1

L25 : Crosstalk-ConcernedPhysical Design (2)

1999. 10 Jun Dong Cho

Sungkyunkwan Univ. Dept. ECEE-Mail : Jdcho@skku.ac.kr

Homepage : vada.skku.ac.kr

2

Min-Crosstalk Top Down Global Routing Algorithm(1)

1

1

2

2

L 2

W 2 L 1

W 1

c ro ss ta lk -c ritic a lreg io n s

Crosstalk-Critical Region : The region disturbed by crosstalk between two wires

Crosstalk generated between random signal net i and j is

m

i i

iij W

Lc

1

m = number of crosstalk-critical regioncrosstalk between two net performed by global routing

3

Min-Crosstalk Top Down Global Routing Algorithm(2)

We decide the routing pattern by the position of net that meets design specification. First, whole chip is divided in 4 plane, performs routing by determined routing pattern. Then, performs dividing previous divided plane into 4 plane, and this process performs recursively.

Channel Density : the maximum number of wire that passes one channel.

13 24 max1 max 2

max1 12 23 34 41 max 2 12 23 34 41

2max{ , , , }, second max{ , , , }

f f f fUpperbound of Channel density

f f f f f f f f f f

T o p L ev e l R o u tin g S ec o n d L ev e l R o u tin g

C hannel

Q 1Q 2

Q 3 Q 4

ijf

iQ jQ

: the number of net which connects terminals between and .

4

Min-Crosstalk Top Down Global Routing Algorithm(3)

1 2 3 4 5

5'4'3'2'1'

The nodes represents information about routing pattern and channel density of each net.

The nodes positioned vertical lines represent different routing pattern of the same net.

We define the information of node as follows.

Graph contains Routing pattern, Channel density and Information on crosstalk

1

d

ijj

i

c

xd

: ij i jwhere c crosstalk between node n and n

in

ijc

d(degree) : the number of node that is not for random node .

The edge represent crosstalk between two nodes, and we consider the crosstalk is 0 when the distance of nets is greater than .

jn

2

5

Min-Crosstalk Top Down Global Routing Algorithm(4)

G = ( V, E ), V = nodes, E = edges;

STEP 1 : sorts the crosstalk between node and in ascending order, construct set Z and X.

STEP 2 : compute for each net.

Z={(n i,n j) | c ij = 0 }

HighLow ...X1

C = sorted list of C ij

X0

X

X log | X|

X2

STEP 3 : choose nodes that has smaller in vertical lines and compute total crosstalk and channel density

STEP 4 : Reconstruct graph

STEP 5 : Iterate STEP 2 ~ STEP 4 until and are equal.

STEP 6 : Choose final result that has minimum crosstalk and meets channel density performed STEP 3.

in jn

;0

;2

,log,,1,0

;

}0),(),({

};0),(),({

1

log210

k

XXXk

XXXXX

candEnnnnX

candEnnnnZ

kk

Xk

ijjiji

ijjiji

ix

,1

d

cx

d

jij

i

;Viallfor

ix

G

;1

;)( '

kk

XZGE k

)(GE

)(GE

6

Min-Crosstalk Top Down Global Routing Algorithm(5)

Experimental Result

Design

APTE

XEROX

AMI49

HP

5773.053

11795.41

48524.73

62.993

21459.81

12987.79

52357.57

57.943

271

10.1

7.9

8.02

13202.77

11728.08

52357.57

56.686

62.5

10.7

0

2.21

23

25

91

7

32

26

95

7

22

24

91

7

23

25

91

7

[1] ours%

diffOptimal

%err

Eq(1)

[7] oursOpt-imal

Crosstalk Channel Density

Design

APTE

XEROX

AMI49

HP

5773.053

11795.41

48524.73

62.993

3799.95

11541.41

48345.79

57.943

34.2

2.14

0.37

8.01

2865.16

11518.16

48345.79

56.686

32.6

0.21

0

2.21

23

25

91

7

32

26

95

7

31

26

94

7

30

26

94

7

[1] ours%

diffOptimal

%err

Eq(1)

[7] oursOpt-imal

Crosstalk Channel Density

<Table 1> Considering Channel Density

<Table 2> Not Considering Channel Density

# of 2-terminal

nets45

71

232

17

# of 2-terminal

nets45

71

232

17

Average - - 31.3 - 12.3 -

Average - - 3.64 - 1.52 -

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An Optimal Track Assignment considering Crosstalk and Power Dissipation

