1. Static Scheduling of Synchronous Data Flow Programs for Digital Signal Processing
S. M. Farhad
PhD Student
Supervisor: Dr. Bernhard Scholz
Programming Language Group
School of Information Technology
University of Sydney

2. Abstract
Synchronous data flow (SDF) differs from traditional data flow
The schedule of SDF nodes can be done at compile time (statically)
Contribution of this paper:
Develop theory for static scheduling of SDF programs on single or multiple processors
- Synchronous data flow differs from traditional data flow in that the amount of data produced and consumed by a data flow node is specified a priori for each input and output

3. Introduction
Need to depart from the simplicity of von Neumann computer architecture
Programming signal processors using large grain data flow languages [W. B. Ackerman 82]
Ease the programming
Enhancing the modularity of code
Describe algorithms in more naturally
Concurrency is immediately evident from program description
- Concurrency is immediately evident from program description so parallel hardware can be used more effectively

4. Data Flow Analysis [W. B. Ackerman 82]
P = X + Y
Q = P/Y
R = X*P
S = R – Q
T = R*P
RESULT = S/T
Many of these instructions can run in parallel as long as some constraints are met
These constraints can be represented by a graph
Node represents instructions
Arrow between nodes represents constraints
So, the permissible computation sequence can be for example (1, 3, 5, 2, 4, 6), (1, 2, 3, 5, 4, 6) and others.
- An arrow from one instruction to another means that the second may not be executed until the first has been completed

5. Sequencing Constraints
(1) P = X + Y
(3) R = X*P
(2) Q = P/Y
(4) S = R - Q
(5) T = R*P
(6) RESULT = S/T

6. The Data Flow Paradigm
A program is divided into pieces (nodes or blocks) which can execute whenever input data are available
An algorithm can be described as data flow graph
Node representing function
Arc representing data paths
Signal processing algorithms can also be described as data flow graph
Node is atomic or non-atomic function
Arc is signal path

7. The Data Flow Paradigm Contd.
The complexity of the functions (granularity) will determine the amount of parallelism available
No attempt to exploit concurrency inside a block
The functions within the blocks can be specified using von Neumann programming techniques
The blocks can themselves represent another data flow graph (hierarchical)
LGDF is ideally suited for signal processing

8. Synchronous Data Flow Graphs
A block is invoked when input available
When it invoked it consumes a fixed number of input samples on each input path and produces fixed number of output samples
A block is synchronous if we can specify a priori its input and output samples when it is invoked
Assuming that the signal processing system repetitively apply an algorithm to an infinite sequence of data

9. A synchronous data flow graphB A C b c e j d f g h i
SDF graph requires buffering the data samples passed between blocks and schedule blocks when data are available (static approach)
This could be done dynamically (runtime supervisor, costly approach)

10. A synchronous data flow graph
SDF graphs can be scheduled statically (at compile time) regardless of the number of processors
No need to have dynamic control
Communication between nodes and processors is set up by the compiler so no runtime control
Thus the LGDF paradigm gives the programmer a natural way for programming with evident concurrency

11. Scheduling an SDF graph
Schedule blocks onto processors in such a way that data is available during its invocation
Assumptions
The SDF graph is non terminating (without dead lock)
The SDF graph is connected
Goal is to find a periodic admissible parallel schedule (PAPS also PASS)
Non terminating is natural for signal processing
if SDF is not connected then each separate graphs can be scheduled separately using subsets of the processors

12. Construction of a PASS Topology matrix e c 1 2 1 i d 2 3 g f 3
Connection to the outside world is ignored
Correctly constructed self loop has equal amount of data produced and consumed so the net difference will be zero
This matrix need not be square in general

13. Construction of a PASS
Replace each arc with FIFO queue to pass data from one block to another (vary)
Vector b(n) contains the queue sizes of all the buffers at time n
For sequential schedule only one block can be invoked at a time
v(n) is the vector of blocks invoked at time n

14. Construction of a PASS 2 1 3 D 2D
The change in the buffer size caused by invoking a node is
A unit delay on an arc from A to B means that a n-th sample consumed by node B is (n-1)-th sample produced by node A
So the first sample consumed by destination block is not produced by the source (part of initial state of arc buffer)

15. Construction of a PASS 2 1 3 D 2D
Because of this initial condition block 2 can be invoked once and block 3 can be invoked twice before block 1 is invoked at all
Delay therefore affect the way the system starts up

16. Construction of a PASS Given this computation model (eqn. 1 - 4)
Find necessary and sufficient conditions for existing a PASS, and hence a PAPS
Find practical algorithms that provably finds a PASS if one exists
Find a practical algorithms that construct reasonable (not necessarily optimal) PAPS, if a PASS exists

17. Necessary condition for existing a PASS
Where s is the number of nodes or blocks in the graph
Definition 1: an admissible sequential schedule is a non-empty ordered list of nodes such that if the nodes are executed in sequence given by , the amount of buffer will remain non negative and bounded. Each node must appear in at least once

18. Quick reminder of rank of a matrix
The rank of a matrix is the maximum number of independent rows
The rank can be calculated by gaussian elimination algorithm
2nd column is the twice of column 1

