Wirelength Estimation based on Rent Exponents of Partitioning and Placement 1

Transcription

1 Wirelength Estimation based on Rent Exponents of Partitioning and Placement 1 Xiaojian Yang, Elaheh Bozorgzadeh, and Majid Sarrafzadeh Synplicity Inc. Sunnyvale, CA xjyang@synplicity.com Computer Science Department University of California at Los Angeles Los Angeles, CA elib,majid@cs.ucla.edu Abstract Wirelength estimation is one of the most important Rent s rule applications. Traditionally, the Rent exponent is extracted using recursive bipartitioning. 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 A preliminary version of this paper appeared in Proc. Int. Workshop on System-Level Interconnect Prediction, pp.25-31, April

2 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 extensive 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 considering occupying probability [4] or recursively applying Rent s rule throughout an entire monolithic system [5]. Extension of basic wirelength estimation, including routing utilization estimation [6], congestion estimation [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 by T tg p, where T and G 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 the intrinsic 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 multiway 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 loglog scale. A linear regression is then applied to find the slope of the fitted line, 2

3 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 placement. 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 partitioning and placement are not identical. However, there exists a relationship between these two exponents. We theoretically analyze and empirically evaluate this relationship. 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] and Dragon [20]) are used in this work. There is no doubt that extracting 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 exponent 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 partitioning. Analyzing an existing placement of a circuit, however, will give a new way 3

4 of measuring the Rent exponent. It is no surprise that the Rent exponents obtained from two methods are different. Partitioning based extraction focuses on the topological structure of the circuit, while placement based extraction concentrates 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. Recursively Bipartitioning Partitioning tree Partitioning Rent s exponent Placement Tool... Placement Placement Rent s exponent Figure 1: Rent exponent extraction from recursive bipartitioning (upper half) and placement (lower half). Algorithm 1 Extract-Rent-by-Partitioning(C) Input: Circuit C V Eµ Output: Rent exponent p 1. Recursively bipartition the original circuits. At each recursive level, calculate the average number of cells per partition and the average number of external nets over all partitions. Save the data pair to G i T i µ where i is the depth of recursive partitioning. Partitioning stops when reaching a given depth n. 2. Apply linear regression on the log-log scaled data pairs: G k T k µ G k 1 T k 1 µ G n T n µ (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, average number of cells and average number of external nets for all subcircuits are 4

5 Algorithm 2 Extract-Rent-by-Placement(C) Input: Circuit C V Eµ Output: Rent exponent p ¼ Place the circuit on two dimensional plane, for i 1 to a given depth n do Divide the core area into 2 i regular regions; Each region contains a group of cells; Compute the average number of cells per group and the average number of external nets over all cell groups. Save the data pair to G i T i µ. end for Apply linear regression on the log-log scaled data pairs: G k T k µ G k 1 T k 1 µ G n T n µ (k is a 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 implementing Extract-Rent-by-Partitioning is as follows: when a subcircuit is partitioned into two smaller subcircuits, the nets which connect 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 circuit, the placement Rent exponent p ¼ is the output of the algorithm Extract-Rentby-Placement(). 5

6 2.2 Relationship between Exponents Since partitioning and placement are related problems, the partitioning Rent exponent 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 placement, 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 size G 1, 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 same y-intercept because the Rent coefficient t 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 ¼ log T Placement Rent s curve p T = t G p T = t G Partitioning Rent s curve log t log G 1 log G Figure 2: Comparison between partitioning Rent exponent and placement Rent exponent If the placement engine is a min-cut class approach, we can derive a relationship between the two Rent exponents. Figure 3 illustrates two different biparti- 6

7 tioning problems. In figure 3(a), the partitioner only considers the interconnects between cells of the subcircuit to be partitioned. We call this problem pure bipartitioning 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 with terminal propagation, which is normally used in min-cut class placement tools, as shown in Figure 3(c). It is the difference between 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 are G 1 and G 2. Let C be the number of cut nets (figure 3(a)). For the bipartitioning process with terminal propagation, let C ¼ be the number of cut nets of bipartitioning. We have C ¼ C because of the effect of the external nets. According to Rent s rule, from Figure 3(a), we obtain: T 1 C T tg p 1 (1) where T is the total number of external nets for subcircuit G 1. T 1 is the number of the external nets which are not 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: T 1 T 2 t G 1 G 2 µ p (2) For simplicity, we assume that in a balanced bipartitioning, G 1 G 2 and T 1 T 2. From equation (1) and (2), we have: T 1 2 p 1 T In the bipartitioning with terminal propagation (Figure 3(b)), there are T 1 external 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 connected to the right side (T 1 2 nets) will drag cells from left to right, thus they 7

