PPoPP15

Cache-Oblivious Wavefront: Improving Parallelism
of Recursive Dynamic Programming Algorithms
without Losing Cache-Efficiency
Yuan Tang ∗
Ronghui You Haibin Kan
Software School, School of Computer Science, Fudan
University
Shanghai Key Laboratory of Intelligent Information
Processing, Fudan University
Shanghai, P. R. China
[yuantang, 11300720164, hbkan]@fudan.edu.cn
Jesmin Jahan Tithi Pramod Ganapathi
Rezaul A. Chowdhury †
Department of Computer Science, Stony Brook
University
Stony Brook, NY 11790, USA
[jtithi, pganapathi, rezaul]@cs.stonybrook.edu
Abstract
State-of-the-art cache-oblivious parallel algorithms for dynamic
programming (DP) problems usually guarantee asymptotically op-
timal cache performance without any tuning of cache parameters,
but they often fail to exploit the theoretically best parallelism at
the same time. While these algorithms achieve cache-optimality
through the use of a recursive divide-and-conquer (DAC) strat-
egy, scheduling tasks at the granularity of task dependency in-
troduces artificial dependencies in addition to those arising from
the defining recurrence equations. We removed the artificial de-
pendency by scheduling tasks ready for execution as soon as all
its real dependency constraints are satisfied, while preserving the
cache-optimality by inheriting the DAC strategy. We applied our
approach to a set of widely known dynamic programming prob-
lems, such as Floyd-Warshall’s All-Pairs Shortest Paths, Stencil,
and LCS. Theoretical analyses show that our techniques improve
the span of 2-way DAC-based Floyd Warshall’s algorithm on an
n node graph from Θ nlog2 n to Θ(n), stencil computations
on a d-dimensional hypercubic grid of width w for h time steps
from Θ (d2h)wlog(d+2)−1 to Θ(h), and LCS on two sequences
of length n each from Θ nlog2 3 to Θ(n). In each case, the total
work and cache complexity remain asymptotically optimal. Exper-
imental measurements exhibit a 3 - 5 times improvement in ab-
solute running time, 10 - 20 times improvement in burdened span
by Cilkview, and approximately the same L1/L2 cache misses by
PAPI.
Categories and Subject Descriptors D.1.3 [Programming Tech-
niques]: Concurrent Programming—Parallel programming; G.1.0
[Mathematics of Computing]: Numerical Analysis—Parallel Al-
∗ Part of the work was done when the author was a visiting scientist at MIT
CSAIL during the summer of 2014.
† Rezaul Chowdhury and Pramod Ganapathi were supported in part by NSF
grants CCF-1162196 and CCF-1439084.
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PPoPP ’15, February 7–11, 2015, San Francisco Bay Area, CA, USA.
Copyright c 2015 ACM 978-1-4503-3205-7/15/02...$15.00.
http://dx.doi.org/10.1145/nnnnnnn.nnnnnnn
gorithms.; G.4 [Mathematical Software]: Algorithm design and
analysis
General Terms Algorithms, Scheduling, Performance.
Keywords cache-oblivious parallel algorithm, cache-oblivious
wavefront, dynamic programming, multi-core, nested parallel com-
putation, Cilk
1. Introduction
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Dynamic programming (DP) [4] algorithms build optimal solutions
to a problem by combining optimal solutions to many overlapping
subproblems. DP algorithms exploit this overlap to explore other-
wise exponential-sized problem spaces in polynomial time, making
them central to many important applications ranging from logistics
to computational biology [20, 30, 33, 38, 45–47, 50, 56].
State-of-the-art cache-oblivious [27] parallel (COP) algorithms
for DP problems [11–13, 15, 16] often trade off parallelism for
better cache performance. Those algorithms typically employ a re-
cursive divide-and-conquer (DAC) approach. This approach allows
an algorithm to achieve asymptotically optimal serial cache perfor-
mance through increased “temporal locality”1 while at the same
time remain oblivious of the parameters of the cache hierarchy.
In other words, these algorithms do not need to tune for different
memory hierarchies, and thus are portable across machines. How-
ever, scheduling tasks at the granularity of task dependency often
limits parallelism by introducing artificial dependencies among re-
cursive subtasks in addition to those arising from the defining re-
currence equations. As a result, most state-of-the-art COP DP algo-
rithms fail to achieve optimal serial cache performance and optimal
parallelism simultaneously.
Performance of a recursive DAC based COP algorithm when
run under a state-of-the-art scheduler on a modern multi-core ma-
chine, depends on both its serial cache complexity and parallelism.
Before we elaborate on the relationship between serial and parallel
performances of these algorithms, we explain below how we will
analyze their performance throughout this paper.
Nested Parallel Computations and Work-Span Model. We use
the “work-span model” to analyze the performance of nested par-
allel computation [8, 49] that covers all recursive DAC-based DP
algorithms in this paper. By Tp and Qp we denote the running
1 Temporal locality — whenever a cache block is brought into the cache, as much
useful work as possible is performed on it before it’s removed from the cache.

time and the cache complexity of the algorithm, respectively, on p
processing cores. The total “work” performed by the algorithm is
given by its running time T1 on a single core. Its theoretical execu-
tion time on an unbounded number of cores is called its “span” or
“critical path length” or “depth”, and is denoted as T∞. The “par-
allelism” of an algorithm is then given by (T1/T∞), which is the
average amount of work it performs per step along the span. While
one can always increase parallelism by increasing the total work T1,
which is not very interesting and often not useful. In this paper, we
focus on reducing the span (T∞) of a given recursive DP algorithm
while keeping its T1 fixed. Table 1 lists the notations and acronyms.
Table 1: Standard acronym and nota-
tions used throughout the paper.
