Placement remains dominant on the overall quality of physical design automation [29, 30]. Based on logic synthesis , back-end design on timing , power [44, 9], routability [38, 8], variability [42, 3] etc. are highly impacted by placement performance. The emerging 3D-IC  challenges the traditional 2D placers [5, 41, 19, 17, 1, 2, 25] to produce 3D circuit layout with minimum wirelength yet limited vertical interconnects (through-silicon vias (TSVs), monolithic inter-tier vias (MIVs), etc.). Innovations of mixed-size 3D-IC placement become quite desirable.
Previous combinatorial 3D-IC placers form two categories. Folding based methods  folds the 2D-IC placement layout to produce 3D solution with local refinement. Partitioning based approaches [7, 18] minimize the usage of vertical resources. Kim et al.  partitions the netlist followed by tier assignment, then applies 2D quadratic placement  simultaneously over all the tiers. Analytic placers achieve better 3D-IC placement performace versus combinatorial algorithms. Goplen et al.  models the 3D-IC placement by a quadratic framework . Hsu et al.  extends the 2D-IC placement prototype  and uses Bell-shape function  to smooth the vertical dimension. Luo et al.  utilizes the 2D algorithm in  and relaxes the discrete tiers via Huber function . However, these modeling functions are only locally smooth. Moreover, their hierarchical cell clustering and grid coarsening would degrade the quality . Separately, prior 3D placement benchmarks [13, 15] are of up to only 210K cells, which are too small to represent modern design complexity. Large-scale bookshelf 3D-IC placement benchmarks become desirable.
In this work, we extend the 2D placers ePlace [25, 23, 22] and ePlace-MS [26, 22] to the 3D domain. Our algorithm is named ePlace-3D and focused on wirelength minimization and density equalization, while other 3D-IC objectives like thermal are not covered. To the best of our knowledge, this is the first work in literature achieving analytically global smoothness along all the three dimensions. In contrast, previous analytic works [10, 27] only ensure (partially) local smoothness in their density functions [34, 12], while their less continuous cell movement would slow down placement convergence and cause more penalty on wirelength. We conduct analytic global placement and stochastic legalization in the entire 3D cuboid domain, which maximizes the search space thus further boost the solution quality. ePlace-3D well demonstrates the applicability of the electrostatic density model eDensity [24, 23] in various physical dimensions. Our specific contributions are listed as follows.
eDensity-3D: an electrostatics based 3D density function ensuring global smoothness.
A 3D numerical solution based on fast Fourier transform (FFT) and improved spectral formulation.
A nonlinear 3D preconditioner to equalize all the moving objects in the optimization perspective.
Interleaving coarse-grained 3D placement with fine-grained 2D placement to enhance efficiency.
The remainder is organized as follows. Section 2 introduces the background knowledge. Section 3 discusses our 3D placement density function eDensity-3D, numerical solution, and nonlinear precondition. Section 4 provides an overview of ePlace-3D algorithm. Experiments and results are shown in Section 5. We conclude in Section 6.
Given a set of objects, net set and 3D cuboid core region , global placement is formulated as constrained optimization. The constraint desires all the objects to be accommodated with zero overlap. Let denote the placement solution, which consists of the physical coordinates of all the objects. The region is uniformly decomposed into 3D bins denoted as set . For every bin , the density should not exceed the target density . The objective is to minimize the total half-perimeter wirelength (HPWL) of all the nets. Let denote the horizontal wirelength of net (similar for ), the total 2D HPWL is . We use , and as dimensional weighting factors. 3D-IC placement needs vertical interconnects, such as through-silicon via (TSV) and monolithic inter-tier via (MIV), to penetrate silicon tiers. Diverse types of connects have different physical and electrical properties. However, ePlace-3D is compatible with any types of connects, which can be reflected on the weight of . In the remainder of this manuscript, we name all types of 3D vertical interconnects uniformly as for simplicity. The number of vertical interconnect units (VI) is computed as how many times silicon tiers have been penetrated, e.g., one vertical connect between tier one and tier three is counted as two . The nonlinear placement optimization is formulated as
Analytic methods conduct placement using gradient-directed optimization. As is not differentiable, we use wirelength smoothing by weighted-average (WA) model .
