Algorithm Representations and Iteration Bound

1
Algorithm Representations and Iteration Bound
2
Dependence Graph (DG)

• The basic representation of an algorithm.
• Shows only dependency among operations.
• No notion of delay is represented.
• No loop, cycle allowed.
• Can be used to represent asynchronous operations.
• Most useful in exploiting inherent parallelism in
  the algorithm

• No implementation or hardware constraint are
  imposed on DG.

3
Data Flow Graph

• Node
• Computation
• Associated with a computing time.
• Direct edge
• data path and delay
• Delay iteration count

• Example
• y(n) ay(n-1) bu(n)
• The delay of 1 u.t. indicates that to compute
  y(n1) in the next iteration depends on result
  y(n) of the present iteration.
• Delay labeled with D or positive integer on edges

4
DFG
x(n)
D
D

• Intra-iteration dependency
• A direct edge without any delay
• Inter-iteration dependency
• Direct edge with 1 or more delays
• Node computing delay labeled with parenthesis.
• Critical path longest path between
• Example critical path delay 422 8 t.u.

M1
M2
(4)
(4)
M0
(4)
y(n)
A1
A0
(2)
(2)

• Recursive DFG contains loops. Must have at least
  one delay element along any loop. Otherwise, the
  algorithm is NON-computable!

5
Loop bound and Iteration bound
D
(2)
(5)
(4)
A
B
C
2D
(2)
(4)
A
B

• TA-B-A (24)/2 3 t.u.
• T? max(24)/2, (245)/1
• max3, 11 11

2D
6
Longest Path Matrix Algorithm

• Paths are from one delay element to another.
• Matrix L(m) dimension d by d, d of delay
  elements
• Element ?ij(m) longest computing time from
  output of di to input of dj with exactly m-1
  delays in between. If no such path exists,
  ?ij(m)-1.
• Let K k k ?1, d, ?ik(1) gt -1 and ?kj(m) gt
  -1, then

• Once L(m) is computed, the iteration bound can be
  found as
• This algorithm can be easily programmed once L(1)
  is given.

7
LPM Example (Fig. 2.2)

• lpm.m

8
Minimum Cycle Mean Algorithm

1. Create a new Graph Gd from the given DFG G. Each
   node of Gd is a delay element in G. Edge from
   node i to node j in Gd is the longest path from
   delay i to delay j in G. So Gd can be constructed
   from non-negative entries of L(1).
2. Compute cycle mean of each cycle in Gd. A cycle
   mean is the average lengths of all eges in a
   cycle
3. Maximum cycle mean of Gd is the maximum cycle
   (loop) bound of all loops in G and hence is the
   iteration bound.

• Algorithm
• Compute Gc whose edge is negative of that of G.
• Choose an arbitrary node s, such that fs(0)0,
  fk(0) ?.
• Iterations
• iteration bound

mcm.m
9
Example (Figure 2.4)
10
LPM and MCM for Fig. 2.4

• LPM Method

• MCM Method
• i-m matrix
• T? 8

m0 m1
i1 -12/2 -8/1 -6
i2 -? -8/1 -8