19. Necessary condition for existing a PASS1
Theorem 1: For a connected SDF graph with s nodes and topology matrix Γ, rank (Γ) = s -1 is a necessary condition for a PASS to exist.
PASS of period p
(3)=> b(p) = b(0) + Γq
where 2 2 1 1 1 2 1 2 3 1 3
q vector tells us how many times we should invoke each node in one period of a PASS
after a period the buffer will end up once again in their initial state

20. Necessary condition for existing a PASS
Since the PASS is periodic, we can write
Since the PASS is admissible, the buffers must remain bounded, by definition 1. The buffers remain bounded if and only if
where O is a vector full of zeros
For q ≠ O, this implies that rank (Γ) < s where s is the dimension of q. But rank (Γ) can be either s or s – 1, and so it must be s – 1 [Lemma 3]

21. Necessary condition for existing a PASS1
Theorem 1 indicates that if we have a SDF graph with a topology matrix of rank s, then the graph is somehow defective and no PASS can be found for it 2 1 1 1 1 2 1 2 3 1 3
Any schedule for this graph will result either in deadlock or unbounded buffer sizes
The rank of the topology matrix indicates s sample rate inconsistency in the graph

22. Necessary condition for existing a PASS
Theorem 2: For a connected SDF graph with s nodes and topology matrix Γ, and with rank(Γ) = s – 1, we can find a positive integer vector q ≠ O such that Γq = O where O is the zero vector.
Definition 2: A predecessor to a node x is a node feeding data to x.

23. Necessary condition for existing a PASS
Definition 3: (Class S algorithm) Given a positive integer vector q such that Γq = O and an initial state for the buffers b(0), the ith node is runnable at a given time if it has not been run times and running it will not cause a buffer size to go negative. A class S algorithm is any algorithm that schedules a node if it is runnable, updates b(n) and stops only when no more nodes are runnable. If class S algorithms terminates before it has scheduled each node the number of times specified in the q vector, then it is said to be deadlocked.

24. Necessary condition for existing a PASS
Theorem 3: Given a SDF graph with topology matrix Γ and given a positive integer vector q s.t. Γq = O, if a PASS of period p = exists, where is a row vector full of ones, any class S algorithm will find such a PASS.

25. Necessary condition for existing a PASS1 1 1 1 1 1 D 2 2
- Networks with insufficient delays in directed loops are not computable 1 1 2 2
(a)
(b)
Two SDF graph with consistent sample rates but no admissible schedule

26. Necessary condition for existing a PASS
Theorem 4: Given a SDF graph with topology matrix Γ and given a positive integer vector q s.t. Γq = O, a PASS of period p = exists if and only if a PASS of period Np exists for any integer N.
Theorem 4 tells us that it does not matter what positive integer vector we use from the null space of the topology matrix, so we can simplify our system by using the smallest such vector, thus obtaining a PASS with minimum period.

27. Class S algorithm given the theorems
Solve for the smallest positive integer vector
Form an arbitrary ordered list L of all nodes in the system
For each , schedule if it is runnable, trying each node once
If each node has been scheduled times, STOP
If no node in L can be scheduled, indicate a deadlock
Else goto 3 and repeate

28. Constructing a PAPS
If a workable schedule for a single processor can be generated then a workable schedule for a multiprocessor system can also be generated
First step is to construct an acyclic precedence graph for J period of the PASS by class S algorithm

29. Construct an acyclic precedence graph by example2 1
This graph is neither acyclic nor a precedence graph
Possible minimum PASS is {1, 3, 1, 3}, {3, 1, 1, 2} or {1, 1, 3, 2} each with period 4.
{2, 1, 3, 1} not a PASS because node 2 is not immediately runnable 3 1 1 D 2 2 1

30. Construct an acyclic precedence graph1 1 2 3 2 1 1 1 3 2 3 1
J=1
J=2

31. Next step constructing a parallel schedule
By critical path method [Adam 74] or by Hu-level scheduling algorithm [T. C. Hu 61]
A level is determined for each node in the acyclic precedence graph, where the level of a given node is the worst case of the total of the runtimes of nodes on a graph from the given node to a terminal node of the graph
The terminal node is a node with no successor
If there is no terminal node then one can be created with zero runtime

32. Hu-level scheduling algorithm6 5 3 1 3 2 1 2 3 2 6 1 3 1 3 1 2 3 3 2 6 3 3 1
J=1
J=2

33. Constructing a parallel schedule
Hu-level scheduling algorithm simply schedules available nodes with the highest level first
When there are more than available nodes with the same highest level than there are processors, a reasonable heuristic is to schedule the ones with the longest runtime first

34. Constructing a parallel schedule3 3 1 3
PROC 1
PROC 1
PROC 2 1 1 2
PROC 2 1 1 2 1 2
J=1
J=2
Two processors, runtime of nodes 1,2,3 are 1, 2,3 time units respectively

35. Limitations of Model
Do not greater scale conditional control flow like general purpose languages
Asynchronous graphs
Connecting to the outside world
Data dependent runtime of blocks

36. Summary
This paper describes the theory necessary to develop a signal processing programming methodology that offers
Programmer convenience
Natural way to describe signal processing
Readily use the available concurrency

37. Question?
Thank you