Here and . , and control the modeling accuracy. Density function relaxes all the constraints in Eq. (1). Most 2D and 3D quadratic placers [19, 18, 6] follow the linear density force formulation by . Nonlinear placers [11, 1, 27, 10] have their dedicated density functions. NTUplace3-3D  leverages bell-shape curve  for local smoothness in 3D domain. mPL6-3D  uses Helmholtz function to globally smoothen the 2D plane and Huber’s function to locally smoothen the vertical dimension. The electrostatics based density function  converts objects to charges. By the Lorentz law, the electric repulsive force spreads charges away towards the electrostatic equilibrium state, which produces a globally even density distribution. Let denote the density cost function, the constraints in Eq. (1) can be relaxed by the penalty factor , while the unconstrained optimization is shown as below.
In this work, we set vertical connects as zero-volumed thus do not consider them in eDensity-3D111Practically, vertical connects can never be zero volumed. However, for academic research we are able to simplify the engineering problems to boost scientific innovations. Similarly, state-of-the-art 2D placement academic works [25, 1, 17, 2, 19, 41, 40] target wirelength only and ignore other objectives like timing, power and routability. As vertical connects may be of large volume thus significantly contribute to the placement density, we will put it in our future work.. Therefore, the optimization of electrostatics will not be affected and can be still achieved based on the movement of netlist objects. Density overflow is used to terminate global placement and denoted as , which is
Here is the total volume of all the movable objects, is the total volume of objects in the bin , and is the total whitespace in bin . The volume of each cell is computed as its planar area multiplied by the depth of each tier.
3 eDensity-3D: 3D Density Function
In this section, we introduce our novel 3D density function eDensity-3D, a fast numerical solution by spectral methods, and approximated 3D nonlinear preconditioner. The key insight is, we treat the third dimension equally as the other two dimensions, such that vertical cell movement will be as smooth as the planar movement in 2D placement. The behavior of eDensity-3D is visualized in Figure 1.
3.1 3D Density Function
Extending the planar function eDensity in , eDensity-3D models the entire placement instance as a 3D electrostatic field. Every placement object (standard cells, macros and fillers) is converted to a positively charged cuboid. The electric repulsive force spreads all the objects away from the high-density region. The 3D density cost is modeled as the total potential energy of the system and defined as below
denotes the electric quantity of the charge and is set as the physical volume of placement object . is the electric potential at charge . Charges with high potential will reduce the placement overlap by moving towards the direction of largest energy descent. Unlike the spatial density distribution (Figure 1) which is coarse and non-differentiable, the electric potential distribution is globally smooth. We use the potential gradient (thus electric field), , to direct cell movement for density equalization. Given a placement layout , we generate the density map , then compute the potential map by solving the 3D Poisson’s equation
Here is the outer unit normal of the placement cube . is the boundary and consists of orthogonal rectangular planes to enclose the placement cuboid. In Eq. (6), the first equation has . Neumann condition by the second equation requires that when any object
reaches any boundary plane, its density force vector will have the component perpendicular to the plane reduced to zero, in order to preventfrom penetrating the plane. The third equation shows that the integral of density and potential within are set to zero to ensure that (1) electric force drives all the charges towards even density distribution rather than pushing them to infinity, which matches the placement objective (2) the 3D Poisson’s equation would have a unique solution by satisfying the Neumann condition. We differentiate the potential on each charge to generate the electric field . The electric (density) force is .
3.2 3D Numerical Solution
Based on the 2D solution in , we solve the 3D Poisson’s equation by spectral methods using frequency decomposition . To satisfy the Neumann condition of zero gradients at the boundaries, we use sinusoidal wave to express the electric field
. We construct an odd and periodic field distribution by negatively mirroring itself w.r.t. the origin, then periodically extending it towards positive and negative infinities. Electric potential and density distributions are then expressed by cosine waveforms, which are the integration and differentiation of the field. Letdenote the 3D coefficients of the density frequency.
eDensity  sets , which equals the discrete index for the th frequency component. However, as we are conducting placement in a continuous domain, the multiplication of and induces inconsistency. In this work, we propose improved spectral methods for the 3D placement density function. Specifically, we set since ranges within . As a result, well matches the original unit of discrete frequency index, and we have all the frequency indexes defined as . As mentioned in Section 2, here , and represent the dimensions of the cuboid placement core region. can be set as any value since will be normalized by . , and range in , which is only half of a cosine function period. In contrast, one complete function period centered at the origin is . Therefore, we have rather than in the above frequency index. We set to remove the zero-frequency component. The spatial density distribution is
To achieve , the solution to the potential can be expressed as
By differentiating Eq. (9), we have the electric field distribution shown as below
Let denote the total number of bins in global placement. Instead of quadratic complexity, above spectral equations can be efficiently solved using FFT algorithms  with complexity.