8 T 1 T 2 G T 1 T 2 C C G 1 G 2 G 1 G 2 (a) (b) (c) Figure 3: Comparison between a pure partitioning (a) and a partitioning with terminal propagation in min-cut placement (b), (c). The former only considers the internal nets, while the latter considers both internal nets and external nets. T 1 and T 2 are the number of external nets which are not cut nets for subcircuit G 1 and G 2, respectively. 8

9 may increase the cut nets of the partitioning. We assume that one such external net increases the number of cut nets by α. 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 by αt 1. Therefore for a partitioned subcircuit, the number of external nets T ¼ after terminal propagation based partitioning is: T ¼ T αt 1 1 α 2 p 1 µt Since T tg p 1 and T ¼ tg p¼ 1 (p and p¼ are partitioning Rent exponent and placement Rent exponent, respectively), we have, Thus we have, p ¼ p logt ¼ logt logg 1 logt logt logg 1 log T ¼ Tµ logg 1 log 1 α 2p 1 µ logg 1 p ¼ p log 1 α 2p 1 µ logg 1 (3) where G 1 should be the number of cells in a subcircuit which corresponds to a data point. In practice we set G 1 to be V 2 5 to avoid the Rent s rule region II 2. Equation (3) shows that the placement Rent exponent (p ¼ ) 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, if p 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. 2 Region 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

10 3 Experimental Validation Equation (3) shows that we can derive placement Rent exponent p ¼ from the partitioning Rent exponent p. 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 benchmark 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 are Capo [18], Feng Shui [19] and Dragon [20]. All of them are recent academic works and they all integrate multi-level hypergraph partitioning, a breakthrough technique in VLSI/CAD partitioning problem. Capo and Feng Shui use recursively bipartitioning approach followed by local improvement. Dragon employs both cut and wirelength minimization in hierarchical placement flow. All experiments are performed on Sun workstations with 400MHz CPU and 128M memory. The depths of both Extract-Rent-bypartitioning 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 result of linear regression on data points collected by recursive bipartitioning. Three upper lines are obtained from placement outputs by Capo, Feng Shui and Dragon. All the slopes of three upper lines are larger than the slope of the partitioning line, 10

11 log T log G points extracted from recursive bipartitioning points extracted from Capo placement points extracted from Feng Shui placement points extracted from Dragon placement fitted line for partitioning fitted line for Capo placement fitted line for Feng Shui placement fitted line for Dragon placement Figure 4: Rent s rule fitted line based on partitioning and placement for benchmark ibm15. The lower line is the result of linear regression on data points from recursive bipartitioning. Three upper lines are from placement outputs. 11

12 supporting the relationship between partitioning Rent exponent and placement Rent exponent discussed in Section 2. Table 1 shows the comparison between partitioning Rent exponent p, derived placement Rent exponent p ¼ which is obtained from Equation (3) 3, and three real placement Rent exponents p ¼¼ 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 exponent p, derived placement Rent exponent p ¼ is closer to real placement Rent exponents, partially supporting the theoretical relationship between two Rent exponents. However, better derivation of placement Rent exponent requires the knowledge of α in Equation (3). ckt #cells #nets Partition Estimated Placement Rent p ¼¼ Rent p Place p ¼ Capo Feng Shui Dragon avqs 21,854 22, avql 25,114 25, golem3 99, , ibm11 68,119 78, ibm12 69,026 75, ibm13 81,018 97, ibm14 147, , ibm15 157, , ibm16 181, , ibm17 182, , ibm18 210, , Table 1: Comparison between partitioning Rent exponent p, derived placement Rent exponent p ¼ and real placement Rent exponent p ¼¼ extracted from three placement tools outputs 3.2 Range of α In the above experiments we set α to be 1, which leads to a simplified model. However, as defined in Section 2.2, α is a coefficient that indicates the effect of 3 We set α 1 in experiments. 12

13 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 of α varies for different circuits. For a given circuit, if we gradually increase α from 0 to 1, we obtain different placement Rent exponent based on Equation (3). Figure 5 illustrates an example of α s effect for circuit ibm15. The solid curve in Figure 5 shows the change of derived placement Rent exponent as α increases. The dashed line represents 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 to α This value is called the expected value of α. It means that if we set α 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 expected α for all of 8 IBM-PLACE circuits. Expected α varies for different circuits, ranging from 0.38 to In general, larger circuits tend to have a smaller expected α. How to obtain a proper α is a non-trivial problem. There could be multiple factors that affect expected α, including percentage of multiterminal nets, quality of partitioning approach and the Rent coefficient (t). In the following sections we still set α 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 exponents can lead to different wirelength estimates. In order to obtain more accurate wirelength estimates, a proper Rent exponent is required. 13