Crosstalk cost-function Where is signal sensitivity between net i,j is overlapped length between net i,j is width between net i,j)(*

ij

ijijij W

LAc

ijA

ijL

ijW

inet of ecapacitanc substrate theis

crosstalk its todue inet of ecapacitanc theis

, where

)(**5.0**5.022

s

x

sxw

wgcycleddlcycleddav

C

C

CCC

NCCTNCTP VV

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Problem Formulation

For Mapping Order for set S T,

)1( subject to minimal, is ),(

ji

ji

ss

Sss

ijw

ijjiijij ANNXw *2),max(* is where

jnet and inet of wireshorizontalbetween distance separative

)(j,net and inet of wireshorizontalbetween length coupled

jnet and inet between noise coupled/

jnet and inet between y sensitivit signal

j)(net inet ofactivitiy switching )(

ij

sxijij

ijijij

ij

ji

D

CCLL

DLX

A

NN

},,,{)( 21 nsssS },,,{ 21 nsssS },,,{ 21 ntttT

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Previous Approach

Track assignment problem is similar to Traveling Salesman Problem(TSP) in general graph algorithm

TSP problem is known as NP-Complete. Brute-Force algorithm :

Single interval clique : Continuous interval clique(k interval) :

Dynamic Programming (greedy approach): In General Cases, Heuristic approach is used.

Proposed Algorithm Single interval clique : Find optimal solution in Continuous interval clique: Propose Heuristic algorithm in

)!(nO

)2( 2 nnO

)log( nnO

)!( knO

)log( nnO

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Special Case I : Containment Interval Clique

The shape of Interval Clique Set is Containment :

We can find mapping order that has minimum crosstalk in

},,,{ 21 nsssS

)log( nnO

Vertical Cut

SS

SS

LL

LL

LL

SS

SS

LL

LL

LL

Containment Interval Clique

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Special Case II : Monotone Interval Clique

The shape of Interval Clique Set is Monotone :

We can find interval mapping order that has minimum crosstalk in

)log( nnO

},,,{ 21 nsssS

Vertical Cut

Monotone Interval Clique

S L

S L

SL

SL

S L

S L

SL

SL

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General Case II : Algorithm 3

Theorem : All Interval Set S consists of Containment

interval clique set and Monotone interval clique set, so

we use below algorithm

< Algorithm 3>Step 1 : Clique-Partition ( )

Step 2 : Apply Algorithm1( ) and Algorithm2( )

Step 3 : Merge_Clique ( )

SSS ,;

SSS ;,

S S

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The case of Single interval clique

(a) Case A - 1 (b) Case A - 2

ll

ss

l l

ls

l s

ss

l l

ll

ss

l l

ls

l s

ss

l l

T1 T1

T2 T2

T1 T1

S3

S4

S2

S1 S1

S2

S3

S4

l l

(c) Case B

ll

s s

sl

s l

T1s s

T1

T2

S1

S2

S3

S4

l : Longs : Short

Procedure Merge_Clique process is only available as below three process.

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The case of Single interval clique : In general case

Conclusion : Using Algorithm 3, We can find interval mapping order that have minimum crosstalk for Single interval clique in general case. In this case computational complexity is

)log( nnO

15

Vertical Crosstalk Crosstalk occurs not only horizontal wires but also occurs vertica

l wires that exist channel Crosstalk by vertical wires has less size than horizontal wire We can find the LONG-SHORT arrangement order by the method

of horizontal wires

l : Longs : Short

l

s

l

s

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Example of Single interval clique : Sepcial case Track no. is 4

Using 45O wire pattern, we can find interval mapping order that has minimum crosstalk for the case that track number is 4.

(a) Long-Long-Short-Short displacement

LONG

SHORT

LONG

SHORT

LONG

LONG

SHORT

SHORT

Crosstalk

(b) After using 45 0 wire

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Continuous interval clique

We can account track assignment problem in general cases of channel routing as track assignment problem of several numbers of divided sub-channel.

We can consider the solution of track assignment problem in general cases of channel routing as Minimization problem of number of LONG-LONG-LONG triple existed in total sub-channel.