Symb. Meaning
DP Dynamic Programming
COP Cache-Oblivious Parallel, used to de-
note original recursive divide-and-
conquer based standard algorithm
COW Cache-Oblivious Wavefront, used to
denote new algorithms proposed in this
paper
DAC divide-and-conquer
FW Floyd-Warshall
APSP All-Pairs-Shortest-Paths
n Input size or input parameter
p Number of processing cores
M Cache or memory size
B Block size or cache line size or I/O
transfer size
T1 Work or serial running time
T∞ Span or critical path length or running
time on an infinite number of cores
Tp Parallel running time on p cores
T1
T∞
Parallelism
Q1 Serial cache complexity
Qp Parallel cache complexity on p cores
Influences of Serial
Cache Complexity and
Parallelism on Paral-
lel Performance. State-
of-the-art schedulers for
shared-memory multi-
core machines guaran-
tee good parallel perfor-
mance provided the al-
gorithm being run shows
both good cache perfor-
mance on a serial ma-
chine and high paral-
lelism (i.e., low span).
For example, the widely
used “randomized work-
stealing scheduler” [1, 9,
26] for distributed caches
provides the following
performance guarantees w.h.p.2:
Tp = O (T1/p+T∞) and Qp = Q1 +O (p(M/B)T∞),
where, p is the number of processing cores, M is the cache size,
and B is the cache line size. Clearly, good parallel cache perfor-
mance (Qp) requires both good serial cache performance (Q1) and
a low span (T∞). The “parallel depth-first scheduler” [7] for shared
caches, on the other hand, guarantees that
Qp ≤ Q1 provided Mp ≥ M1 +Θ(pT∞),
where, Mp is the size of cache shared by p cores, and M1 is the size
of cache on a serial machine. The more recently proposed “space-
bounded schedulers” [8, 12, 14, 17, 49] guarantee:
Tp = O (T1/p+T∞) and Qp = O (Q1),
which is achieved by trading off parallelism for reduced cache
misses. Again, the lower the values of Q1 and T∞, the better the
parallel performance.
With the increase of core count on multi-core machines, and the
emergence of many-core processors with cache hierarchies such as
the “Intel MIC” [35], the need for algorithms with both optimal
cache complexity and high parallelism continues to grow.
The Tradeoff between Q1 and T∞ in Recursive DAC-based DPs.
We explain the tradeoff using the longest common subsequence
(LCS) problem as an example.
Given two sequences S = s1,s2,...,sm and T = t1,t2,...,tn ,
we define X(i, j), (0 ≤ i ≤ m,0 ≤ j ≤ n) to be the length of the
longest common subsequence (LCS) of S and T. LCS can be
computed using the following defining recurrence [18] (A similar
recurrence applies to the “pairwise sequence alignment with affine
2 For an input of size n, an event E occurs w.h.p. (with high probability) if, for any
α ≥ 1 and c independent of n, Pr(E) ≥ 1 − c
nα . The larger the value of n, the closer
Pr(E) is to 1, and limn→∞ Pr(E) = 1.
(a) 2-way recursive DAC algorithm
with Q1(n) = O n2/(BM) and
T∞(n) = O nlog2 3 .
(b) Straightforward parallel looping
(no tiling) algorithm with Q1(n) =
O n2/B and T∞(n) = O (n).
Figure 1: Do we need to tune LCS between parallelism and cache effi-
ciency? 2-way recursive DAC algorithm has best cache complexity bound
but worst span, straightforward parallel looping algorithm has best span but
worst cache performance, any r-way (r ∈ (2,n)) recursive DAC algorithm is
in between. Note that the solid arrows in diagram denote real dependencies
arising from defining recurrence and dashed arrows denote artificial depen-
dencies introduced by the algorithm. Timeline t is the computing direction.
gap cost” problem [31]):
X(i, j) =
0 if i = 0 ∨ j = 0
X(i−1, j −1)+1 if i, j > 0 ∧ si = tj
max{X(i, j −1),X(i−1, j)} if i, j > 0 ∧ si = tj
A 2-way recursive DAC algorithm (see Figure 1a) for the LCS
problem was given in [15]. We assume for simplicity of exposition
that m = n = 2k for some integer k ≥ 0. The algorithm splits the DP
table X into four equal quadrants: X00 (top-left), X01 (top-right), X10
(bottom-left), and X11 (bottom-right). It then recursively computes
the quadrants in the following order: X00 first, then X01 and X10
in parallel, and finally X11, i.e. X00;X01||X10;X11. A recursive call
on an n × n sub-matrix uses only Θ(n ) space. When the space
needed to solve a subproblem is small enough to fit into the cache,
the algorithm won’t incur any cache misses in addition to those
needed to bring the subproblem into the cache and write it back to
the memory. Thus the algorithm fully exploits the temporal cache
locality, and the resulting cache complexity Q1(n) = O n2/(BM)
can be shown to be optimal [15]. However, the span T∞(n) =
Θ nlog2 3 is suboptimal since a straightforward looping algorithm
that computes all entries of a diagonal in parallel and proceeds in
this way diagonal by diagonal has Θ(n) span (though the cache
complexity is Ω n2/B ).
In general, assuming m = n = rk for integers r ≥ 2 and k ≥ 0,
and performing an r×r decomposition at each level and recursively
computing the r2 subproblems, we have an r-way recursive DAC
algorithm. For such an approach, Q1 can be computed as follows.
Q1(n) =
O (n/B+1) if n ≤ αrM,
r2Q1(n/r)+Θ(1) if n > αrM;
where, αr ∈ (0,1] is a constant automatically and implicitly deter-
mined by the r-way recursive DAC strategy 3. The recurrence for
3 αr is the ratio of the subtask size and the cache size in the first level of recursion in
which each individual subtask fits into the cache. This αr is only used during the anal-
ysis of algorithm, it does not appear anywhere in the algorithm or its implementation.

