Sources of Constraints in Computations

1. g f r(x) x p2 h p1 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

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. f g h p B A BLE ALE x fd gd hd pd xd 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

4. B A ALE BLE x xd f h p g g f r(x) x p2 gd hd pd fd h p1 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

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 applies 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 ok 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 annd p2 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