reduced implementation (fused)
example, abstracted from a quantum chemistry model, thefusion for memory reduction. lower dimensional slices o
Figure 1: Example illustrating use of loop ar- by only storing
extents along indices are the largest, while the extents recomputing the slices as needed. This is th
178
g indices are the smallest. Therefore, the size of tempo- problem we address in this paper. We exten
ussian, NWChem, PSI, and MOLPRO. In particular, they com- The operationproposed concept of a fusion here is a and de
minimization problem encountered graph gen-
array
se the bulk of the computation with thetotal memoryapproach
would dominate the coupled cluster requirement. eralization of the well known matrix-chain multiplication problem,
4.
Optimization System
Algebraic Transformations
Memory Minimization
Space-Time Transformation
Data Locality Optimization
Data Distribution and Partitioning
5.
Fusion Graph
can be used to facilitate enumeration of all possible compatible fusion conﬁgurations
for a given computation tree.
The potential for fusion of a common loop among a producer-consumer pair of loop
nests is indicated in the fusion graph through a dashed edge connecting
the corresponding vertices.
ceaf Althoug
E +ceaf number of
theory gro
bk number of
X +ij Y +bk and there
size of the
The fus
T T T1 T2 problem, w
aei j cf i j
the fusion
gorithm w
f1 f2 and ﬁnd th
cebk af bk
and and
the size of
Figure 5: Fusion graph for unfused operation-minimal form of unable to r
loop in Figure 2. arrays wo
6.
Example (1)
for a, e, c, f for a, e, c, f A desirable solution would be somewhere in bet
for i, j for i, j
X += Tijae Tijcf fused structure of Fig. 2 (with maximal memory req
Xaecf += Tijae Tijcf maximal reuse) and the fully fused structure of Fig.
for a, f for b, k
T1 = f (c,e,b,k) imal memory requirement and minimal reuse). Thi
for c, e, b, k
T1cebk = f (c,e,b,k) T2 = f (a,f,b,k) Fig. 4, where tiling and partial fusion of the loops
Y += T1 T2 The loops with indices are tiled by splitting
for c, e
for a, f, b, k E += X Y indices into a pair of indices. The indices with a super
T2afbk = f (a,f,b,k) sent the tiling loops and the unsuperscripted indices
array space time
for c, e, a, f
X 1
intra-tile loops with a range of , the block size used
for b, k each tile , blocks of and of size
T1 1
Yceaf += T1cebk T2afbk puted and used to form product contributions to th
T2 1
for c, e, a, f ceaf
components of , which are stored in an array of siz
Y 1
E += Xaecf Yceaf E +ceaf As the tile size is increased, the cost of function
E 1
for decreases by factor , due to the reuse e
Figure 3: Use of redundant computation to allow full fusion. ever, the size of the neededb ktemporary array for in
X +ij (the space needed for can actually be reduced back
Y +bk
for a , e , c , f fusing its producer loop with the loop producing E,
for a, e, c, f requirement cannot be decreased). When becom
for i, j the size of physical memory, expensive paging in an
Xaecf += Tijae Tijcf T T T
array space time will be required for T1 Further, there are diminishi
. T2
for b, k aei j c freuse of
i j e
for c, e X and after becomes comparable
T1ce = f (c,e,b,k) T1 the loop producing now becomes the dominant on
for a, f T2 expect that as f1 increased, performance will imp
is f2
T2af = f (a,f,b,k) Y level off and then deteriorate. The optimum value of
cebk af bk
for c, e, a, f E 1
(a) Fully fused computation from Fig. 3. at the various levels o
depend on the cost of access
Yceaf += T1ce T2af hierarchy.
for c, e, a, f The computation consideredgraphs just one com
E += Xaecf Yceaf Figure 6: Fusion here is showing re
7.