span is
T∞(n) =
Θ(1) if n = 1,
(2r −1)T∞(n/r)+Θ(1) if n > 1.
The recurrences above solve to:
Q1(n) = O min n2
,
n2r
BM
+
n2r2
M2
and T∞(n) = Θ nlogr (2r−1)
.
Observe that while Q1(n) is the lowest (and optimal) when
r = 2, it grows as r increases and becomes as bad as O n2 when r
exceeds M (can be slightly improved though 4). On the other hand,
T∞(n) has the worst value when r = 2, but it improves as r increases
and reaches optimal value Θ(n) when r = n.
The change in behavior of Q1 and T∞ with the change of r can
be explained as follows. When r is small, artificial dependency
relations due to scheduling at the granularity of task dependency
prevents many completely independent subtasks from executing in
parallel. For example, consider the 2-way DAC algorithm in Fig-
ure 1a in which the solid arrows denote real data dependencies aris-
ing from the defining recurrence and the dashed arrows denote ar-
tificial dependencies introduced by the algorithm. The dependency
of X10,00 on X00,11 is artificial, while the dependency of X11,00 on
X00,11 is real, and so on. As r increases, the subproblem sizes de-
crease, and the number of artificial dependencies keeps decreasing
which leads to increased parallelism. But large r is bad for Q1 as
that leads to smaller subproblems, as a result, the largest subprob-
lems that fit into the cache is often much smaller than the size of
the cache leading to cache under-utilization and loss of temporal
locality.
In the light of discussion above, traditional wisdom may suggest
that one should strike a balance point between span and cache com-
plexity in order to get good performance in practice. Apparently,
the intuition behind balance is that we can not get both optimal at
the same time.
Our Contributions.
• [Algorithmic Framework] We introduce parallel “Cache-
Oblivious Wavefront” (COW) algorithms that perform divide-
and-conquer of the DP table in exactly the same way as standard
cache-optimal recursive DAC-based DP algorithms, but con-
sider a recursive subtask ready for execution as soon as all its
real dependency constraints are satisfied. By performing divide-
and-conquer the same way as original 2-way DAC-based algo-
rithm, COW algorithms retain the recursive execution order of
subtasks derived from the same parent task. As a consequence,
COW algorithms inherit the cache-obliviousness and (serial)
cache-optimality properties of the standard recursive DAC-
based algorithms. By scheduling a recursive subtask ready for
execution based only on its real dependency constraints, COW
algorithms lead to potentially improved span. To distinguish,
we name original recursive DAC-based standard algorithm as
“cache-oblivious parallel” (COP) algorithm, and the new pro-
posed algorithm “cache-oblivious wavefront” (COW) because
the overall execution pattern of COW algorithms usually pro-
ceed like propagating a wave of executing and ready-to-execute
subtasks through a dynamically unfolding recursive DAC tree.
Depending on how long it takes to align a spawned subtask to
the advancing wavefront, we have devised two different tech-
niques for efficient wavefront alignment of subtasks without
wasting valuable computing resources (i.e., cores).
4 Q1(n) can be improved to O n2
/B by suitably fixing the order of execution of
subtasks.
1. [Eager COW] We present in Section 2 algorithms when
subtasks can start executing within O (1) time of spawning.
We use such algorithms for Floyd-Warshall’s APSP and
stencil computations among others.
2. [Lazy COW] We present in Section 3 algorithms when
subtasks require ω(1) time to get ready for execution after
they are spawned. We use such algorithms for LCS, among
others.
• [Theoretical Analyses] We provide theoretical analyses of po-
tentially improved span in Sections 2 and 3 for Eager and Lazy
COW algorithms respectively. The improvement in span (T∞)
comes without increasing the total work (T1) and / or serial
cache complexity (Q1) of the original COP algorithm.
• [Experimental Results] We have implemented several COW
algorithms and compared them in Section 4 with standard 2-
way COP algorithm, tiled parallel loop, and standard parallel
loop implementations of the same DP on a number different
hardware platforms including 16-core and 32-core machines.
Experimental measurements of absolute running times, relative
speedups w.r.t. direct parallel loops, burdened span by Cilkview
[34], and L1/L2 cache misses by PAPI [10] validate our claims.
We conclude the work in Section 6 with a discussion on the
limitations of current approach.
2. Eager Recursion to Simulate a COW
This section introduces an algorithmic technique, called the Eager
recursion, to simulate a COW algorithm for a wide range of cache-
optimal recursive DAC DP algorithms including the ones that solve
path problems over closed semirings [13, 15] and perform stencil
computations [23–25, 53].
Semiring serves as a general framework for solving path prob-
lems in directed graphs [3], and both Floyd-Warshall’s (FW) algo-
rithm [22] for finding All-Pairs Shortest Paths (APSP) and War-
shall’s algorithm [55] for finding transitive closures [55] are instan-
tiations of this algorithm. A stencil, on the other hand, defines the
value of a grid point in a d-dimensional spatial grid at time t as a
function of neighboring grid points at recent times before t. A sten-
cil computation computes the stencil for each grid point over many
time steps, and has numerous applications [6, 19, 21, 23, 25, 36,
37, 39, 42–44, 51, 53, 57].
The main insight motivating the design of the Eager recursion
technique came from the following observation. Cache-optimal re-
cursive DAC algorithms schedule the execution of recursive sub-
tasks at the granularity of task dependency. As a result, they often
delay the execution of subtasks because of some artificial depen-
dencies arising from such a coarse-grain ordering even when the
real dependency constraints are already satisfied.
