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Wirelength Estimation based on Rent Exponents ofPartitioning and Placement 1

Xiaojian Yang, Elaheh Bozorgzadeh, and Majid Sarrafzadeh

Synplicity Inc.Sunnyvale, CA 94086

[email protected]

Computer Science DepartmentUniversity of California at Los Angeles

Los Angeles, CA 90095elib,[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 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.

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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-

tion [7], 3-D design performance analysis [8, 9], interconnect fan-out distribution

[10], are also of value for physical design automation tools.

Rent’s rule correlation is commonly presented byT � tGp, whereT andG

are the number of external nets and the number of cells for a block, respectively.

t is often called Rent coefficient, which is the average number of pins per cell.

The Rent exponent,p, is the feature parameter of the circuit. Hagen, et. al.,

studied Rent exponents of circuits by comparing different partitioning approaches

[11]. They proposed theintrinsic Rent exponent which indicates the minimum

Rent exponent obtained by an optimal partitioning method. Furthermore, it is

argued that the Rent exponent is a measure of partitioning approach. Smaller Rent

exponent means that the partitioning approach used to obtain this Rent exponent

is better. Other related work includes the proposal of Region III [12], the local

variation of the Rent exponent [13] and Rent exponent prediction [14].

One of the fundamental issues in Rent’s rule study is the extraction of the

Rent exponent from a given circuit. Traditionally, Rent exponents were obtained

by partitioning circuits and analyzing the partitioned subcircuits. In [1] a multi-

way partitioning algorithm is used to generate partitioning instances. For each

instance, the average subcircuit size and the average number of pins (external

nets) per subcircuit are calculated and the result represents a data point on a log-

log scale. A linear regression is then applied to find the slope of the fitted line,

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which is the Rent exponent of the circuit. A similar strategy was employed in

[11].

In this paper we propose a different method of extracting the Rent exponent

for a given circuit, that is, achieving the Rent exponent from an existing place-

ment. This is to better understand the notion of Rent parameters, and is not to

suggest that the Rent parameters should be obtained from placement. The issue

of the Rent exponents in partitioning and placement was studied in [15, 16, 17].

In this work we try to evaluate the relationship between the Rent exponents from

empirical point of view. We argue that Rent exponents extracted from partition-

ing and placement are not identical. However, there exists a relationship between

these two exponents. We theoretically analyze and empirically evaluate this re-

lationship. All the experiments are conducted on mid-size or large benchmark

circuits in order to provide useful information close to real world. To take the

variety of placement tools into account, three recent placement tools (Capo [18],

Feng Shui [19] andDragon [20]) are used in this work. There is no doubt that ex-

tracting the Rent exponent from placement is much slower than from partitioning.

Furthermore, the Rent exponent is indeed useless after placement stage. However,

studies on this issue will provide a different point of view on Rent’s rule and its

applications.

The rest of the paper is organized as follows: Section 2 defines the Rent ex-

ponent for partitioning and placement. The relationship between two different

Rent exponents is analyzed. Section 3 presents experimental results to support the

claim in section 2. In section 4, wirelength estimation methods based on Rent’s

rule are evaluated. Section 5 gives the conclusion of the paper.

2 Rent Exponents for Partitioning and Placement

2.1 Extracting Rent Exponents

Conventional approaches of extracting the Rent exponent are based on partition-

ing. Analyzing an existing placement of a circuit, however, will give a new way

3

of measuring the Rent exponent. It is no surprise that the Rent exponents ob-

tained from two methods are different. Partitioning based extraction focuses on

the topological structure of the circuit, while placement based extraction concen-

trates on the geometrical information of the placed circuit. Figure 1, algorithm 1

and algorithm 2 explain the two different methods to extract the Rent exponent.

BipartitioningRecursively

...... ...Partitioning tree Partitioning Rent’s exponent

ToolPlacement ...

Placement Rent’s exponentPlacement

Figure 1: Rent exponent extraction from recursive bipartitioning (upper half) andplacement (lower half).

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

4

Algorithm 2 Extract-Rent-by-Placement(C)Input: CircuitC � �V�E�Output: Rent exponentp�

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().

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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-

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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.

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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.

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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.

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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).

ckt #cells #nets Partition Estimated Placement Rentp��

Rentp Placep� Capo Feng Shui Dragonavqs 21,854 22,124 0.449 0.529 0.535 0.526 0.521avql 25,114 25,384 0.449 0.527 0.536 0.520 0.522

golem3 99,932 143,379 0.556 0.624 0.615 0.600 0.575ibm11 68,119 78,843 0.608 0.682 0.693 0.682 0.667ibm12 69,026 75,157 0.648 0.723 0.708 0.694 0.683ibm13 81,018 97,574 0.600 0.672 0.689 0.668 0.648ibm14 147,088 147,605 0.622 0.690 0.679 0.650 0.663ibm15 157,861 183,684 0.599 0.665 0.669 0.630 0.640ibm16 181,633 188,324 0.609 0.690 0.674 0.627 0.651ibm17 182,359 186,764 0.645 0.712 0.704 0.650 0.671ibm18 210,323 201,560 0.600 0.664 0.658 0.615 0.632

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.

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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.

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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.

Circuit Partitioning Derived placement Average placement ExpectedαRent exponent Rent exponent Rent exponent

ibm11 0.608 0.680 0.681 0.98ibm12 0.648 0.723 0.695 0.55ibm13 0.600 0.674 0.668 0.93ibm14 0.622 0.690 0.664 0.55ibm15 0.599 0.665 0.646 0.65ibm16 0.609 0.673 0.651 0.57ibm17 0.645 0.708 0.675 0.38ibm18 0.600 0.664 0.635 0.48

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.

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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

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Partitioning Placementckt p est. WL est. WL placement p�� est. WL est. WL real WL

(Donath’s) (Davis’s) tool (Donath’s) (Davis’s)Capo 0.693 800 583 489

ibm11 0.608 562 442 Feng Shui 0.682 764 562 494Dragon 0.667 718 534 483Capo 0.708 1024 699 738







ibm18 0.600 1207 933 Feng Shui 0.615 1605 1173 1247Dragon 0.632 1300 989 1090

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

ibm11 0.608 0.682 558 489 494 483ibm12 0.648 0.723 734 738 744 646ibm13 0.600 0.672 641 676 716 605ibm14 0.622 0.690 977 913 841 822ibm15 0.599 0.665 1037 1242 1175 1196ibm16 0.609 0.690 1189 1207 1090 1078ibm17 0.645 0.712 1453 1661 1651 1653ibm18 0.600 0.664 1204 1108 1247 1090

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)

Capo Feng Shui Dragon Capo Feng Shui Dragon Capo Feng Shui Dragonibm11 0.693 0.682 0.667 435 442 423 489 494 483ibm12 0.708 0.694 0.683 655 654 567 738 744 646ibm13 0.689 0.668 0.648 505 510 487 676 716 605ibm14 0.679 0.650 0.663 827 759 740 913 841 822ibm15 0.669 0.630 0.640 951 882 890 1242 1175 1196ibm16 0.674 0.627 0.651 1025 972 961 1207 1090 1078ibm17 0.704 0.650 0.671 1483 1315 1364 1661 1651 1653ibm18 0.658 0.615 0.632 1088 953 967 1108 1247 1090

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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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


Bounding box Wirelength by different placement tools

Wire

leng

th


(b) Normalized bounding box wirelengths

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2


Routed Wirelength by different placement tools

Wire

leng

th


(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

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