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Continuous interval clique

Algorithm 4 [ time ] Step 1 : run wirelength-based left-edge algorithm and Interval cli

que partitioning [ time] Step 2 : interval type definition (LONG,SHORT)[ time] Step 3 : find maximum LONG-SHORT ordered interval pair by usi

ng maximum-edge weight matching [ time] Step 4 : make subchannel that have minimum LONG-LONG-LON

G ordered interval triple by using minimum-edge weight matching [

)log( nnO

)( 3mO

)log( nnO

)( 3mO

)log()()log( 3 nnOmOnnO

)( 3 nm

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Experimental Result : Single interval clique

Crosstalk Avg. Length Avg. Power Avg.

MaxTrack

L.E. Our Ratio L.E. Our Ratio L.E. Our Ratio

9 55 34 0.618 167 148 0.886 279 216 0.774

11 86 53 0.616 249 217 0.871 422 309 0.732

13 123 76 0.618 346 304 0.879 594 420 0.707

17 218 134 0.615 589 507 0.860 1025 748 0.729

21 339 207 0.610 895 751 0.839 1573 1168 0.742

Avg.(%)

38.5 % 13.3 % 26.3 %

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Experimental Result : Continuous interval clique

Left-edge Algorithm (wire length-based)

Our Algorithm Brute-force method Benchmark circuit

(# track/# net) Crosstalk Execute Time(sec)

Crosstalk/# Clique

Execute Time(sec)

Crosstalk Execute Time(sec)

Deu(19/72) 1910 0.17 1715/13 0.44 N/A -

Deu1(16/48) 813 0.11 712 /11 0.27 N/A -

Deu2(14/24) 144 0.11 112 / 9 0.22 N/A -

Deu3(16/47) 1273 0.16 1235/ 9 0.24 N/A -

Deu4(8/28) 123 0.08 105 / 5 0.13 N/A -

Deu5(4/6) 15 0.06 11 / 2 0.06 13 6.64

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Experimental Result : Deutsch’s Difficult Routing Problem

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References and Suggested Readings [1] Currie M, Sobolewski R, Hsiang TY. High-frequency crosstalk in superconducting microstrip waveguide i

nterconnects. IEEE Transactions on Applied Superconductivity, V.9 N.2 P.3, 3602-3605, 1999 [2] Chou M, White JK. EFFICIENT FORMULATION AND MODEL-ORDER REDUCTION FOR THE TRANSIENT SIM

ULATION OF THREE-DIMENSIONAL VLSI INTERCONNECT, IEEE Transactions on Computer-Aided Design of Integrated Circuits & Systems, V.16 N.12, 1454-1476, 1997

[3] Vittal A, Mareksadowska M. CROSSTALK REDUCTION FOR VLSI. IEEE Transactions on Computer-Aided Design of Integrated Circuits & Systems, V.16 N.3, 290-29856, 1997.

[4] Yen-Tai Lai, Chi-Chou Kao, Wu-Chien Shieh. A Quadratic Programming Method for Interconnection Crosstalk Minimization. Proceedings of the 1999 IEEE International Symposium on Circuits and Systems -, 270-273, 1999

[5] Zemo Yang, Samiha Mourad. Deep Submicron On-chip Crosstalk. Proceedings of the 16th IEEE Instrumentation and Measurement Technology, 1788-1793, 1999

[6] Lee, Mankoo. Fringing and coupling interconnect line capacitance model for VLSI on-chip. Proceedings of the IEEE International Symposium on Circuits and Systems, 1996

[7] Hai Zhou and D.F.Wong. Crosstalk-Constrained maze Routing Basd on lagrangian Relaxation. Proceedings of the 1997 IEEE International Conference on Computer Desin : VLSI, 1997

[8] Prashant Saxena, C. L. Liu. Crosstalk Minimization using Wire Perturbation. In Proc. Design Automation Conference, 1999

[9] Hai Zhou, D. F. Wong. Global Routing with Crosstalk Contstraints , In Proc. Design Automation Conference, 1998

[10] Hsiao-Ping Tseng, Louis Scheffer, Carl Sechen, Timing and Crosstalk Driven Area Routing,In Proc. Design Automation Conference, 1999

[11] Tilmann Stohr, Markus Alt, Asmus Hetzel, Jurgen Koehl, Analysis, Reduction and Avoidance of Crosstalk on VLSI Chips, International Symposium on Physical Design, 1998