The Eager recursion technique tries to execute a subtask as soon
as all its real dependency constraints are satisfied while still follow-
ing the recursive DAC scheme. For example, suppose X and Y are
two tasks in such an algorithm and subtask Xi of X has a real depen-
dency on subtask Yj of Y, and there are no other real dependencies
between X and Y. Let’s assume for simplicity that each subtask of
X and Y takes only O (1) time to execute. The standard algorithm
will not let X execute until all subtasks of Y completes execution
though Yj has perhaps finished much earlier and there was no need
of extra delay for X. In our Eager approach, task X is spawned at
the same time as Y. Instead of the task X waiting for the comple-
tion of the entire task Y, subtasks of X are spawned in parallel with
those of Y. Subtask Xi will be busy waiting for the completion of
Yj by continuously checking the wavefront data structure. When Yj
completes execution it updates the wavefront and within constant

time Xi (and thus X) can advance. This Eager approach works ef-
ficiently provided the structure of computation guarantees that real
dependency constraints of X are satisfied within O (1) time of the
start of execution of Y.
(a) Data dependency pattern of 2D
FW
(b) Procedure A of 1D FW in
2-way DAC algo.
(c) Data dependency pattern
of 1D FW
(d) Procedure B of 1D FW in both 2-way
DAC algo. and Eager recursion
(e) Procedure A of 1D FW in Eager re-
cursion.
Figure 2: Data dependency pattern of 1D/2D FW. The solid arrows indicate
the real dependencies from defining recurrence. The dashed arrows indicate
the artificial dependencies introduced by the algorithm. Dark-shaded cells
denote the cells to update (write) in current procedure and light-shaded cells
denote the dependent (read) cells. Notation A(X) denotes a computation on
data block X with all dependent diagonal cells self-contained and B(X,Y)
denotes a computation on data block X with all dependent diagonal cells
contained in a completely disjoint data block Y.
Eager Recursion on a Simple Example. We will explain our
approach using a simple synthetic benchmark as an example. This
benchmark, called 1D FW, is a simplification and abstraction of
original 2D FW algorithm [22] in the sense that they have a similar
data dependency pattern from defining recurrence equations. The
defining recurrence of 1D FW for 1 ≤ i,t ≤ n is as follows which
assumes that d(0,i) for 1 ≤ i ≤ n are already known.
d(t,i) = d(t −1,i)⊕d(t −1,t −1) (1)
Figures 2c and 2a show the dependency pattern of 1D FW and
2D FW side-by-side to exhibit the similarity. In both figures, dark-
shaded cells denote the cells to update (write), and the light-shaded
cells denote the dependent (read) diagonal cell from previous time
step (t − 1). We can see that the update of any cell d(t,i) in 1D
FW depends on both the diagonal cell from previous time step, i.e.
d(t −1,t −1), and the cell of the same space position from previous
time step, i.e. d(t −1,i). The serial 1D FW problem can be solved
in Θ n2 time using Θ(n) space.
Based on the dependency pattern in Figure 2c, we have a COP
algorithm based on 2-way recursive DAC strategy adapted from
[13] to solve the 1D FW with optimal work and optimal cache com-
plexity. This algorithm comprises two recursive functions named A
and B. A(X) denotes a computation on data block X with all de-
pendent diagonal cells self-contained, and B(X,Y) denotes a com-
putation on data block X with all dependent diagonal cells included
in a completely disjoint data block Y. The recursive structure of al-
gorithm is as follows:
A(X) : A(X00);B(X01,X00);A(X11);B(X10,X11)
B(X,Y) : B(X00,Y00)||B(X01,Y00);B(X10,Y11)||B(X11,Y11).
Figures 2b and 2d are graphical illustrations of the algorithm.
Observing that A is serialized by the task-level dependency, all
parallelism come from B.
For this 2-way COP algorithm, we have following recurrences
to compute the span (T∞(n) = T∞,A(n)), and serial cache complexity
(Q1(n) = QA(n)), respectively:
T∞,A(n) = 2T∞,A(n
2 )+2T∞,B(n
2 )
T∞,B(n
2 ) = 2T∞,B(n
4 )
QA(n) = 2QA(n
2 )+2QB(n
2 )
QB(n
2 ) = 4QB(n
4 )
Assuming that T∞,A(1) = Θ(1) and T∞,B(1) = Θ(1). For n ≤
εM, we have QA(n) = O n
B and QB(n) = O n
B to indicate that
the recursive calculation of cache complexity should terminate as
soon as A and B fit into the cache, though this terminating condi-
tion is oblivious to the algorithm design and implementation. The
recurrences solve to: T∞(n) = T∞,A(n) = O (nlog2 n), and Q1(n) =
QA(n) = O n2/(BM) .
While the cache complexity is optimal, the span is not because
the span of straightforward parallel looping algorithm solving this
problem is only O (n). The reason behind the suboptimal span
can be understood by examining Figure 2b, in which solid arrows
represent the real data dependency originating from the defining
recurrence Equation (1) and dashed arrows represent the artificial
dependency introduced by the algorithm. For example, at each
recursion level of A, the computation of first cell of X01 will have
to wait on the last cell of X00 and the first cell of X10 will wait
on the last cell of X11. These dependencies do not arise from the
defining recurrence but are introduced by the structure of algorithm,
more precisely, the scheduling of subtasks at the granularity of task
dependency. If we can reduce those artificial dependencies, we can
possibly improve the span asymptotically.
The basic idea behind Eager recursion is to spawn all sibling
subtasks at the same recursion level in A simultaneously, and intro-
duce atomic operations to guard the data dependency on diagonal
cells. The execution pattern of A changes to:
A(X) : A(X00)||B(X01,X00);A(X11)||B(X10,X11).

