Wirelength Estimation based on Rent Exponents of Partitioning and Placement 1 Xiaojian Yang, Elaheh Bozorgzadeh, and Majid Sarrafzadeh Synplicity Inc. Sunnyvale, CA 94086 [email protected]Computer Science Department University of California at Los Angeles Los Angeles, CA 90095 elib,[email protected]Abstract Wirelength estimation is one of the most important Rent’s rule applica- tions. Traditionally, the Rent exponent is extracted using recursive bipar- titioning. However, the obtained exponent may not be appropriate for the purpose of wirelength estimation. In this paper, we propose the concepts of partitioning-based Rent exponent and placement-based Rent exponent. The relationship between these two exponents is analyzed and empirically verified. Experiments on large industrial circuits show that for wirelength estimation, the Rent exponent extracted from placement is more appropriate than that from partitioning. 1 This work was supported by NSF under Grant #CCR-0090203. A preliminary version of this paper appeared in Proc. Int. Workshop on System-Level Interconnect Prediction, pp.25-31, April 2001. 1
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Wirelength Estimation based on Rent Exponents ofPartitioning and Placement 1
Xiaojian Yang, Elaheh Bozorgzadeh, and Majid Sarrafzadeh
Wirelength estimation is one of the most important Rent’s rule applica-tions. Traditionally, the Rent exponent is extracted using recursive bipar-titioning. However, the obtained exponent may not be appropriate for thepurpose of wirelength estimation. In this paper, we propose the conceptsof partitioning-based Rent exponent and placement-based Rent exponent.The relationship between these two exponents is analyzed and empiricallyverified. Experiments on large industrial circuits show that for wirelengthestimation, the Rent exponent extracted from placement is more appropriatethan that from partitioning.
1This work was supported by NSF under Grant #CCR-0090203. A preliminary version of thispaper appeared inProc. Int. Workshop on System-Level Interconnect Prediction, pp.25-31, April2001.
1
1 Introduction
Rent’s rule was first described by Landman and Russo in 1971 [1]. It relates the
number of external connections and the number of cells for a given block in a
partitioned circuit. Rent’s rule has been observed on many real designs. It has ex-
tensive applications in VLSI design. A priori wirelength estimation is one of the
most important applications of Rent’s rule. The classical work [2, 3] gives good
estimates for post layout interconnect wirelength. More recent work improves
the estimation by consideringoccupying probability [4] or recursively applying
Rent’s rule throughout an entire monolithic system [5]. Extension of basic wire-
length estimation, including routing utilization estimation [6], congestion estima-
Algorithm 1 Extract-Rent-by-Partitioning(C)Input: Circuit C � �V�E�Output: Rent exponentp
1. Recursively bipartition the original circuits. At each recursive level, calculate the av-erage number of cells per partition and the average number of external nets over allpartitions. Save the data pair to�Gi�Ti� wherei is the depth of recursive partitioning.Partitioning stops when reaching a given depthn.
2. Apply linear regression on the log-log scaled data pairs:�Gk�Tk���Gk�1�Tk�1�� �����Gn�Tn� (k is a given number around 4-6)
3. Return the slope of the fitted line by linear regression.
In the first method Extract-Rent-by-Partitioning, a partitioning algorithm is
used to recursively bisection the original circuits. At each bisection level, aver-
age number of cells and average number of external nets for all subcircuits are
Place the circuit on two dimensional plane,for i� 1 to a given depthn do
Divide the core area into 2i regular regions;Each region contains a group of cells; Compute the average number of cells per groupand the average number of external nets over all cell groups.Save the data pair to�Gi�Ti�.
end forApply linear regression on the log-log scaled data pairs:�G k�Tk���Gk�1�Tk�1�� �����Gn�Tn� (k isa given number around 4-6 to skip Region II)Return the slope of the fitted line by linear regression
recorded. This pair of numbers form a point on a log-log plane. After achieving
enough points, a linear regression is performed to obtain the Rent exponent.
