Statistical Clock Tree Routing for Robustness to Process Variations - - PowerPoint PPT Presentation

Statistical Clock Tree Routing for Robustness to Process Variations

Uday Padmanabhan1, Janet M. Wang1 and Jiang Hu2

1University of Arizona at Tucson 2Texas A&M University

Outline

Previous research Motivation of this project Variation aware delay model Statistical centering algorithm Examples Conclusion

Clock Tree Routing

Previous Research

Layout embedding method to achieve

zero skew

Deferred Merging Embedding technique

(DME)

Nearest neighbor based abstract tree

method (NNA)

Bounded skew tree (BST) algorithm Minimal Skew Violations (MinSV)

Permissible Range

Align the mean to the center

Skew and Permissible Range

Basic Concepts

Existing Issues

Most of the research works are for

deterministic clock tree routing

Some research works consider corner

cases

not work well with statistical clock skew

Possible Strategies

Skew can be non-Gaussian distributed:

asymmetric

Only mean and variance are not enough Our methodology:

include high order moments center measures include: Mode, Median and

Mean

A Task List

Variation aware delay model Parameter reduction based on ANOVA

and OPCA

Choose among Mean, Median and Mode

as Central Measures

Select wire length based on Central

Measures

Variation aware abstract tree topology

Delay model

( ) 2 ( ) 2 ( )

k i k i k i

FED k d d i i d f i d j i T i T j S i i i a j j i P j E i i i i f j i P j E i i i j i P j S i i

d Ar c l w Br c l C r cL l l w D c l w w t l l E c l w t l F cL w t ρ ρ ρ

∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

= + + + + + + +

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

Delay Mean and variance

, , , 2 3 2 2 3

| 1 1 [( 2 ) 2 1 ( 2 ) ]

w t d rd L cL

T FED w t r c f L w f L a t

d c l lc E F t t c l lc D c l E F w w

µ µ µ µ

µ ρ ρ µ ρ ρ ρ µ

= = = =

= + + + + + +

2 2 2 2 2 2 2 2 2

( ) ( ) ( ) ( )

F E D F E D T r d c L d L F E D F E D w t

d d r c d d w t σ σ σ σ σ ∂ ∂ + ∂ ∂ ∂ ∂ + + ∂ ∂ ฀

First Order Estimates

Delay Mean and Variance

il

Til Tk k P

µ µ

∈

= ∑

2 2

il

Til Tk k P

σ σ

∈

= ∑

Slr Til Tjr

µ µ µ = −

2 2 2 Slr Til Tjr

σ σ σ = +

S1 S2 S3 S4 n5 n6 n7

Path

Delay Median

Edge Delay Median: Path Delay Median: Skew Median:

Delay Mode

Matching the delay mode by Weibul function:

ANOVA

1, 7 1 5 5 7

1 1 2 2 , 1 1 , 2 2

( , , , ) ( , ) ( , )

s n

P s n n n

w t w t w t w t µ µ µ = +

S1 S2 S3 S4 n5 n6 n7

Path

1, 7 5, 7 1 5 5 7

1 1 , 1 1 ,

( , , ) ( , )

s n

P n n s n n n

w t w t µ µ µ µ = +

Statistical Centering Based Layout Embedding

Statistical Centering Based Layout Embedding

Mean, Median and Mode based design

where

Variation Aware Abstract Tree Topology

Extend the Nearest Neighbor Algorithm

(NNA)

Length is positive dependent on the

distance between nodes

The minimal composite distance

Process variation model

Experimental Results

Experimental Results

Experimental Results

Run Time Comparison

Conclusions

A Statistical centering approach A Fitted Elmore delay model with

analysis of variance and principle component analysis

A topology generation algorithm which

takes the width and thickness into consideration