A graphical illustration is shown in Figure 2e. The recursion of
B doesn’t change since there are no artificial dependencies there.
Note that B(X01,X00) slightly lags behind A(X00) in the same
parallel block of A(X) because B can not start computation until
sibling A finishes its first base case due to the dependency on
diagonal. Since the dependency between siblings in A occurs only
on diagonal cells, and there is a strict (monotonically increasing)
timing order in computing diagonal cells, atomic operations just
need to be imposed on one integer that records the latest time
step of computed diagonal cells. This data structure (one integer
in the case of 1D FW) that records the progressing information
and protected by atomic operations is called wavefront because the
progress of algorithm usually proceeds alike a wavefront.
The span recurrence of A under Eager recursion changes to
T∞,A(n) = T∞,B(n/2)+T∞,B(n/2)+O (1)
, because B and A within the same parallel block overlaps and B
depends on the diagonal data produced by A but not vice versa
(Figure 2e). The O (1) term in the recurrence represents the de-
pendency cost of B (atomic busy-waiting) along time dimension
on the diagonal cells computed by A at each recursion level, and
is constant because if we assume perfect scheduling and constant
base case size, B needs to wait for only one diagonal base case of
A before it can start execution as shown in Figure 2e. The span
T∞(n) = T∞,A(n) of the entire algorithm then solves to O (n), which
matches the bound of straightforward parallel looping algorithm
and is optimal. The improvement of performance in practice is also
significant as shown in Section 4.
The cache recurrence of A under Eager recursion changes to
QA(n) = 2QA(n/2)+2QB(n/2)+O (n),
where the additional term O (n) accounts for the cache miss over-
head during atomic busy-waiting time of all first-row subtasks in
B on A for the first diagonal base case (see Figure 2e). Note
that in the original 2-way COP algorithm adapted from [13], due
to the dependency of B on diagonal cells computed by A, the
cache complexity of fetching the diagonal cells from A to B ex-
ists anyway (included in QB(n)). The Eager recursion algorithm
just charges explicitly an additive O (1) (assuming base case size
is constant) cost on this dependency to all first-row sub-tasks in B.
The cache recurrence of B doesn’t change. The recurrence solves
to O n2/(BM)+nlog(n/M) . If we assume n/logn >> BM, the
complexity reduces to O n2/(BM) , the same as original algo-
rithm.
We have measured the L1/L2 cache misses of the benchmarks in
Section 4 using PAPI [10], the results of which validate our claim.
Claim 1. The COW algorithm for 1D FW has only data depen-
dency originating from the defining recurrence Equation (1).
Proof of the claim is obvious from the algorithm, so is omitted.
Similar ideas apply to original 2D FW and stencil computation.
Claim 2. The COW algorithm for 2D FW improves the span from
O nlog2 n in [13] to O (n) with the same cache complexity bound.
The COW algorithm for d-dimensional Stencil improves the span
from Θ (d2h)wlg(d+2)−1 in [53], where h is the height and w is
the width of the hypercubic computing space, respectively, to O (h)
with the same cache complexity bound.
3. Lazy Recursion to Simulate a COW
This section introduces Lazy recursion – an algorithmic technique
to simulate a cache-oblivious wavefront efficiently for a range of
recursive DP algorithms for which the “eager” recursion technique
presented in Section 2 may end up wasting too much computing
resources because of the busy-waiting in practice5. This set of DP
problems include LCS [15, 16, 18], pairwise sequence alignment
with affine gap cost [31], sequence alignment with gaps (GAP
problem) [15, 28, 29, 56], and the parenthesis problem [12, 29].
Why is the Simple Eager Technique not Efficient for LCS?
(a) Procedure A of LCS in 2-way
COP algo.
(b) Procedure A of LCS in Eager
COW.
(c) Procedure A of LCS in Lazy COW.
Figure 3: Comparison between classic 2-way COP, Eager, Lazy COW
algorithms for LCS. The solid arrows indicate the real dependencies from
defining recurrence. The dashed arrows indicate the artificial dependencies
introduced by algorithm.
A classic 2-way COP algorithm (see Figure 3a) proceeds as
follows: it divides the input task at each recursion level into four
equally sized subtasks and executes them in diagonal order along
timeline t. The execution pattern is
A(X) : A(X00);A(X01)||A(X10);A(X11).
This execution pattern creates artificial data dependencies among
subtasks as shown by the dashed arrows in Figure 3a. For example,
the first cell of X10 depends on the last cell of X00, which is an
artificial dependency. The dependencies arising from the defining
recurrence are drawn as solid arrows.
If we simply adopt the Eager recursion technique introduced in
Section 2 and execute the pattern as 6
A(X) : A(X00)||A(X01)||A(X10);A(X11),
we still have two problems (see Figure 3b):
1. Artificial dependencies still exist, e.g., the first cell of X11 will
have to wait for the last cells of X10 and X01 to be computed.
5 Though does not matter in theory because when we compute T∞ we assume an
infinite number of computing cores anyways
6 note that A(X11) still have to lag behind A(X00) because of the strict data dependency
of the first cell of X11 on the last cell of X00

Figure 4: Execution details of procedure A of LCS in Lazy recursion.
2. X01 and X10 are spawned early, but can not start computing their
first cells until X00 has updated at least half of its cells. In other
words, the simple Eager recursion causes non-constant busy-
waiting time (O (n) in this case) of X01 and X10 on X00 along
time dimension. The non-constant busy-waiting time implies
that there will be fewer computing cores for producer X00 to
produce the data on wavefront to unlock consumers X01 and
X10. In practice, we do not have an infinite number of com-
puting cores. All computing cores should perform useful work
instead of busy-waiting unless the busy-waiting takes only con-
stant time. Note that when we say waiting time, we only count
the waiting time along time dimension because the computation
along space dimension are parallelized.