To extract the Rent exponent from placement, we first place the circuit using
existing placement tools. Then we divide the layout area into several regions and
analyze the subcircuit in each region. The average number of cells and average
number of external nets for all regions are recorded. This dividing step continues
to a given depth. Then we obtain the Rent exponent by linear regression on the
recorded points.
A detailed step of implementingExtract-Rent-by-Partitioning is as follows:
when a subcircuit is partitioned into two smaller subcircuits, the nets which con-
nect the outside cells are not considered. For multi-terminal nets, part of the net
will be reserved and the external pins are ignored.
We define the terms for partitioning-based Rent exponent and placement-based
Rent exponent:
Definition 1 For a given circuit and a bipartition approach, the partitioning Rent
exponent p is the output of the algorithm Extract-Rent-by-Partitioning().
Definition 2 For a given circuit and a wirelength optimized placement of the cir-
cuit, the placement Rent exponent p� is the output of the algorithm Extract-Rent-
by-Placement().
5
2.2 Relationship between Exponents
Since partitioning and placement are related problems, the partitioning Rent ex-
ponent and placement Rent exponent might also be related. Partitioning tends
to minimize the number of cut nets for two subcircuits, which in turn leads to a
small number of external nets for a subcircuit. While in a wirelength driven place-
ment, the cells which are tightly connected are placed closer. There is no effort on
reducing the crossing nets between two regions.
As shown in Figure 2, for a given subcircuit with sizeG1, the number of
external nets in placement is larger than that in partitioning. Two straight lines
represent linear regression results for partitioning and placement. Both lines share
the samey-intercept because the Rent coefficientt is fixed for a given circuit.
Therefore the slope of the line which is obtained by partitioning is smaller than
the slope of the other line, which is done by placement.
p � p�
log Glog G1
log t
T = t G
T = t G
p’
p
Placement Rent’s curve
Partitioning Rent’s curve
log T
Figure 2: Comparison between partitioning Rent exponent and placement Rentexponent
If the placement engine is a min-cut class approach, we can derive a relation-
ship between the two Rent exponents. Figure 3 illustrates two different biparti-
6
tioning problems. In figure 3(a), the partitioner only considers the interconnects
between cells of the subcircuit to be partitioned. We call this problempure bi-
partitioning problem. In Figure 3(b), external nets, which connect cells of this
subcircuit to other subcircuits, are also included into partitioning problem. This
is the bipartitioning problem withterminal propagation, which is normally used
in min-cut class placement tools, as shown in Figure 3(c). It is the difference be-
tween these two bipartitioning approaches which explains the difference between
partitioning Rent exponent and placement Rent exponent.
In the pure bipartitioning problem without terminal propagation, assuming the
sizes of the subcircuits after partitioning areG1 andG2. Let C be the number of
cut nets (figure 3(a)). For the bipartitioning process with terminal propagation, let
C� be the number of cut nets of bipartitioning. We haveC � � C because of the
effect of the external nets. According to Rent’s rule, from Figure 3(a), we obtain:
T1�C � T � tGp1� (1)
whereT is the total number of external nets for subcircuitG1. T1 is the number of
the external nets which arenot cut nets.t is Rent coefficient, the average number
of pins per cell.p is the partitioning Rent exponent.
We assume that all the nets are two-terminal nets. Applying Rent’s rule on the
original subcircuit before partitioning, we obtain:
T1�T2 � t�G1�G2�p (2)
For simplicity, we assume that in a balanced bipartitioning,G1 � G2 andT1 �
T2. From equation (1) and (2), we have:
T1 � 2p�1T
In the bipartitioning with terminal propagation (Figure 3(b)), there areT1 ex-
ternal nets connected to other subcircuits. These nets connect to cells that are
located either to the left or to the right of original circuit. The external nets con-
nected to the right side (T1�2 nets) will “drag” cells from left to right, thus they
7
G1 G2
C’
(a) (b)
G1 G2
C
2T’1
GT1 T2T’
(c)
Figure 3: Comparison between a pure partitioning (a) and a partitioning withterminal propagation in min-cut placement (b), (c). The former only considers theinternal nets, while the latter considers both internal nets and external nets.T1 andT2 are the number of external nets which are not cut nets for subcircuitG1 andG2,respectively.