14 0.7 p as a function of α for circuit ibm15 derived p as α changes average p from three placement outputs 0.65 Rent exponent α Figure 5: Derived placement Rent exponent p as a function of α (the solid curve). The dashed line reprensents the average placement Rent exponent of three different placement Rent exponents extracted by three placers. The intersection of solid and dashed lines corresponds to α Circuit Partitioning Derived placement Average placement Expected α Rent exponent Rent exponent Rent exponent ibm ibm ibm ibm ibm ibm ibm ibm Table 2: Partitioning Rent exponent, placement Rent exponent derived from Equation, average placement Rent exponent by three placers, and the expected α computed by these exponents. 14

15 4.1 Different Rent Exponents in Estimation The authors in [11] show that the Rent exponent of a circuit depends on the partitioning approach from which it is derived. Similar situation exists in extracting placement Rent exponent. If we use different placement algorithms, we will obtain different placement Rent exponents. Likewise, it is expected that the placement Rent exponents do not have much variation from different placement algorithms. In the wirelength estimation work [2, 4], the authors adopt a hierarchical placement model and assume that Rent s rule holds for all subcircuits at each hierarchical level. In [5] the wirelength distribution is derived from the number of interconnects 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 partitioning 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 placement Rent exponents for each circuit. With these exponents, we estimate the total wirelength based on existing wirelength distribution models. Both classic Donath 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 placement 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 comparison, 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 is G, and the 4 We refer it as Davis s model while the authors of [5] are J. A. Davis, V. K. De and J. Meindl. 15

16 global routing grid is n n, then the scaled estimated wirelength WL is, n WL WL ¼ Ô G where WL ¼ is the estimated wirelength. Table 3 shows a comparison between the estimated wirelength and real wirelength 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 wirelength 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 assumption 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 wirelength 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 exponents, 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

17 Partitioning Placement ckt p est. WL est. WL placement p ¼¼ est. WL est. WL real WL (Donath s) (Davis s) tool (Donath s) (Davis s) Capo ibm Feng Shui Dragon Capo ibm Feng Shui Dragon Capo ibm Feng Shui Dragon Capo ibm Feng Shui Dragon Capo ibm Feng Shui Dragon Capo ibm Feng Shui Dragon Capo ibm Feng Shui Dragon Capo ibm Feng Shui Dragon Table 3: Partitioning Rent exponent p and wirelength estimates by two estimation methods (Donath s and Davis s), comparing with placement exponent p ¼¼ by three different placement tools (Capo, Feng Shui and Dragon), and the wirelength estimates based on p ¼¼. The final column is the real wirelength output by global router. Both estimated and real WL (wirelength) are in 10 3 grid units of global routing. 17

18 Circuit Recursively Bipartitioning Partitioning Rent Exponent ( p ) Derivation of Placement Rent Exponent Estimated Wirelength Wirelength Estimation Based on Placement Rent Exponent Placement Rent Exponent ( p" ) Figure 6: A new approach for wirelength estimation. The difference between this approach and previous ones is that it derives placement Rent exponent from partitioning 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 no perfect 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 algorithm. 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 exponents for different placement tools. Table 5 lists the Rent exponent, total bounding box wirelength and total routed 18

19 ckt partitioning derived placement estimated real WL ( 10 3 units) Rent exp. p Rent exp. p ¼ WL by p ¼ Capo Feng Shui Dragon ibm ibm ibm ibm ibm ibm ibm ibm Table 4: Partitioning Rent exponent p, derived placement Rent exponent p ¼ and estimated total wirelength based on p ¼, comparing with the routed total wirelength from 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 WL ckt ( 10 3 grid units) ( 10 3 grid units) Capo Feng Shui Dragon Capo Feng Shui Dragon Capo Feng Shui Dragon ibm ibm ibm ibm ibm ibm ibm ibm Table 5: Placement Rent exponents derived from layouts by three different placement tools, with the normalized total bounding box wirelength and normalized total 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

20 trary cases. However, the difference are relatively small in these cases. The correlation 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 factors 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 estimates, designers ought to adjust estimation model and the Rent exponent extraction 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. References [1] B. Landman and R. Russo. On a Pin Versus Block Relationship for Partitions of Logic Graphs. IEEE Transactions on Computers, c-20: , [2] W. E. Donath. Placement and Average Interconnection Lengths of Computer Logic. IEEE Transactions on Circuits and Systems, 26(4): , April [3] M. Feuer. Connectivity of random logic. IEEE Transactions on Computers, C-31(1):29 33, Jan [4] D. Stroobandt and J. Van Campenhout. Accurate Interconnection Length Estimations for Predictions Early in the Design Cycle. VLSI Design, Special Issue on Physical Design in Deep Submicron, 10(1):1 20,