3.3 3D Nonlinear Precondition
Theoretically, preconditioning improves convergence rate rather than solution quality. However, as placement is a highly nonlinear, non-convex and ill-conditioned problem, the Hessian matrix with improved condition number would reshape the search direction for the nonlinear solver to follow. As a result, preconditioning would open the gate for unexplored high-dimension search space, while surprising quality enhancement would be expectable.
Preconditioned mixed-size placement should tolerate the huge physical and topological differences between all the standard cells, macros, and dummy fillers. In , the nonlinear preconditioner for 2D placement is modeled as
Here are all the nets incident to the object , is the 2D area of the object . In 3D placement, we use to denote the volume of instead. The preconditioned gradient then improves and accelerates the placement. Our studies show that Eq. (11) relies on the assumption of . However, the third dimension weakens and breaks the above assumption. As a result, dominates and makes fillers and macros with small spread faster than standard cells, as Eq. (12) shows
Instead, we propose a new preconditioner as below
The noise factors introduced by is resolved, where all the objects are being equalized in the optimizer’s perspective and simultaneously spread over the entire domain. Experiments show that our 3D preconditioner reduces the global placement iterations by and improves the wirelength by over all the 16 MMS benchmarks.
Complexity significantly impacts the placement runtime. In each iteration, we traverse all the bins to reset their density in time, then traverse all the placement objects in time to update the superimposed density map. By Eq. (7), (9) and (10), five times of 3D FFT computation are invoked, which costs time. By our grid sizing strategy in Eq. (14), is limited to constant. The overall complexity is thus ,
In ePlace-3D, the placement domain is geometrically transformed from to . We set the density resolutions to make the placement domain uniformly decomposed into cubic bins. Let denote the total volume of and denote the average area of all standard cells. The grid sizing is set as
Here every standard cells are accommodated by one bin. Placement quality (efficiency) is determined by the value of . In this work, we constantly set .
4 ePlace-3D: Overview
ePlace-3D is built upon the infrastructure of ePlace-MS . Figure 2 shows the flowchart of our algorithm. Given a placement instance, our algorithm minimizes the quadratic wirelength over the 3D domain to produce the initial solution . To approach the optimum solution in the end, we make as minimum-wirelength violation-tolerant.
Our 3D-IC global placement is visualized in Figure 3. Unconnected fillers [1, 25] are inserted to populate up extra whitespace. All the fillers are equally sized by the average dimensions of all the standard cells. The optimum solution of 3D global placement will have all the cells, macros and fillers orient towards discrete tiers. Otherwise, some cuboid placement sites will be partially wasted, degrading the solution quality. Figure 3(f) illustrates the beauty of our approach, i.e., the analytic 3D placer is visually approaching density evenness from the vertical dimension, which ensures negligible quality overhead during tier assignment. We use Nesterov’s method  as the nonlinear solver and determine the steplength by .
A multi-tier 2D-IC mixed-size global placement follows by assigning all the macros and standard cells to the closest tiers and separately filling the remaining whitespace on each tier with fillers. Planar placement is conducted simultaneously over all the tiers. As wirelength smoothing is homogeneous over all the tiers (with the same ), heterogeneous grid sizing is not feasible as density force is dependent on resolution by Eq. (10). We set all the tiers with the same density resolution , which is the maximum of that of all the tiers by Eq. (14) with . In practice, we have . Figure 4 illustrates the progression.
Our 3D-IC macro legalizer generates a legal macro layout with zero macro overlap and small wirelength overhead. The algorithm is stochastic based on simulated annealing . A 3D-IC standard-cell global placement follows to mitigate the quality loss due to sub-optimal macro legalization. We assign standard cells to their closest tiers and conduct simultaneous 2D-IC standard-cell placement on all the tiers. The standard-cell layouts of all the tiers are locally refined. Figure 5 shows the respective placement progressions, more details can be found in . The detailed placer from  is then invoked for a tier-by-tier standard-cell legalization and detailed placement from the bottom to the top tier.
|Categories||NTUplace3-3D ||mPL6-3D ||ePlace-3D|
In general, we have fine-grained 2D placement interleaved with coarse-grained 3D placement, which achieves a good trade-off between quality and efficiency. On average of all the ten IBM-PLACE circuits, the application of 2D refinement reduces the wirelength by more than .