An ideal execution pattern for LCS is illustrated in Figure 3c. In
the recursive execution, as soon as X00 updates half of its cells up
to the middle line, X01 and X10 start executing, and as soon as X01
and X10 update half of their cells, X11 starts running. So there are
overlaps of execution on timeline among all four subtasks at each
recursion level.
For the ideal algorithm, the span recurrence becomes
T∞(n) = 2T∞(n/2) = O (n),
because the execution time of X01 and X10 completely overlap with
those of X00 and X11. The cache recurrence doesn’t change because
the DAC behavior doesn’t change.
3.1 Lazy Recursion
We devised a Lazy recursion technique as shown in Figure 4 to
simulate the ideal algorithm in Figure 3c.
Conceptually, the Lazy recursion approach perform the same
divide-and-conquer as classic 2-way COP algorithm but sched-
ule the execution of subtasks across different levels of DAC tree
aligned to a wavefront (proof in Claim 3). In the case of LCS, re-
ferring to Figure 4, at each recursion level, except the last X11 each
subtask pushes its X11 subsubtask one level up in the DAC tree to
execute in parallel with the subtask’s siblings. The execution pat-
tern shown in Figure 4 is for one level of Lazy recursion
A(X) : A(X00,Γ);
A(X00,11)||A(X01,Γ)||A(X10,Γ);
A(X01,11)||A(X10,11)||A(X11)
where X00,Γ denotes all sub-tasks in X00 except X00,11 and so
on. We prove later in this section that by infinite levels of Lazy
recursions, the execution of all subtasks across different levels of
the DAC tree are aligned to a wavefront along time dimension.
The span recurrence of LCS (T∞(n) = T∞,2(n)) under Lazy
recursion is:
T∞,2(n) = 2T∞,Γ(n/2)+T∞,2(n/2)
T∞,Γ(n) = 2T∞,Γ(n/2)
The explanation of span recurrence is as follows. T∞,2 denotes
the span of procedure that computes a task without pushing its
X11 up the DAC tree, and T∞,Γ denotes the span of procedure that
computes a subtask with its X11 pushed up the DAC tree and has
the execution of X11 overlapped with its siblings. Since in the latter
case, the X11 quadrant of a subtask always executes in parallel with
that subtask’s siblings which are geometrically larger, the execution
time of that X11 quadrant is not counted in T∞,Γ. We can see from
Figure 4 that for the initial recursion, span of all subtasks except the
one handling X11 are counted as T∞,Γ. For subsequent recursions,
except that the span of X11,11,...,11 will be given by T∞,2, all other
subtasks will be executed in an overlapping fashion and will have
T∞,Γ type span. So, the overall span of algorithm reduces to O (n).
In Section 4 we will see that the performance improvement in
practice matches our theoretical predictions.
We argue that the cache recurrence doesn’t change because if we
put the child X11 back to its corresponding T∞,Γ parent, the number,
the shapes, and the sizes of subtasks at each recursion level doesn’t
change from the original 2-way COP algorithm. Note that the cache
recurrence counts only the number, the shapes and the sizes of
subtasks at each recursion level. If we assume an infinite number of
processing cores and perfect scheduling, the atomic busy-waiting
overhead of each base case will be an additive O (1) because each
base case needs to wait for only two (2) other base cases from
previous time step to start its own computation. For example, at
the bottom (last two levels) of the DAC tree, the computation of
a base case X11 only depends on base cases X01 and X10 from the
same parent recursion. In Section 4 we report L1/L2 cache misses,
measured by PAPI [10], incurred by our Lazy COW algorithm
showing that its cache performance is comparable to that of the
standard 2-way COP algorithm.
Claim 3. Lazy algorithm for LCS aligns subtasks across different
levels of the DAC tree with a conceptual wavefront (see Figure 4)
that sweeps through the entire problem space along timeline t.
Proof. Omitted due to page limitation.
4. Experimental Evaluation
In this section we report experimental results showing how well the
COW simulation techniques, i.e., Eager and Lazy recursion, per-
form in practice. We compared our COW algorithms with parallel
loops (without tiling), blocked parallel loops (with tiling), and clas-
sic 2-way recursive DAC (COP) implementations. Note that COW
technique is an algorithmic improvement over classic COP algo-
rithm, so the natural counterpart of our COW algorithms are the
classic COP algorithms. We list as well the comparison with paral-
lel loops and blocked parallel loops just for references. The bench-
mark problems that are used to evaluate our techniques are given in
Table 3.
We feel that programmability of the COW algorithms are not
within current scope. Without proper primitive support in either
nested parallel programming model or Cilk runtime system, pro-
gramming of the COW algorithms are quite tricky and requires

Table 2: Machine specifications.
Name Intel32 Intel16
System Intel Xeon E5-4650 Intel Xeon E5-2680
Clock 2.70 GHz 2.70 GHz
#Cores 4×8 2×8
L1 data cache 32 KB 32 KB
Last-level cache 20 MB 20 MB
Memory 1 TB 32 GB
OS CentOS 6.3 CentOS 6.3
Compiler icc v14.0 icc v14.0
hacking into Cilk runtime system. That’s also the reason why we
do not give out the pseudo-code in this paper. We plan to investi-
gate on primitive support as our future work.
Due to page limitations, we only include in this extended ab-
stract some selected charts for 1D FW and LCS. Other charts and
charts of other problems show similar patterns and trends, so are
omitted.
