Handbook of algorithms for physical design automation part 37 doc

Most analytical placers contain some local optimization methods that are executed between the partitioning steps and that allow cells to leave the regions they are assigned to.. However,

= x2 1∈ R 2 ∈ R 1 = t − Sx2

we only have to compute the entries of x2by solving the following unconstrained problem on n − k

variables:

min



t − Sx2

x2

T

A



t − Sx2

x2



− 2bT



t − Sx2

x2



By ignoring all constant summands in the objective function, we get the equivalent problem

min xT

where U := −S

I

and v : = UT

b − At

0

The matrix UTAU is positive definite if A is positive

definite, but usually UTAU will not be sparse Therefore, for an efficient solution, an explicit

com-putation of UTAU must be avoided Fortunately, the conjugate gradient method (see Section 17.2.4)

only requires to multiply UTAU with a vector, which can be done by three single multiplications of a

sparse matrix and a vector Hence, provided that the number of constraints is small compared to the number of cells, the conjugate gradient method will efficiently solve the problem (Equation 17.2) Prescribing the centers of gravity of the cell groups is an efficient way to spread the cells over the chip area However, we cannot be sure that all cells are placed inside their region, which can be

a problem for ensuing partitioning steps Moreover, the constraints may be too strong if we do not demand an even distribution of the cells If we allow a higher area utilization in some regions, it will often be reasonable to place cells in their region in such a way that their center of gravity is far away from the center of the region

17.5.2 SPLITTINGNETS

A second way to reflect the result of partitioning in the QP, proposed by Vygen (1997), consists of splitting nets at the borders of regions In this approach, we assume that the chip area is partitioned

in a grid-like manner by vertical and horizontal cutlines that cross the whole chip

Suppose that we have boundsµ ≤ x(c) ≤ ν for the x-coordinate of a cell c For each cell c

that is connected to c but is placed in a window to the right of b (i.e., ν is a lower bound on the x-coordinate of c), we replace the connection to cby an artificial connection between c and a fixed pin with x-coordinate ν Analogously, connections to cells c that will be placed to the left ofµ are

replaced by connections to a fixed pin with x-coordinate µ Connections to fixed pins outside the

bounds of a cell are also split

Note that this splitting is done for x- and y-coordinate independently, so for x-coordinates only the vertical boderlines and for the y-coordinates only the horizontal borderlines between the windows

are considered In particular, in contrast to standard terminal propagation, it is possible (and in fact

will happen quite often) that a connection has to be split for the computation of the x-coordinates but not the y-coordinates, and vice versa This splitting of the nets forces each cell to be placed inside

the region that it is assigned to

However, a problem that has to be addressed in this approach is the following: it may happen that in a region all cells (or most of them) have their external connection to only one direction In

FIGURE 17.7 Effect of the constrained QP before the partitioning step.

that case, a QP solution will place all of them at one border or even in one corner of the region Such

a placement is obviously useless for the next partitioning step based on cell positions Vygen (1997) proposes to make use of center-of-gravity constraints (see Section 17.5.1) to modify the placements

in these cases Figure 17.7 illustrates how this works The left picture shows the placement with minimum quadratic netlength (splitting connections at the borderlines as described above) without any additional center-of-gravity constraints

Based on this we compute a new center of gravity for each region in which the current center

of gravity of the cells in the region is closer to the border than it would be possible in any disjoint placement The new center of gravity is (approximately) the closest possible position in a disjoint placement Then, a new global QP is solved forcing the centers of gravity of the cell groups in these regions to the new prescribed positions The right-hand side of Figure 17.7 shows the result It demonstrates that in particular in the outer regions of the chip area this step changes the placement significantly

17.6 FURTHER TECHNIQUES

17.6.1 REPARTITIONING

In a pure recursive partitioning approach, cells may never leave their regions However, especially cell assignments in the first paritioning steps may be suboptimal because they are based on placements

in which the cell positions may not differ enough Therefore, there is need for techniques that are able to correct bad decisions in partitioning Most analytical placers contain some local optimization methods that are executed between the partitioning steps and that allow cells to leave the regions they are assigned to

In Gordian (see Kleinhans et al., 1991), cells are moved toward their regions by solving a constrained QP (see Section 17.5.1) As this constrained QP does not force the cells to be placed inside their window, groups of cells that are assigned to different windows may be mixed with each other In such situations, Kleinhans et al (1991) reassign cells locally Let us consider the case when

a window is partitioned by a vertical cutline (the case of a horizontal cutline is handled analogously)

If after the constrained QP one of the cells assigned to the left window is placed to the right of a cell assigned to the right window, then the two cell subsets are merged and are partitioned once again (using this time the positions of the constrained QP) The old assignment is always replaced by the new assignment Note that only pairs of cell groups are considered that belong to the same window before the previous partitioning, so this reassignment is the last chance for a cell to leave its window

orders of the arrays of regions) as long as it yields a considerable improvement of the weighted netlength

