1. Sources of Constraints in Computations
Data dependencies in the computation
Timing properties of the individual computations
Number and kind of components are connected
Constraints on the control structure
Suppose we want to compute
r(x) = p(g(f(x)), p(f(x), h(x)))
p, g, f, and h are separate combinational functions
p requires two inputs, g, f, and h require only one
If we are free to connect things, we can compute the function as a combinational function as shown below h f g p1 p2 x
r(x)

2. Computing using Constrained Resources
In the previous implementation, we are using multiple resources of the same kind (like p)
That is expensive
Moreover, many resources are not computing at most of the time even if we had requirement to compute
Datapaths are designed to sequentialize computation and minimize the requirement of resources
An example of multiple functional unit data path is shown in the next slide

3. Multiple Functional Units and Computation
A Data path may have multiple functional units
It may also have some restriction on how to move data around
For example consider the following data path
All lines in this data path are n-bit wide
A and B are registers
f, g, h, and p are functional units
Any output can be enabled
A and B reg can be loaded
X is input
Output is from B fd f gd g hd h pd p xd x B
BLE A
ALE

4. Computation Organization
To perform the given computation, we do the following steps
load x into reg A
Begin operation of module f and wait for it to complete
Completion depends on the propagation delay of module f
Load the output of f into reg B, enable fd and BLE signals
What next…. is dictated by the structure of data path and required computation f fd xd x B
BLE A
ALE h hd g gd p pd h f g p1 p2 x
r(x)

5. Precedence Constraints (Relations)
Precedence constraint dictates a certain ordering
Function g cannot starts until f is completed
p1 cannot start until both f and h are computed
p2 cannot starts until g and p1 are completed
Let us denote fs and ff as start and finish of function f
Then ff < gs or ff precedes gs and the corresponding events must occur in that order
Two events can be simultaneous as is the case for f and h
When there is no confusion, we will drop superscript
f < g means f must occur before g
f <= g and g <= f implies that the two can be simultaneous
For the example, f < g, f < p1, h < p1, g < p2, and p1 < p2

6. Other Dependencies and Precedence Graph
Data dependencies reflect structure of computation and the flow of information among its operations
These are inherent in the algorithms
There is a data dependency from r to s (r < s) if some value produced by r is used or required as an input to s
Device specification reflect physical constraints imposed by the component timing constraints
A precedence graph is a graph showing order of computation and data dependencies
It specifies a partial ordering on the set of operations
Neither h < g or g < h applies for our example
This represents a degree of freedom
However, note that f < p2 apples but is not included as it is captured in some other dependency

7. Some Properties of Precedence Graph
A precedence graph provides a useful means for visualizing the potential concurrency allowed by a set of precedence constraints
For example g and P1 may take place simultaneously (provided f and h has been computed)
f and h can take place simultaneously (but cannot be written back simultaneously due to resource constraint)
A data dependence graph is a precedence graph depicting only the constraints due to data dependencies
Device specification: Timing constraints fs <= ff indicate another f cannot start until the first one finishes
Data dependencies and device specifications represent essential precedence constraints, we cannot avoid them

8. Data Path Topology
A third source of constraint is number and kind of components and the way they are connected
Precedence relations may not be adequate to express topological constraints
For example, in our data path, we have only one p module
Therefore p1 and p2 may not proceed simultaneously
It is on in this case as we cannot even start them together
Use of a shared bus dictates that at most one value can be written back at one time
f and h results with input x must be written at different times
This may not be a problem if transfer time is short
However, watch out for contamination due to input change
Functional units, registers, and buses are common conflicts

9. Data Path Topology (Contd.) and Serialization
Mutual exclusion constraint typically arise from conflict over resources, e.g., p3 ( p1(x), p2(x) ) would allow p1 an dp2 to go simultaneously but will be prohibited due to single p
Precedence graph has no mechanism to express mutual exclusion like p1 and p2 need to be mutually exclusive above
We can reduce resource conflicts by replicating resources
Else, serialization of a computation can be used to impose additional precedence constraints
Serialization provides a total order over computation
For example for the given computation, we can serialize it as
x < f < g < h < p1 < p2 < r(x) or x < f < h < g < p1 < p2 < r(x) or x < f < h < p1 < g < p2 < r(x) or x < h < f < g < p1 < p2 < r(x) or x < h < f < p1 < g < p2 < r(x)
Serialization, once done, severely restrict our choice