The methodology of our experiments is as follows. For a “fair”
comparison, we use the same base case function throughout differ-
ent algorithms of the same DP problem. For blocked loops, 2-way
COP and COW, we also coarsened them to the same base case size.
So the way how subtasks are scheduled becomes the main if not the
only difference between COW algorithm, 2-way COP algorithm,
and blocked parallel looping algorithm. To validate our claim that
the COW algorithm improves parallelism without giving up cache
efficiency, we measure the burdened span of all algorithms using
Cilkview [34] and L1/L2 cache misses using PAPI [10]. All mea-
surement results reported in this section are “min” of at least three
(3) independent runs. We experimented on the hardware platforms
listed in Table 2.
Table 3: Benchmark problems that use Eager and Lazy approaches. COP is
a recursive DAC based cache-oblivious parallel algorithm, and ∗ represents
this paper.
Eager Lazy 2-way COP
Problem recursion recursion implementation
Floyd-Warshall 1D [*] − [∗]
Floyd-Warshall 2D [22, 55] − [13]
Stencil Programs [23–25] − [52]
LCS [15, 18] − [15]
(a) (b)
Figure 5: “Burdened span” by Cilkview of 1D FW and LCS. “Burdened
span” is counted in number of binary instructions on the critical path. In
figures, we fix base case size but change problem sizes.
Burdened Span. Figure 5 plot the burdened span measured by
Cilkview [34]. Vertical axis is “burdened span (×1e6)” in log scale,
and horizontal axis is the “side length n” (the problem size is then
n2) in log scale. Span is the length of critical path in the directed
acyclic graph (DAG) representation of a parallel program. The
(a) Fixed problem size with vari-
able base case sizes
(b) Fixed base case size with vari-
able problem sizes
(c) Fixed problem size with vari-
(d) Fixed base case size with vari-
able problem sizes
Figure 6: Comparison of cache misses (by PAPI) among parallel loops,
blocked loops, 2-way COP, and COW algorithms. bsize is the base case
size. Both vertical and horizontal axes are log scale in all sub-figures.
shorter the span, the better the parallelism an algorithm has. Bur-
dened span is theoretical span plus scheduling overhead. In gen-
eral, Figures 5a and 5b reveal that COW algorithms (solid line with
empty square) always have the best span, which matches our the-
oretical prediction. Parallel loops (solid line with solid diamond)
always have the worst burdened span because the loops are par-
allelized with granularity of 1, which is too fine-grain to amor-
tize the scheduling overhead. In Figures 5a and 5b, blocked loops
(solid line with empty diamond) always have shorter span than 2-
way COP (solid line with solid square) because theoretical span
of blocked loops for 1D FW and LCS are both O (n) but theo-
retical span of 2-way COP for 1D FW and LCS are O (nlogn)
and O nlog2 3 , respectively. Note that Cilkview [34] measures the
span by counting the number of binary instructions on the critical
path, hence the results in span charts (e.g. Figures 5a and 5b) may
slightly deviate from theoretical predictions especially when the
problem size is small.
Cache Misses. Figure 6 plot the L1 cache misses measured by
PAPI [10] on Intel32. Vertical axis is “L1 cache misses (×1e6)” in
log scale, and horizontal axis is “base case size” in log scale. We
only show the charts of L1 cache misses. The charts of L2 cache
misses show similar patterns, so are omitted. Figures 6b and 6d
fixed the base case size to a small constant and varies side length,
so are the problem size, to see how the L1 cache misses changes
accordingly. Figures 6a and 6c, on the contrary, fixed a reasonably
large problem size and varies base case sizes from small to large,
to see how the L1 cache misses changes accordingly. By above
two comparisons, we have a rough overview of a 3D chart of how
L1 cache misses change with different problem sizes and different
base case sizes for 1D FW and LCS problems. From the cache
charts, we can see that blocked loops, 2-way COP, and COW algo-
rithms have roughly the same L1 cache misses. Statistics of other

(a) Fixed base case size with vari-
able problem sizes
(b) Fixed base case size with vari-
able problem sizes
(c) Fixed problem size with vari-
(d) Fixed problem size with vari-
(e) (f)
Figure 7: Comparison of absolute performance (updated points / second) on
Intel32 and relative performance (speedup with respect to parallel loops) on
Intel16 and Intel32. In all sub-figures of absolute performance, the vertical
axis is linear scale and horizontal axis is log scale. In all sub-figures of
relative performance, both the vertical and horizontal axes are log scale. In
charts of relative performance, the suffix 32 is the speedup on Intel32 and
the suffix 16 is the speedup on Intel16.
problems on different machines or different level of cache show
similar patterns, so are omitted.
Absolute Performance (Points Updated per Second). Figure 7
plot absolute performance (updated points/second) on Intel32, and
relative performance (speedup w.r.t. parallel loops) on Intel32 and
Intel16. Similar to the methodology of showing cache charts, Fig-
ures 7a and 7b show the absolute performance (updated points
/ second) when we fixed base case size to a small constant and
changed problem size (side length n actually) from small to large.
Figures 7c and 7d show the absolute performance when we fixed
a reasonably large problem size but changed base case sizes from
small to large. Absolute performance were measured using the
clock gettime(CLOCK MONOTONIC,...) function in Linux. In
figures showing absolute performance, vertical axis is “updated
points/second (×1e9)” in linear scale and horizontal axis is “side
length (n)” in log scale. The absolute performance charts show
that on Intel32 blocked loops always have better performance than
2-way COP when using exactly the same base case function. We
conjecture that on modern multi-core machine, parallelism may
play a more important role in actual performance. All performance
charts show that our COW algorithms beated blocked loops in al-
most all cases, because cache-oblivious wavefront algorithms catch
up in parallelism. In Figures 7c and 7d, we can see with larger base
case sizes, both COW algorithms and blocked loops tend to have
better performance.