Repartitioning enables the cells to leave the region in which they are currently placed It has also been used by Huang and Kahng (1997) in a minimum-cut-based placer and by Xiu and Rutenbar (2005) in their warping approach

17.6.2 PARALLELIZATION

Analytical placement methods that use recursive partitioning allow a parallel implementation of most parts of the algorithm Sometimes, placement and partitioning in one region does not depend

on another region, so both regions can be handled in parallel However, it should be mentioned that many analytical placers apply a global optimization before a partitioning step where all cells are placed simultaneously For example, in Gordian (Kleinhans et al., 1991), the placements with minimum quadratic netlength (with different center-of-gravity constraints) can hardly be parallelized Nevertheless, even some parts of these global optimization steps allow a parallel computation if the assignment of the cells to their windows is used as hard constraints Assume, for example, that

we want to compute the x-coordinate of a cell c for which we have the constraints µ ≤ x(c) ≤ ν for

some numbersµ and ν Then, if we minimize linear netlength, the x-coordinate of c can be computed

without knowing the x-coordinates of the cell that have to be placed to the left of µ or to the right

ofν Thus, the x-coordinates in different columns given by the regions can be computed in parallel

(and analogously for the y-coordinates).

Such a parallel computation is possible as well if quadratic netlength is minimized and connections are split at the borderlines of regions (see Section 17.5.2)

Also multisection can be done in parallel for separate regions Moreover, local optimization steps like repartitioning that are often quite time consuming can be performed efficiently in parallel (see Brenner and Struzyna, 2005)

17.6.3 DEALING WITHMACROS

Analytical placers can handle cells of different sizes and shapes However, recursive partitioning has to stop when cells are too big compared to the region size Hence, for larger macros only a few partitioning steps can be made Then, macros have to stay more or less at their position

In Gordian (Kleinhans et al., 1991), a region is only partitioned if it contains a sufficient number

of cells, so in the presence of macros the region sizes may differ over a large range at the end of global placement Finally, macros are legalized together with the standard cells

Other analytical placers such as BonnPlace (Vygen, 1997; Brenner and Struzyna, 2005) place the macros legally as soon as they are too big compared to the region size and fix them before continuing with the recursive partitioning

17.7 CONCLUSION

Analytical placement is the dominant strategy for placement today Decomposing the task into minimizing netlength and partitioning with respect to area constraints is natural Using quadratic

placement and multisection as the two main components has the advantage that both subproblems can be solved almost optimally very efficiently even for the largest netlists Moreover, this approach has nice stability features and works well in a timing-closure framework Therefore, this approach

is widely used in industry for many of the hardest placement problems

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18 Force-Directed and Other

Continuous Placement

Methods

Andrew Kennings and Kristofer Vorwerk

CONTENTS

18.1 Introduction 347

18.2 Basic Elements of Force-Directed Placement 349

18.2.1 Quadratic Optimization Preliminaries 349

18.2.2 Force-Based Spreading 351

18.3 Alternative Techniques for Spreading Cells 352

18.3.1 Fixed Points and Bin Shifting 352

18.3.1.1 Fixed Points in mFAR 353

18.3.1.2 Fixed Points in FastPlace 357

18.3.2 Frequency-Based Methods 359

18.4 Enhancements 361

18.4.1 Interleaved Optimizations 361

18.4.2 Multilevel Optimization 364

18.5 Nonquadratic, Continuous Methods 365

18.5.1 Placement via Line Search 365

18.5.2 APlace and the Log-Sum-Exp Approximation 366

18.5.3 mPL and Its Generalization of Force-Directed Placement 369

18.6 Other Issues 371

18.7 Conclusions 373

References 374

18.1 INTRODUCTION

Force-directed methods have been studied over the past four decades as a means of placing cells These methods employ forces to move cells into positions of shorter wirelength or smaller delay The use of forces was borne out of the physical analogy with Hooke’s law in which cells connected by nets can be viewed as exerting attractive spring forces on one another If the cells in such a system could move freely, they would move in the direction of their forces until the system achieved equilibrium at

a minimum energy state Unfortunately, a minimum energy placement is most often not valid as cells have physical dimensions that are ignored in the spring analogy Consequently, additional repulsive forces are applied to perturb the cell positions and remove overlap Force-directed methods, in general, purge cell overlap over many placement iterations, while trading off attractive and repulsive forces

to achieve a placement in which cells are placed with little overlap For example, the progress made

by a force-directed placer on circuit IBM04 from the ICCAD04 mixed-size placement benchmark suite [1] is illustrated in Figure 18.1