8
may increase the cut nets of the partitioning. We assume that one such external
net increases the number of cut nets byα. α is a real number between 0 and 1. It
represents the possibility that an external net increases the number of cut nets by
one.
The same situation exists on the right subcircuit. Thus the result of partitioning
with terminal propagation will increase byαT1. Therefore for a partitioned subcir-
cuit, the number of external netsT � after terminal propagation based partitioning
is:
T � � T �αT1 � �1�α �2p�1�T
SinceT � tGp1 and T � � tGp�
1 (p and p� are partitioning Rent exponent and
placement Rent exponent, respectively), we have,
p�� p �logT �� logt
logG1� logT � logt
logG1
�log�T ��T �
logG1
�log�1�α �2p�1�
logG1
Thus we have,
p� � p�log�1�α �2p�1�
logG1(3)
whereG1 should be the number of cells in a subcircuit which corresponds to a
data point. In practice we setG1 to be�V ��25 to avoid the Rent’s rule region II2.
Equation (3) shows that the placement Rent exponent (p�) is larger than the
partitioning Rent exponent (p). It should be noted that the analysis is based on
some simplifications (e.g. two-teminal nets). The valid range of Equation (3)
is limited. For example, ifp is close either 0 or 1, the equation does not give
meaningful result. However, for ordinary circuits and ordinary partitioning Rent
exponents, this equation approximately derives a placement Rent exponent which
can be used for certain estimation purposes.2Region II corresponds to a few top-most levels of the partitioning or placement where the
number of cells and the number of external nets do not follow the Rent’s rule.
9
3 Experimental Validation
Equation (3) shows that we can derive placement Rent exponentp� from the par-
titioning Rent exponentp. The following experiments are conducted to evaluate
the relationship.
3.1 Derivation of placement Rent exponent
We experimentally extract both partitioning exponent and placement exponent for
a set of circuits. The circuits are chosen from MCNC and IBM-PLACE bench-
mark suits. IBM-PLACE benchmarks are obtained by modifying ISPD98 IBM
partitioning benchmark suits [21]. Experimental circuit sizes range from 21,000
cells to 210,000 cells. For partitioning Rent exponent, we use hMetis [22] as
the partitioning tool. Unbalance factor is set to 1% in each bipartitioning call.
For placement Rent exponent, three different placement tools are used to place
the circuit and placement Rent exponents are extracted from the placed circuits.
The placement tools used in this work areCapo [18], Feng Shui [19] andDragon
[20]. All of them are recent academic works and they all integrate multi-level hy-
pergraph partitioning, a breakthrough technique in VLSI/CAD partitioning prob-
lem. Capo and Feng Shui use recursively bipartitioning approach followed by
local improvement.Dragon employs both cut and wirelength minimization in hi-
erarchical placement flow. All experiments are performed on Sun workstations
with 400MHz CPU and 128M memory. The depths of both Extract-Rent-by-
partitioning and Extract-Rent-by-placement are set to be 14, i.e., 14 data points
are collected from partitioning or placement to do linear regression. The first 5
points are discarded in order to avoid effects caused by Rent’s rule region II. Thus
the linear regression is actually carried out on 9 data points for each circuit.
Figure 4 shows a sample extraction on ibm15 circuit. The lower line is the re-
sult of linear regression on data points collected by recursive bipartitioning. Three
upper lines are obtained from placement outputs byCapo, Feng Shui andDragon.
All the slopes of three upper lines are larger than the slope of the partitioning line,
10
2
3
4
5
6
7
8
9
10
2 4 6 8 10 12
log G
log
T
points extracted from recursivebipartitioning
points extracted from Capoplacement
points extracted from FengShui placement
points extracted from Dragonplacement
fitted line for partitioning
fitted line for Capo placement
fitted line for Feng Shuiplacement
fitted line for Dragonplacement
Figure 4: Rent’s rule fitted line based on partitioning and placement for bench-mark ibm15. The lower line is the result of linear regression on data points fromrecursive bipartitioning. Three upper lines are from placement outputs.
