longest path matrix algorithm

NCU EE -- DSP VLSI Design. Chap. 2 Tsung-Han Tsai 1

Chapter 2 Iteration Bound


Iteration Bound

&Introduction&Critical path&Loop Boundt Important Definitions and Examples

&Iteration Boundt Important Definitions and Examplest Techniques to Compute Iteration Bound


Critical Path&The path with longest computation time and zero delay.& the minimal computation time required for one iteration of

DFG&Speed of the DSP system:t depends on the “critical path comp. time”

&The loop with the maximum loop bound


Loop Bound&The lower bound on the loop computation time

loop)ofdelaysofnumber(n time)computatioloop(

BoundLoop==

=l

l

wt


Loop Bound (cont’d)

& If no delay element in the loop, thent Delay-free loops are non-computable, see the example


Iteration Bound&The critical loop is the loop with maximum loop bound which

is the bound for the DSP program

& Iteration bound= Max { 6/2 , 11/1 }=11


Recursive&A non-recursive DFG contains no loops&A recursive DFG at least one loop T∞

&A recursive DFG has a fundamental limit on the speed =Iteration Bound

& Iteration bound= Max { 4/2 , 5/3, 5/4 }=2


Iteration Bound (cont’d)

&Algorithms to compute iteration boundt Longest Path Matrix (LPM)tMinimum Cycle Mean (MCM)


Algorithms to Compute Iteration Bound

& Longest Path Matrix Algorithm (LPM)t d: # of delays (=4 here )t Compute L(1)~L(4)

t l, i, j: The longest computation time of all paths from delay element dito delay element dj that pass through exactly m-1 delays

t K: int in [1~d]t


Examples for LPM1


Examples for LPM (cont’d)

•Example 2:

}2

16,

212

,18

,14

max{

16161212

8844

2

)2()1(

=

=

=

=

∞T

LL

d


Algorithms to Compute Iteration Bound

&Minimum Cycle Mean Method (MCM)1. Construct the new graph Gd and Gd

’

Ø transform from DFGØ decide the weight of each edge

2. Compute the maximum cycle meanØ construct the series of d+1 vectors f(m)

Ø find the max cycle mean

3. find the min cycle mean between each cycle


& delay ⇒ node& longest path length (computation time) ⇒ weight w(i,j)

Example to MCM


Example to MCM (cont’d)

& longest path lengtht path which pass through no delayst longest : two loops that contain Da and DB

Ø max { 6,4 } = 6

t cycle mean = 6/2=3



&Cycle mean= average length of the edge in c (cycle = loop )

&We need the maximum one

Gincyclesin thesedelaysofnumberthecindelaysecontain thG thatincyclesallofn timecomputatiomax



&Compute the cycle meant weights of the edges x-1

& hence called MCM



&We will find d+1 vectors , f(m) m=0, 1,~, d ( dimension = d×1 )

&s=1

⇒

∞∞∞

=

0

)0(f



&

& I ={ i : path from node i to j exists }

∞∞∞

=

0

)0(f

∞∞

∞

=0)1(f

∞

∞−

=0

4

)2(f

∞−−

=

0

45

)3(f

∞−−−

=458

)4(f



&The iteration bound is given by:




Multirate DFGs& Iteration bound of Multirate Data-flow Graphst Construct a SRDFG that is equivalent to the MRDFGt Compute the iteration Bound of the equivalent SRDFG using the LPM

algorithm

Ø OUV ,IUV : # of samples I/O of the node per invocationØ iUV : delayØ kU,kV : # of invocation per iteration

vUVUUV klkO =


Transformation of Multi Rate DFG


Multi Rate DFG to Single Rate DFG


Conclusion&When the DFG is recursive, the iteration bound is the

fundamental limit on the minimum sample period of ahardware implementation of the DSP program.

&Two algorithms to compute iteration bound, LPM and MCM,were introduced.

&The iteration bound of a multirate DFG can also be determined.

longest path matrix algorithm

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