347

(a) (b)

(d) (c)

FIGURE 18.1 Typical progression of a force-directed placement method for the circuit IBM04from the ICCAD04mixed-size placement benchmark suite: (a) initial placement, (b) after roughly 1/3 through placement, (c) after roughly 2/3 through placement, and (d) before legalization and detailed improvement The fairly nonoverlapping placement before legalization is obtained without the use of partitioning

Force-directed methods differ from other placement methods, including simulated annealing, minimum-cut, and analytic methods Simulated annealing typically begins with an initial feasible (or nearly feasible) placement and applies iterative improvement Conversely, force-directed methods typically begin with no initial placement and construct the placement as they progress Minimum-cut and analytic methods are also constructive, but rely on partitioning of the placement area to remove cell overlap Force-directed methods, however, do not use partitioning, but rather eliminate cell overlap through the introduction of repulsive forces

The earliest implementations of force-directed methods were examined in the 1960s [2], and many adaptations of these methods remain in use today Although many variations exist, it is a proper understanding of the similarities and differences between the methods that can lead to either

a successful or unsuccessful implementation

In this chapter, we examine force-directed methods by considering how some of these work We

do not make comparisons between the methods, but rather attempt to illustrate the issues, similarities, and differences between the various implementations described in the literature We also examine other continuous placement methods (i.e., methods that do not rely on partitioning to remove cell overlap) that, although not seemingly force-directed, still share characteristics with force-directed methods This chapter is organized as follows In Section 18.2, we describe the traditional force-directed method that employs quadratic optimization to minimize wirelength and additional constant forces to remove cell overlap We describe methods that use techniques other than constant forces

to eliminate cell overlap in Section 18.3 This includes fixed-point and frequency-based methods Quadratic optimization is not the best choice for high-quality placements Many force-directed methods are used in combination with other optimization strategies—we touch upon these interleaved optimizations in Section 18.4 In Section 18.5, we describe other continuous placement techniques and describe their relationships with force-directed methods Section 18.6 describes several issues facing force-directed methods, and Section 18.7 offers concluding remarks

18.2 BASIC ELEMENTS OF FORCE-DIRECTED PLACEMENT

Placement typically begins with a circuit netlist modeled as a hypergraph Gh(Vh , Eh) with vertices

Vh = {v1, v2, , v n , v n+1, , v n +p } representing circuit cells and hyperedges Eh = {e1, e2, , e m} representing circuit nets The set{v1, v2, , v n } represents movable cells and the set {v n+1, , v n +p}

represents preplaced cells and I/O pads Each vertex v i has dimensions w i and h i that represent the width and height of its corresponding circuit cell, respectively Let(x i , y i ) denote the coordinates

of the center of vertex v i Placement information is then captured in the x- and y-directions by two

placement vectors x= (x1, x2, , x n ) and y = (y1 , y2, , y n ).

Placement seeks to optimize objectives, including the minimization of total interconnect length, routing congestion, power consumption, and timing requirements, subject to the constraint that cells cannot overlap Of course, the simultaneous optimization of these different objectives is difficult Wirelength is one of the most commonly employed measures of quality—the minimization of wire-length tends to be simpler and also aids in the minimization of other objectives The most commonly used measurement of wirelength in modern placement is the half-perimeter wirelength (HPWL),

which, for any given net e ∈ EH, is the minimum rectangle that encloses all cells on net e and can be

written as

HPWL(e) = max

i,j ∈e,i<j |x i − x j| + max

The total wirelength of the circuit is given by

e ∈EHHPWL(e) Although HPWL is a convex function,

it is neither strictly convex nor differentiable because of the absolute distances|x i −x j | and |y i −y j|— its direct and efficient minimization is difficult Placement focuses on minimizing an approximation

of HPWL subject to the constraint that no cells may overlap In practice, HPWL is a reasonably close approximation to the final, routed wirelength [3]

18.2.1 QUADRATICOPTIMIZATIONPRELIMINARIES

Kraftwerk[4], perhaps the best-known force-directed placer, introduced a quadratic approxima-tion [5] to HPWL and many other placers have since followed its lead In quadratic placement, the circuit hypergraph is transformed into a weighted graph Such a transformation necessitates that each hyperedge be modeled as a set of two-pin nets using a suitable net model Typical net models include

clique or star models, as shown in Figure 18.2 for a five-pin net In the clique model, each k-pin net

is replaced by k (k − 1)/2 two-pin nets In the star model, a star node is added for each net to which

all pins of the nets are connected If the weight of a k-pin hyperedge is W , it is common to weight the set of two-pin nets using a weight such as W /(k − 1) [6].