11
supporting the relationship between partitioning Rent exponent and placement
Rent exponent discussed in Section 2.
Table 1 shows the comparison between partitioning Rent exponentp, derived
placement Rent exponentp� which is obtained from Equation (3)3, and three real
placement Rent exponentsp�� extracted from outputs of three different placement
tools. Note that the Rent exponents produced by different placement tools are
not the same. However, they do not vary much for a given circuit. Comparing
with partitioning Rent exponentp, derived placement Rent exponentp� is closer
to real placement Rent exponents, partially supporting the theoretical relationship
between two Rent exponents. However, better derivation of placement Rent ex-
ponent requires the knowledge ofα in Equation (3).
Table 1: Comparison between partitioning Rent exponentp, derived placementRent exponentp� and real placement Rent exponentp�� extracted from three place-ment tools’ outputs
3.2 Range of α
In the above experiments we setα to be 1, which leads to a simplified model.
However, as defined in Section 2.2,α is a coefficient that indicates the effect of3We setα � 1 in experiments.
12
the external nets in partitioning. The larger this coefficient, the more cut nets
appear in partitioning with terminal propagation, the larger difference between
partitioning Rent exponent and placement Rent exponent.
Theoretically,α is a number between 0 and 1. the value ofα varies for differ-
ent circuits. For a given circuit, if we gradually increaseα from 0 to 1, we obtain
different placement Rent exponent based on Equation (3). Figure 5 illustrates an
example ofα’s effect for circuit ibm15. The solid curve in Figure 5 shows the
change of derived placement Rent exponent asα increases. The dashed line rep-
resents the average placement Rent exponent of three different placement Rent
exponents extracted by three placers. The intersection of the solid and the dashed
line corresponds toα � 0�65. This value is called the expected value ofα. It
means that if we setα in Equation (3) to be this value, the derived placement Rent
exponent is close to the real exponent extracted from placement outputs.
Applying the same approach on other circuits, we obtain the expected value
of α for every circuit. Table 2 shows the average placement Rent exponent and
the expectedα for all of 8 IBM-PLACE circuits. Expectedα varies for differ-
ent circuits, ranging from 0.38 to 0.98. In general, larger circuits tend to have a
smaller expectedα. How to obtain a properα is a non-trivial problem. There
could be multiple factors that affect expectedα, including percentage of multi-
terminal nets, quality of partitioning approach and the Rent coefficient (t). In the
following sections we still setα to be 1 for simplicity.
4 Wirelength Estimation
In Section 3 we have shown the difference between the partitioning Rent exponent
and the placement Rent exponent. In wirelength estimation, the total wirelength
or the average wirelength is a function of the Rent exponent. Different Rent expo-
nents can lead to different wirelength estimates. In order to obtain more accurate
wirelength estimates, aproper Rent exponent is required.
13
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.55
0.6
0.65
0.7p’ as a function of α for circuit ibm15
α
Ren
t exp
onen
tderived p’ as α changes average p’ from three placement outputs
Figure 5: Derived placement Rent exponent p’ as a function ofα (the solid curve).The dashed line reprensents the average placement Rent exponent of three differ-ent placement Rent exponents extracted by three placers. The intersection of solidand dashed lines corresponds toα � 0�65.
Table 2: Partitioning Rent exponent, placement Rent exponent derived from Equa-tion, average placement Rent exponent by three placers, and the expectedα com-puted by these exponents.
14
4.1 Different Rent Exponents in Estimation
The authors in [11] show that the Rent exponent of a circuit depends on the par-
titioning approach from which it is derived. Similar situation exists in extracting
placement Rent exponent. If we use different placement algorithms, we will ob-
tain different placement Rent exponents. Likewise, it is expected that the place-
ment Rent exponents do not have much variation from different placement algo-
rithms.