21 [5] J. A. Davis, V. K. De, and J. Meindl. A Stochastic Wire-Length Distribution for Gigascale Integration(GSI) - Part I: Derivation and Validation. IEEE Transactions on Electron Devices, 45(3): , Mar [6] P. Chong and R. K. Brayton. Estimating and Optimizing Routing Utilization in DSM Design. In International Workshop on System-Level Interconnect Prediction. ACM, April [7] X. Yang, R. Kastner, and M. Sarrafzadeh. Congestion Estimation During Top-down Placement. In International Symposium on Physical Design, pages ACM, April [8] K. C. Saraswat, S. J. Souri, K. Banerjee, and P. Kapur. Performance Analysis and Technology of 3-D ICs. In International Workshop on System-Level Interconnect Prediction, pages ACM, April [9] R. Zhang, K. Roy, C. K. Koh, and D. B. Janes. Stochastic Wire-Length and Delay Distributions of 3-Dimensional Circuits. In International Conference on Computer-Aided Design, pages IEEE, November [10] P. Zarkesh-Ha, J. A. Davis, W. Loh, and J. D. Meindl. Prediction of Interconnect Fan-Out Distribution Using Rent s Rule. In International Workshop on System-Level Interconnect Prediction, pages ACM, April [11] L. Hagen, A. B. Kahng, F. J. Kurdahi, and C. Ramachandran. On the Intrinsic Rent Parameter and Spectra-Based Partitioning Methodologies. IEEE Transactions on Computer Aided Design, 13(no.1):27 37, Jan [12] D. Stroobandt. On an Efficient Method for Estimating the Interconnection Complexity of Designs and on the Existence of Region III in Rent s Rule. In Proceedings of the Ninth Great Lakes Symposium on VLSI, pages IEEE, March [13] H. Van Marck, D. Stroobandt, and J. Van Campenhout. Towards An Extension of Rent s Rule for Describing Local Variations in Interconnection Complexity. In Proceedings of the Fourth International Conference for Young Computer Scientists, pages , [14] P. Christie. Managing Interconnect Resources. In International Workshop on System-Level Interconnect Prediction, pages ACM, April [15] P. Christie and D. Stroobandt. The Interpretation and Application of Rent s Rule. IEEE Transactions on VLSI Systems, 8(6): , [16] P. Verplaetse, J. Dambre, D. Stroobandt, and J. Van Campenhout. On partitioning vs. placement Rent properties. In International Workshop on System-Level Interconnect Prediction, pages ACM, March

22 [17] X. Yang, E. Bozorgzadeh, and M. Sarrafzadeh. Wirelength Estimation based on Rent Exponents of Partitioning and Placement. In International Workshop on System-Level Interconnect Prediction, pages ACM, April [18] A. E. Caldwell, A. B. Kahng, and I. L. Markov. Can Recursive Bisection Alone Produce Routable Placements?. In Design Automation Conference, pages IEEE/ACM, June [19] M. C. Yildiz and P. H. Madden. Global Objectives for Standard Cell Placement. In Proceedings of the Great Lakes Symposium on VLSI, pages 68 72, March [20] M. Wang, X. Yang, and M. Sarrafzadeh. Dragon2000: Fast Standard-cell Placement for Large Circuits. In International Conference on Computer- Aided Design, pages IEEE, [21] C. J. Alpert. The ISPD98 Circuit Benchmark Suite. In International Symposium on Physical Design, pages ACM, April [22] G. Karypis, R. Aggarwal, V. Kumar, and S. Shekhar. Multilevel Hypergraph Partitioning: Application in VLSI Domain. In Design Automation Conference, pages IEEE/ACM,

23 Rent exponents by different placement tools Capo Feng Shui Dragon 0.7 Rent exponent ibm11 ibm12 ibm13 ibm14 ibm15 ibm16 ibm17 ibm18 (a) Placement Rent exponents Bounding box Wirelength by different placement tools Capo Feng Shui Dragon Wirelength ibm11 ibm12 ibm13 ibm14 ibm15 ibm16 ibm17 ibm18 (b) Normalized bounding box wirelengths Routed Wirelength by different placement tools Capo Feng Shui Dragon Wirelength ibm11 ibm12 ibm13 ibm14 ibm15 ibm16 ibm17 ibm18 (c) Normalized routed wirelengths Figure 7: (a) Placement Rent exponents derived from layouts by three different placement tools(capo, Feng Shui and Dragon). (b) Total bounding box wirelength in grid units by three placement tools. (c) Total routed wirelength in grid units by three placement tools. In (b) and (c) wirelengths are normalized by dividing the average value of three placement tools. 23