1. Parallel processing is not easy
Good schedules for parallel implementations of composite tasks are hard to find.
Parallel implementation
Mapping and scheduling
Optimization for makespan
or throughput
User’s idea on
Parallel Computations
(e.g. STAP)
Executable file
for a given parallel computer

2. Parallel processing can be made easy with ALPS software framework/*
This is a .cpp file of the partially adaptive STAP methods that the user had in mind.
This is optimized for minimum latency or maximum throughput on a given parallel machine. */
#include <alps.h>
...
ALPS CodeGen
User’s idea on
Parallel Computations
(e.g. STAP)
ALPS Library
Various STAP heuristics
constructed from ALPS building blocks
Find optimal configuration that minimizes
overall makespan or maximizes throughput
3. ALPS Scheduler
1. Task Graph
2. ALPS Benchmarker & Modeler
Modeling execution times of
basic building blocks under various configurations

3. Task Graph: PRI-overlapped STAP1
Task Graph: PRI-overlapped STAP
User describes the algorithm using task graphs
ALPS software framework
determines the implementation parameters such as number of processors for each task and return the executable code
Target cube
Data cube
Steering vector
read
fft1d
STAP
join
write
split with
overlap

4. ALPS Benchmarker and Modeler2
ALPS Benchmarker and Modeler
Measures of actual timings yi for a set of input parameters
Candidate terms {gk} such as the size of data cube or the size of
processor cube are generated based on the input parameters
Model coefficients xk’s are found using least-squares
Ttask ≈ ∑(xkgk)
Solved incrementally by adding a best column to the model at a time. g1 g2 g3
gn-1 gn … y1 y2 ym x1 x2 xn 2
minx W(Ax-b) = minx W

5. Ordinary least-squares
minx W(Ax-b) 2
Ordinary least-squares
W = I
COV-weighted least-squares (BLUE estimator)
W = R-1/2 where R is the covariance matrix
one of the best, but requires lots of replications hence not practical
VAR-weighted least-squares
W = D-1/2 where D=diag(R)
INV-weighted least-squares
W = diag(1/b)
Used ordinary, VAR-weighted and INV-weighted least-squares with 5 replications for each measurement

6. 10% better 20% better 30% better 40% better 26.6% error from ordinary
0.1
0.2
0.3
0.4
0.5
0.6
0.7
26.6% error
from ordinary
least-square
10% better
20% better
From weighted
least-square
30% better
40% better
7.5% error
from INV-weighted
least-square
From ordinary least-square

7. 3 ALPS Scheduler Schedules tree-shaped task graphs
(Tree-shaped graphs) = (linear chains) (joining nodes) A B C
rather simple
complicated B C A B C A A C B
processors t t t
Data-parallelism
only [3]
Task-parallelism
preferred [4]
Other possibly
better schedules

8. We consider both data-parallel (series) and task-parallel (parallel) schedules for a joining node
Scheduling a joining node with two incoming arcs
Use dynamic programming considering two incoming arcs in series and in parallel
Scheduling a joining node with more than two incoming arcs
1. Sort the incoming arcs based on makespan when maximally parallelized
2. Schedule two incoming nodes at a time 10 A 10 A 1 B 8 C 8 5 C D 1 B 5 D

9. Makespan: achieved vs. estimated
Experiments on 8-node linux cluster: executable codes were generated by ALPS software framework
Makespan: achieved vs. estimated 1 2 4 8 20 40 60 80
Makespan (sec)
Makespan (sec)
estimated
achieved 30 20
achieved
estimated 10
Data-
parallelism
only [3]
Task-
parallelism
preferred [4]
Our
scheduling
algorithm
Number of processors used

10. Schedules found in makespan minimization
Target cube (32x2048x32)
Data cube (32x2048x32)
Steering vector (32x1x32)
Data-parallelism only [3]
~31 (sec) P1 P8
Task-parallelism preferred [4]
~24 (sec) P1 P8
Our scheduling algorithm
~24 (sec) P1 P8

11. Throughput: achieved vs. estimated1 2 4 8
0.05
0.1
0.12
Throughput (1/sec)
Throughput (1/sec)
estimated
0.1
achieved
estimated
achieved
0.05
Data-
parallelism
only [3]
Task-
parallelism
preferred [4]
Our
scheduling
algorithm
Number of processors used

12. Schedules found in throughput maximization
Target cube (32x2048x32)
Data cube (32x2048x32)
Steering vector (32x1x32)
Data-parallelism only [3] and our scheduling algorithm
~77 (sec) P1 P8
Schedule on 1 processor replicated 8 times
Task-parallelism preferred [4]
~45 (sec) P1 P8
Schedule on 4 processors replicated 2 times

13. Conclusion
The schedules found by ALPS framework are as good as or often better than those computed by other published algorithms
The relative error between the estimated and the achieved makespan/throughput were around 10% confirming that the modeled execution time is useful in predicting performance
Weighted least-squares produced estimates 20—30% better than ordinary least-squares in terms of relative errors; may be useful when modeling the unreliable timings.

14. Related works Frameworks Scheduling algorithms [1] PVL and [2] S3P
We use text version of task graph as front-end and source code generation as back-end without users dealing with C++ code
We cover larger scheduling problem space (tree-structured task graphs) than S3P.
Scheduling algorithms
[3] J. Subholk and G. Vondran: chain of tasks only
[4] D. Nicol, R. Simha, and A. Choudhary: s-p task graph, but task- parallelism is preferred
(… and many other works; refer to reference in the abstract)

15. References
[1] Eddie Rutledge and Jeremy Kepner, “PVL: An Object Oriented Software Library for Parallel Signal Processing”, IEEE Cluster 2001.
[2] H. Hoffmann, J. V. Kepner, and R. A. Bond, “S3P: automatic, optimized mapping of signal processing applications to parallel architectures,” in Proceedings of the Fifth Annual High-Performance Embedded Computing (HPEC) Workshop, Nov
[3] J. Subhlok and G. Vondran, “Optimal Use of Mixed Task and Data Parallelism for Pipelined Computations”, Journal of Parallel and Distributed Computing, vol. 60, 2000.
[4] D. Nicol, R. Simha, and A. Choudhary, “Optimal Processor Assignments for a Class of Pipelined Computations”, IEEE Transaction on Parallel and Distributed Systems, vol. 5(4), 1994.