5 Experiments and Results
We implement ePlace-3D using C programming language in the single-thread mode and execute the program in a Linux machine with Intel i7 920 2.67GHz CPU and 12GB memory. There is no benchmark specific parameter tuning in our work. VI are controlled by the weighting factor based on capacitance ratio. By , one TSV (VI) has the capacitance of at 45nm tech-node. ITRS annual reports  show that unit capacitance of interconnects at intermediate routing layers is constantly across various tech-nodes. Placement row height is at 45nm tech-node ( M1 half-pitch, ten M1 tracks per row), capacitance becomes for 2D interconnect spanning one-row height. Based on the length units for each benchmark, as well as our geometric transformation of the placement core region to be as discussed in Section 3.4, we compute the respective capacitance ratio of one VI versus one unit wirelength and use it as the VI weight. Specifically, we have
Notice that the focus of this work is the algorithm framework of 3D placement, not the accurate weight modeling of vertical connects. The weighting factor can be adjusted by VLSI designers for their particular needs, e.g., vertical connects of different electric and physical attributes (TSVs, MIVs, super contacts, etc.).
We conduct experiments on IBM-PLACE  standard-cell benchmarks without macros or blockages, all of which are derived from real IC design. We include two state-of-the-art 3D-IC placers, mPL6-3D 222Although mPL6-3D has extension to thermal-aware placement, its experiments on the IBM-PLACE cases are based on their original prototype driven by only wirelength and density but not thermal., and NTUplace3-3D , in our experiments on IBM-PLACE. As other categories of algorithms (e.g., folding and partition based approaches) have been outperformed by analytic placement in literature, we do not include them in our experiments. We have obtained the binary of NTUplace3-3D from the original authors and executed it on our machine for experiments333There is a small quality gap on NTUplace3-3D between our local experiment results and that published in , which may be due to the differences in computing platforms.. mPL6-3D is not available (as notified by the author), so we cite the performance from their latest publication . We use exactly the same benchmark transformation as that by mPL6-3D and NTUplace3-3D. I.e., we insert four silicon tiers into each benchmark, scale down each tier to of the original 2D placement area, add whitespace to each tier, and keep the aspect ratio of each tier to be the same as the original 2D design. As a result, all the experiments on the three placers, including those from , are conducted on exactly the same IBM-PLACE-3D benchmarks. As HPWL and VI are being computed in exactly the same way, the performance comparison among the three placers are fair. The results on IBM-PLACE cases are shown in Table 1. On average of all the ten circuits, ePlace-3D outperforms mPL6-3D and NTUplace3-3D with and shorter wirelength together with and fewer VIs. ePlace-3D runs faster than mPL6-3D but slower than NTUplace3-3D, nevertheless, the improvement on wirelength () and VI () is significant.
|tiers||ePlace-MS ||ePlace-3D w/ 2 tiers||ePlace-3D w/ 3 tiers||ePlace-3D w/ 4 tiers|
To validate the scalability of ePlace-3D, we also conduct experiments on the large-scale modern mixed-size (MMS) benchmarks  with on average 829K and up to 2.5M netlist objects. MMS benchmarks was first published in DAC 2009. The circuits inherit the same netlists and density constraints from ISPD 2005  and ISPD 2006  benchmarks but have all the macros freed to place. The original planar placement domain is geometrically transformed to be of , and silicon tiers, each tier is equally downsized to keep both the aspect ratio and total silicon area unchanged. All the standard cells and macros keep their original dimensions and span only one tier. MMS circuits have all their fixed objects with zero area (volume) and outside the placement boundaries, and we geometrically transform them to the boundary of the bottom (first) tier. Also, as macros are all free to move, we skip the geometrical transformation of the fixed macro layout from 2D to 3D, which is sub-optimal and usually causes quality loss. Similar to mPL6-3D  and NTUplace3-3D , we add extra whitespace to each tier, in order to relieve the placement dilemma due to the increased area ratio between large macros and silicon tiers444 BIGBLUE3, NEWBLUE2 and NEWBLUE3 have very large macros. For the tier insertion of two, three and four, we add , and whitespace to each tier to make sure that the largest macro can be accommodated.. There are benchmark-dependent target density for eight out of the sixteen MMS circuits. Detailed circuit statistics can be found in Table 1 of . We create evaluation scripts to compute the total wirelength, number of vertical interconnects, and legality of the produced 3D-IC placement solution. The results on the MMS benchmarks are shown in Table 2. Notice that here HPWL is the original half-perimeter wirelength. It is not penalized by the amount of density overflow, since the density overflow in 3D domain is of one more dimension thus hard to compare with that of 2D domain. The binary of NTUplace3-3D does not work with these benchmarks, while the binary of mPL6-3D is not available for use. As a result, we compare the 3D MMS placement solutions with the best published (golden) 2D results in literature . By using two, three and four tiers, ePlace-3D outperforms the golden 2D placement with on average , and shorter wirelength. On the other side, the average ratio between the number of vertical interconnect units versus the number of placement objects (standard cells and macros) are only , and , respectively. These vertical connect ratios are much smaller than the average VI ratio on IBM-PLACE, which are more than for all the three placers in Table 1. Due to the introduction of the third dimension, the search space of placement optimization is substantially enlarged. However, the runtime increase is just , which indicates high efficiency of ePlace-3D.