In the wirelength estimation work [2, 4], the authors adopt a hierarchical place-
ment model and assume that Rent’s rule holds for all subcircuits at each hierar-
chical level. In [5] the wirelength distribution is derived from the number of in-
terconnects between gates that are a given distance away. In these approaches, the
partitioning Rent exponent and the placement Rent exponent are not distinguished
from each other. By definition, wirelength estimation requires the placement Rent
exponent. In the real world, however, wirelength is often estimated using par-
titioning Rent exponent since it can be obtained easily. In general, wirelength
estimates using the partitioning Rent exponent tend to under-estimate the total
wirelength. This can be observed by the following experiments.
In Section 3 we have obtained the partitioning Rent exponent and three place-
ment Rent exponents for each circuit. With these exponents, we estimate the total
wirelength based on existing wirelength distribution models. Both classic Do-
nath’s method [2] and the recent Davis’s distribution model [5]4 are used in this
work.
The estimation results are compared with real wirelength given by the global
router. Since we have three placement outputs, we also have three corresponding
global routing results. For simplicity, the number of rows in standard cell place-
ment is set to be the power of 2 (128 in the experiments). We also assume that the
grid in global routing is a square with unit width and unit height. For better com-
parison, the estimated total wirelength is scaled to the length in terms of global
routing grid units. Specifically, if the number of cells in a circuit isG, and the4We refer it as Davis’s model while the authors of [5] are J. A. Davis, V. K. De and J. Meindl.
15
global routing grid isn�n, then the scaled estimated wirelengthWL is,
WL �WL�n�G
whereWL� is the estimated wirelength.
Table 3 shows a comparison between the estimated wirelength and real wire-
length after global routing. For each circuit, two estimation methods (Donath’s
and Davis’s) are used on four Rent exponents (one partitioning Rent exponent and
three placement Rent exponents). Placements of circuit are obtained using three
different placement tools. For each placement output the corresponding global
routing result is reported.
It is generally believed that Donath’s classic work over-estimates the total
wirelength for most circuits. Therefore we focus on wirelength estimates by
Davis’s wirelength distribution model. From Table 3 we observe that the wire-
length estimates based on the partitioning Rent exponent are always smaller then
the real wirelength. While wirelength estimates based on the placement Rent
exponents are closer to the real results. This observation supports the previous as-
sumption that wirelength estimation should be based on placement Rent exponent
rather than partitioning Rent exponent.
4.2 Using Derived Placement Rent Exponent
The fact that placement Rent exponent is more appropriate suggests a new wire-
length estimation approach, as shown in Figure 6. For a given circuit, we first
extract its partitioning Rent exponent using traditional recursively bipartitioning.
Then placement Rent exponent is derived by the relationship between two ex-
ponents, which was discussed in Section 2.2. Now we can estimate wirelength
using existing models and derived placement Rent exponent. The motivation is to
exploit the advantage of partitioning Rent exponent (easy to be obtained), while
avoid its inaccuracy in estimating wirelength.
Table 4 shows the estimated total wirelength based on derived placement Rent
exponent, compared with real wirelength after placement and global routing. For
16
Partitioning Placementckt p est. WL est. WL placement p�� est. WL est. WL real WL
Table 3: Partitioning Rent exponentp and wirelength estimates by two estima-tion methods (Donath’s and Davis’s), comparing with placement exponentp �� bythree different placement tools (Capo, Feng Shui andDragon), and the wirelengthestimates based onp��. The final column is the real wirelength output by globalrouter. Both estimated and real WL (wirelength) are in 103 grid units of globalrouting.
17
BipartitioningPartitioning Rent
Exponent ( p )Recursively
Placement RentExponent ( p" )
Rent ExponentBased on Placement
Wirelength Estimation
WirelengthEstimated
Circuit
Derivation ofPlacement
Rent Exponent
Figure 6: A new approach for wirelength estimation. The difference betweenthis approach and previous ones is that it derives placement Rent exponent frompartitioning Rent exponent, and then uses this derived exponent to do estimation.
most circuits, wirelength estimates based on derived placement Rent exponent are
closer to real wirelength than those based on partitioning Rent exponent.
