Data dependencies in the computation

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

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

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

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 lt 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 lt g means f must occur before g
• f lt g and g lt f implies that the two can be
  simultaneous
• For the example, f lt g, f lt p1, h lt p1, g lt p2,
  and p1 lt 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 lt 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 lt g or g lt h applies for our example
• This represents a degree of freedom
• However, note that f lt 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 lt 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 lt f lt g lt h lt p1 lt p2 lt r(x) or x lt f lt h
  lt g lt p1 lt p2 lt r(x) or x lt f lt h lt p1 lt g lt p2
  lt r(x) or x lt h lt f lt g lt p1 lt p2 lt r(x) or
  x lt h lt f lt p1 lt g lt p2 lt r(x)
• Serialization, once done, severely restrict our
  choice