We also study the trends of HPWL and VI by linearly sweeping the number of tiers and exponentially sweeping the VI weight. We select eight out of the sixteen MMS benchmarks (ADAPTEC1, ADAPTEC4, BIGBLUE1, BIGBLUE2, BIGBLUE3, BIGBLUE4, NEWBLUE6, NEWBLUE7), all of which could accommodate the maximum macro block after inserting ten tiers. Keeping the same aspect ratio, the area of each tier is scaled down by ten times with the insertion of extra whitespace. Figure 6 shows that ePlace-3D is able to reduce the total 2D wirelength by up to (with the insertion of up to ten tiers), while VI is roughly scaled up by the number of tiers.
VI weight sweeping is conducted on all the sixteen MMS benchmarks. Figure 7 shows the trends of average HPWL and VI by dividing the normal VI weight by up to 32 times (i.e. 0.03125). The total 2D HPWL saturates at the reduction of , while VI is scaled up by roughtly .
Our 3D-IC placement algorithm shows significant quality improvement while limited runtime overhead. BIGBLUE4 and NEWBLUE7 are the largest circuits with 2.2M and 2.5M cells, and they consume the longest runtime on the 3D-IC placement. However, compared to the respective golden 2D placement solutions, the runtime ratio is upper-bounded by , which is still less than the average runtime ratios of for two tiers, for three tiers and for four tiers, respectively. To this end, ePlace-3D shows good scalability and acceptable efficiency on the large cases.
In this work, we do not test ePlace-3D on circuits with fixed macros, as geometrically transforming the 2D floorplan into 3D is difficult and usually error-prone. However, ePlace-3D shows high performance and scalability on MMS benchmarks with lots of movable large macros, which is more difficult to place than fixed-macro layouts. As a result, we are confident on the performance of ePlace-3D on any circuits with fixed macros. The advantage of 3D tier insertion vanishes if there are large macros to accommodate (BIGBLUE3, NEWBLUE3, etc.). Transformation of 2D planar macros into 3D cuboid macros would resolve this issue and ensure the consistent benefits by inserting more tiers. However, it is beyond this work and will be covered in future.
We propose the first electrostatics based placement algorithm ePlace-3D, which is effective and efficient for 3D-ICs with uniform exploration over the entire 3D space. Our 3D-IC density function leverages the analogy between placement spreading and electrostatic equilibrium, while global and uniform smoothness is realized at all the three dimensions. Our balancing and preconditioning techniques prevent solution oscillation or divergence. The interleaved 3D coarse-grained optimization followed by 2D fine-grained post processing obtains a good trade-off between quality and efficiency. The experimental results validate the high performance and scalability of our approach, indicating the benefits of placement smoothness. In future, we will develop 3D density function to address the volume of vertical interconnects (VI). We would also like to explore advanced technology for 3D-IC placement/routing with patterning and graph coloring technology .
The authors acknowledge (1) Prof. Dae Hyun Kim and Prof. Sung Kyu Lim for providing the 3D-IC flow scripts and IWLS testcases (2) Dr. Meng-Kai Hsu and Prof. Yao-Wen Chang for providing the binary of NTUplace3-3D (3) Dr. Guojie Luo and Prof. Jason Cong for providing the binary of mPL6-3D (4) the support of NSF CCF-1017864.
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