The selection of the net model is an implementation decision It is typical to use a clique model for small nets with few pins, but to switch to the star model for nets with a large number of pins The clique model results in a denser quadratic optimization problem, whereas the star model tends

to improve the sparsity of the problem, but requires additional dummy cells to represent the star nodes At first glance, it might appear that the choice of net model can influence the solution to the placement problem and this observation is generally true However, it has been demonstrated [7–9] that weights for the clique and star models can be selected such that the placement solutions are

(a) (b)

FIGURE 18.2 Models for a five-pin net in which the net is modeled as a (a) clique or as a (b) star In both

cases, the edges in the net model can be weighted (From Viswanathan, N and Chu, C.C.-N., IEEE Trans CAD

24, 5, 722, 2005 Copyright IEEE 2005 With permission.)

identical regardless of the model used Without any loss of generality, we will assume a clique model for all nets in the remainder of our discussion.∗

A quadratic placement can be obtained by minimizing the unconstrained objective function given by

(x, y) =

i,j

w i,j [(x i − x j )2+ (y i − y j )2] (18.2)

where w ij represents the weight on the two-pin edge connecting cells i and j in the circuit’s weighted

graph representation This objective function is separable into
(x, y) =
(x) +
(y) and can be

written in matrix notation (x-direction only) given by

(x) = 1

2x

TQxx + cT

The n x × n xHessian matrix Qxencapsulates the connections between pairs of movable cells and is symmetric positive definite.†Vector cxencapsulates connections between movable cells and fixed cells Finally, the constant is a result of connections to and between fixed cells Equation 18.3 is minimized by solving the positive definite system of linear equations given by

and is typically solved using any number of iterative solvers including CG, SYMMLQ, GMRES, BICGSTAB, and so forth [10] Matrix preconditioning is also used to improve the overall efficiency

of the iterative solver, including ILU, drop tolerance, and random walk preconditioners [10,11] Quadratic optimization is often referred to as force-directed placement Such an analogy follows if the weighted two-pin nets in the circuit’s graph approximation are viewed as springs We can consider the netlist as a system of objects connected by springs with different spring constants (weights) Minimizing the quadratic objective is equivalent to putting the system into a force-equilibrium state

in which the resultant force on each movable cell owing to all of the connected spring forces is

zero This result stems from Equation 18.4, in which the ith row of the system of linear equations represents the resultant force on cell i being set to zero.

∗ We note that models based on Steiner trees [12], multipin net decomposition [13], and bounding box [14] have also been suggested in the literature; we refer interested readers to these sources for more information.

† Positive definiteness requires the presence of fixed cells [5,15] Further, fixed cells are required for a unique solution to the optimization problem.

Unfortunately, solving Equation 18.4 results in a cell placement with significant cell overlap For example, Figure 18.1a shows the placement for circuitIBM04from theICCAD04mixed-size placement benchmark suite [1] after the solution of an unconstrained quadratic program—significant cell overlap clearly exists

18.2.2 FORCE-BASEDSPREADING

BothKraftwerkandFDP[16] apply additional constant forces to reduce cell overlap The force

Equation 18.4 is extended with an additional constant force vector fxyielding

The vector fx is used to perturb the placement such that cell overlap is reduced It is easy to show that the additional forces do not restrict the solution space and that any given placement can satisfy

Equation 18.5 by proper selection of fx[4]

Cell overlap is not removed just by solving a single perturbation of Equation 18.4 by Equa-tion 18.5 Instead, the cell overlap is removed over numerous iteraEqua-tions with the addiEqua-tional constant forces being updated at each iteration to reflect the changing distribution of cells throughout the placement area Hence, the additional constant forces are accumulated over iterations and the force

equation at any given iteration i can be written as

Qxxi + cx+

i−1



k=1

fk

x + fi

The additional constant force is divided into two parts, namely those forces accumulated over previous

placement iterations 1 through i −1 and a current constant force computed at iteration i Equivalently,

the additional constant force computed at any given iteration is broken into two specific components, namely (1) a stabilizing force that holds the current placement in equilibrium (represented by the accumulation of forces from previous iterations) and (2) a perturbing force computed for a given placement to further reduce cell overlap

In addition to the requirement that the additional forces be used to distribute cells evenly throughout the placement area, Ref [4] specifies additional requirements that the forces must satisfy:

1 Force on a cell depends only on its position

2 Overutilized regions of the placement area are sources of forces

3 Under-utilized regions of the placement area are sinks of forces

4 Forces should not form circles

5 Forces should be zero at infinity

Given these requirements, the force f (x, y) acting on a cell at position (x, y) within the placement

region is computed using Poisson’s equation given by

∂2φ(x, y)

∂x2 +∂2φ(x, y)

where

d (x, y) is a measure of density at position (x, y)

κ is a constant of proportionality

φ(x, y) is a scalar function such that ∇φ(x, y) = f (x, y)