Relative Performance (Speedup w.r.t. Parallel Loops). Figures
7e and 7f plot relative performance (speedup w.r.t. parallel loops) of
various algorithms on Intel32 and Intel16. Vertical axis is “speedup
w.r.t. parallel loops” in linear scale and horizontal axis is “side
length (n)” in log scale. Suffix 32 indicates the results on Intel32
and suffix 16 is used for results on Intel16. The general trend in rel-
ative performance (speedup) charts is that COW algorithms benefit
when there is not enough parallelism (compared with the available
number of computing cores) for classic 2-way COP algorithms.
This pattern matches theoretical predictions because the key point
of COW algorithms is to improve the span with approximately the
same cache complexity as 2-way COP algorithms.
5. Related Work
Recursive DAC algorithms, both serial and parallel, most of them
having optimal serial cache complexity have been developed, im-
plemented, and evaluated for LCS [11, 15], pairwise sequence
alignment [16], Floyd-Warshall’s APSP [13], stencil computation
[23–25, 52] etc.
Tiling approaches and their expressions, such as overlapped
tiling, dynamic partitioning, hierarchically tiled arrays [5, 32] are
heavily studied. Meng et al. [41] studies the approach of tiling plus
inter-thread locality. The main difference of our work from their
approach is that while we focus on a (cache, processor)-oblivious
approach, Meng et al.’s approach is (processor, cache, thread)-
aware. While retaining provably optimal cache complexity bounds
is the main focus of our work, Meng et al.’s work does not focus on
providing such theoretic guarantees.
Priority update [48] simulates a CRCW (Concurrent Read Con-
current Write) memory model when the updating operation can be
prioritized to reduce memory contention among multiple writes to
a single memory location. The primitive can be used to improve
parallelism and overall performance of certain parallel algorithms.
In all the cases we have considered, each memory location can have
only one writer at any time but can have multiple readers. We still
maintain the CREW (Concurrent Read Exclusive Write) memory
model and doesn’t require any special property from the updating
operation. In addition, we are more concerned about avoiding artifi-
cial dependencies introduced by algorithm to minimize span while
keeping optimum cache performance.
Maleki et al. [40] presented in their paper that certain dynamic
programming problem called “Linear-Tropical Dynamic Program-
ming (LTDP)” can possibly obtain extra parallelism based on rank
convergence, a property by which the rank of a sequence of matrix
products in the tropical semiring is likely to converge to 1. LTDP
includes a class of important dynamic programming problems,
such as Viterbi, LCS. The parallel LTDP algorithm works well in
practice though the worst case is sequential. The LTDP algorithm
also doesn’t provide theoretical guarantee on cache-obliviousness,
which is the key property of our algorithms.
Hybrid r-way DAC algorithms with different values of r at dif-
ferent levels of recursion have been considered in [12]. These al-
gorithms can reach parallel cache complexity matching the best se-

quential cache complexity, but the algorithms then become compli-
cated to program, processor-aware, and often cache-aware.
Current implementations of the Eager/Lazy COW simulation
techniques rely on atomic operations on the data dependency path
to guard the correctness of algorithms in addition to the fork-join
primitives. Atomic operations are also commonly used to imple-
ment parallel task graph execution systems such as Nabbit [2],
BDDT [54], etc. The key difference between COW algorithms and
these task parallel graph execution systems is that these systems
usually unroll the entire execution beforehand and execute all sub-
tasks in a parallel looping fashion and as a result may lose cache
efficiency. In case of a COW algorithm, recursion unfolds dynami-
cally on-the-fly, and it inherits the recursive execution order among
subtasks divided from the same parent task. So the COW algorithm
retains the cache-obliviousness and cache-optimality properties of
the original 2-way recursive DAC algorithm.
6. Conclusion
We have proposed two algorithmic techniques to achieve locality-
preserving improvements in parallelism of standard recursive
DAC-based cache-oblivious parallel DP algorithms. Our tech-
niques work by performing the same divide-and-conquer as origi-
nal COP algorithms but scheduling the execution of subtasks across
different levels of DAC tree as soon as their real data dependency
constraints are satisfied. The proceeding of our COW algorithms
usually looks like a conceptual wavefront sweeping through a dy-
namically unfolded DAC tree. The resulting algorithms are called
Cache-Oblivious Wavefront (COW) algorithms. They often achieve
asymptotically shorter span than the original 2-way DAC based
COP algorithm without losing the cache-obliviousness and cache-
optimality properties of that algorithm. We have provided theoreti-
cal analyses as well as experimental results on several DP problems
to validate our claims. However, COW algorithms still have the
same recursive function call overhead as classic COP algorithms.
In other words, while these algorithms do not need to tune for par-
allelism and cache efficiency, they still have to tune for base case
size.
Current implementations of COW algorithms rely on atomic
operations to guard the correctness of the algorithm in addition to
the fork-join primitives. Atomic operations do not work well with
current theory of scheduling and the nested parallel programming
model. Getting rid of atomic instructions from COW algorithms
without degrading the performance guarantees they provide will be
one of our future research goals.
Acknowledgments
We thank Prof. Baifeng Wu and Prof. Xiaoyang Wang at Fudan
University for general help on research environment. We thank
Prof. Charles E. Leiserson at MIT CSAIL for coining the name
of “Cache-Oblivious Wavefront”. We thank all our reviewers for
their insightful comments and helpful discussions. This work used
the Extreme Science and Engineering Discovery Environment
(XSEDE), which is supported by National Science Foundation
grant number OCI-1053575.
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