1 l25 : crosstalk-concerned physical design (2) 1999. 10 jun dong cho sungkyunkwan univ. dept. ece...
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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 -
7
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
8
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
9
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
10
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
11
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
12
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
13
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.
14
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
16
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
17
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.
18
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
19
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 %
20
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
21
Experimental Result : Deutsch’s Difficult Routing Problem
22
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
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