T2 1
for c, e, a, f components of , which are stored in
Y 1
E += Xaecf Yceaf
Example (2)
E 1 As the tile size is increased, the
for decreases by factor , du
Figure 3: Use of redundant computation to allow full fusion. ever, the size of the needed temporar
(the space needed for can actually
for a , e , c , f fusing its producer loop with the loo
for a, e, c, f requirement cannot be decreased). W
for i, j the size of physical memory, expens
Xaecf += Tijae Tijcf
array space time will be required for . Further, the
for b, k
for c, e X reuse of and after becom
T1ce = f (c,e,b,k) T1 the loop producing now becomes
for a, f T2 expect that as is increased, perfor
T2af = f (a,f,b,k) Y level off and then deteriorate. The op
for c, e, a, f E 1
depend on the cost of access at the
Yceaf += T1ce T2af hierarchy.
for c, e, a, f ct et at f t c e a f
E +ceaf The computation considered here
E += Xaecf Yceaf
term, which in turn is only one
bk be computed. Although developers
Figure 4: Use of tiling and partial fusion to reduce recomputa- bk
naturally recognize and+bk
Y perform some
tion cost. Y +bk X +ij
lective analysis of all these computati
implementation is beyond the scope o
developments in optimizing compiler
reuse of the stored T2
T1 integrals T
in and (each element of
T and T1 T2
et at a e i j ct f t f i j
niﬁcant strides in data locality optimi
is used times). However, it is impractical cdue to the
existing work that addresses the kind
huge memory requirement. With and , the size
mization required in the context we c
f1 of , is f2 bytes and the size of , is bytes. f1 f2
k By fusing together pairs of producer-consumer loops in the compu- t et c e b k
af bk c at f t a f b k
tation, reductions in the needed array sizes may Partially fused computation from Fig. 4.
n from Fig. 3. (b) be sought, since the
fusion of a loop with common index in the pair of loops allows the
4. SOLUTION APPROA
elimination of that dimension of the intermediate array. It can be
ure 6: Fusion graphs showing redundant compution and tiling. GRAPH
8.
so be made fu-
lem discussed in the previous sub-section, requiring that selective
ng fusion edges
Space-Time Tradeoff Exploration
search strategies be developed.
ully fused with
In this paper, we develop a two-step search strategy for explo-
oducer loop for
ration of the space-time trade-off:
n edge (say for
hains for and Search among all possible ways of introducing redundant
loop indices in the fusion graph to reduce memory require-
sented graphi- ments, and determine the optimal set of lower dimensional
ve been added intermediate arrays for various total memory limits. In this
es correspond- step, the use of tiling for partial reduction of array extents is
omplete fusion not considered. However, among all possible combinations
n the scopes of of lower dimensional arrays for intermediates, the combina-
that in fact the tion that minimizes recomputation cost is determined, for a
of or to speciﬁed memory limit. The range from zero to the actual
the additional memory limit is split into subranges within which the op-
partial-overlap timal combination of lower dimensional arrays remains the
same.
hm [16, 14] to
s the total stor- Because the ﬁrst step only considers complete fusion of loops,
s. A bottom-up each array dimension is either fully eliminated or left intact,
intains a set of i.e. partial reduction of array extents is not performed. The
merging solu- objective of the second step is to allow for such arrays. Start-
onﬁgurations at ing from each of the optimal combinations of lower dimen-
y required un- sional intermediate arrays derived in the ﬁrst step, possible
s imposed by a ways of using tiling to partially expand arrays along previ-
uration is infe- ously compressed dimensions are explored. The goal is to
g” with respect further reduce recomputation cost by partially expanding ar-
memory. At the rays to fully utilize the available memory
ry requirement
9.
Space-Time optimization
Dimension Reduction for Intermediate Arrays
search among all possible combination
memory and recomputation costs
Partial Expansion of Reduced Intermediates
resort to array expansion
for determining the best choice for array
expansion costs
10.
Result
15 over
10
clus
1 the d
Memory limit
L
sion
Memory usage (words)
10
10
2 and
anis
3 repe
fusi
10
5
4
at m
5 tion
and
refe
0 6
10 0 5 10 15 20 25 T
10 10 10 10 10 10
Recomputation cost (floating point operations) ory
alig
by f
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