However, 100% accurate wirelength estimation does not exist. As shown in
Table 3, even the real placement Rent exponent does not always lead to an accurate
wirelength estimate. Wirelength estimates vary with different placement tools.
In addition, parameters in global routing (e.g. routing capacity) also affect total
wirelength. A good wirelength estimate is only meaningful in a given context. In
general there is noperfect wirelength estimation independent of place and route
tool.
4.3 Placement Quality and Rent Exponent
In [11] the Rent exponent is regarded as a metric of quality of partitioning algo-
rithm. It is interesting to know whether there is a similar correlation between the
placement quality and the Rent exponent of placement. Previously the quality of
placement is measured by the total bounding box wirelength or the wirelength
after global routing. Therefore we compare placement wirelength and Rent expo-
nents for different placement tools.
Table 5 lists the Rent exponent, total bounding box wirelength and total routed
18
ckt partitioning derived placement estimated real WL (�103units)Rent exp.p Rent exp.p� WL by p� Capo Feng Shui Dragon
Table 4: Partitioning Rent exponentp, derived placement Rent exponentp� andestimated total wirelength based onp�, comparing with the routed total wirelengthfrom three placement outputs.
wirelength for three placement approaches. For consistency, both bounding box
wirelength and routed wirelength is reported in grid units of global routing. The
global router is based on maze routing including rip-up and re-route. The capacity
of global routing edges is set to a value such that the number of nets which are
ripped-up and re-routed is less than 10% of the total nets. This is to reduce the
influence of the global routing on the placement.
placement Rent exponent total bounding box WL total routed WLckt (�103 grid units) (�103 grid units)
Table 5: Placement Rent exponents derived from layouts by three different place-ment tools, with the normalized total bounding box wirelength and normalizedtotal routed wirelength.
Figure 7 shows the comparison more clearly. For most circuits the smaller
Rent exponent relates to less total wirelength. Some other circuits show the con-
19
trary cases. However, the difference are relatively small in these cases. The cor-
relation exists for both bounding box wirelength and routed wirelength. Thus
we conclude that the Rent exponent of placement is a good metric of placement
quality.
5 Conclusion
Wirelength estimation for large circuits is a complex problem. A number of fac-
tors can affect the accuracy of estimating, including the approach to obtain the
Rent exponent, the placement algorithm used in the design flow and the quality
or parameters of the global router. In order to obtain accurate wirelength esti-
mates, designers ought to adjust estimation model and the Rent exponent extrac-
tion method according to the place and route tool they employ. Precise wirelength
estimation needs extensive experimental data as well as theoretical formulation.
Our work is a step toward understanding this process.
6 Acknowledgments
The authors wish to thank Dr. Dirk Stroobandt for his precious comments.
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22
0.5
0.55
0.6
0.65
0.7
0.75
0.8
ibm11 ibm12 ibm13 ibm14 ibm15 ibm16 ibm17 ibm18
Rent exponents by different placement tools
Ren
t exp
onen
t
Capo Feng ShuiDragon
(a) Placement Rent exponents
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
ibm11 ibm12 ibm13 ibm14 ibm15 ibm16 ibm17 ibm18
Bounding box Wirelength by different placement tools
Wire
leng
th
Capo Feng ShuiDragon
(b) Normalized bounding box wirelengths
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
ibm11 ibm12 ibm13 ibm14 ibm15 ibm16 ibm17 ibm18
Routed Wirelength by different placement tools
Wire
leng
th
Capo Feng ShuiDragon
(c) Normalized routed wirelengths
Figure 7: (a) Placement Rent exponents derived from layouts by three differentplacement tools(Capo, Feng Shui andDragon). (b) Total bounding box wirelengthin grid units by three placement tools. (c) Total routed wirelength in grid units bythree placement tools. In (b) and (c) wirelengths are normalized by dividing theaverage value of three placement tools.23