For the moment we ignore the algorithm used to define the coarsening and concentrate just on the question of how to place the modules, once a placement of their parent clusters has been
Trang 1ec∈Rm ˜q(ec) ≡ q(˜s + Pec) =
2(˜s + Pec) Q (˜s + Pec) − b (˜s + Pec)
It is easily shown that ecminimizes˜q(ec) if and only if it satisfies
PTQPec= PT
Galerkin coarsening thus defines the linear coarsening operator PT: Rn → Rmas the transpose of
the linear interpolation operator P, as simple substitution of Equation 19.5 into Equation 19.4 yields Equation 19.6 on premultiplication by PT
By confining coarse-level iterative improvement to the perturbation e = Pec to˜s, relaxation
at the coarse-level is in effect decoupled from relaxation at the adjacent finer level The combined approximation˜s + e is thus not restricted to the range of P and can represent a broader set of solution candidates for fixed P than is possible if relaxation at the coarser level and subsequent interpolation are applied to the entire coarse-level solution scrather than just its perturbation ec
In both theory and practice, coarsening and interpolation are so closely related that defining one essentially characterizes the other, even if nonlinear or discrete methods are used
To give the simplified model concreteness for placement, consider the sample netlist H illustrated
in Figure 19.4 The coarsening of H on the right-hand side of the figure shows modules 1, 2, and 3
mapped to cluster 0, module 4 mapped to cluster 1, and modules 0 and 5 mapped to cluster 2 For the moment we ignore the algorithm used to define the coarsening and concentrate just on the question
of how to place the modules, once a placement of their parent clusters has been computed.∗Because
in this example there are six modules at the finer level and three modules at the adjacent coarse level, linear interpolation operators for this example are represented as constant 6× 3 matrices Let
Pconst denote the matrix for piecewise-constant linear interpolation, in which each module simply
inherits its parent cluster’s position Alternatively, let Pavgdenote the matrix for linear interpolation
in which each module is placed at some linear combination of its parent’s position and the positions
of other clusters containing modules with which it shares nets; let w ijdenote the weight that module
5
4
2 1
3
0
Coarsened H
2
0,5
1 0
Netlist H
e1
e5
e4
e4
e3
e3
e2
e2
FIGURE 19.4 Simple six module netlist H with five nets and its coarsened approximation.
∗ For concreteness in this example, we describe finer-level objects as modules and coarser-level objects as clusters, but the example applies to any two adjacent levels of hierarchy—the finer-level objects might also be clusters of still finer-level objects, etc.
Trang 2Piecewise-constant interpolation (a)
(b) Nonpiecewise-constant interpolation
C
FIGURE 19.5 Piecewise constant versus nonpiecewise-constant interpolation (declustering) Each component
within a cluster is placed at a weighted average of the locations of all clusters containing other components to which it is connected
i assigns to the position of cluster j, and assume weights are normalized such that w ij= 1 if module
i is contained by cluster j With modules indexed 0, 1, 2, …, 5 and clusters indexed 0, 1, 2, as shown
in the figure, Pconstand Pavgtake the following forms
Pconst=
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠ and Pavg=
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
Each is applied separately to the xc and yc coordinate vectors of the coarse placement to obtain
corresponding x and y coordinate vectors of the fine placement, e.g., x = Pavgxcand y = Pavgyc ∗(see also Figure 19.5)
A common objection to clustering is that its associations may be incorrect and therefore lead subse-quent iterations to the wrong part of the solution space To reduce the likelihood of poorly chosen clusters, the notion of a cluster can be generalized by weighted aggregation Rather than assign each cell to just one cluster, we can break it into a small number of weighted fragments and assign the fragments to different coarse-level vertices; these are no longer simple clusters and are instead called aggregates During interpolation, a cell’s initial, inherited position is then typically determined by that of several aggregates as a weighted average [16] Clustering, also called strict aggregation, is a special case of weighted aggregation To our knowledge, weighted aggregation is not currently used
by any published placement algorithm
19.2.4 SCALABILITY
The scalability of the multilevel approach is straightforward to obtain and understand Provided
relaxation at each level has order linear in the number Naof aggregates at that level, and the number
∗Formally, the matrix for the interpolation operator on s = (x, y) ∈ R n is P = diag(P x , P y ), where P xis the interpolation
matrix for the x coordinates, and P for y In these examples, P = P.
Trang 31 Their use is limited to subsets of bounded size, e.g., by sweeps over overlapping windows
of contiguous clusters at the current aggregation level
2 Starting placement for iterations at each level is good enough that the termination criteria for relaxation at that level can be satisfied after a strictly controlled number of iterations
In general, unstructured global optimization problems, these restrictions might sharply limit an algorithm’s ability to explore the solution space in any scalable way Placement, however, exhibits sparse local structure, where most variables are related through the objectives and constraints to only a small constant number of other variables In the presence of such structure, both analysis and results obtained in practice have confirmed that these restrictions need not impair final quality of results [9]
Consider a Fourier decomposition of the error e = ˆx − x∗ in a system of equations N (x) = 0 (x∗denotes an exact solution, ˆx is a computed approximation to x∗) By localized relaxation, we mean iterative optimization over a small subset of coupled variables, assuming all other variables are held fixed; e.g., Gauss Seidel (Equation 19.3) The fundamental observation is as follows [2]:
Observation 1 Localized relaxation tends to rapidly reduce high-frequency components of the
error in the system of equations Lower-frequency components of the error are reduced much more slowly.
There are two key consequences of the observation, both of which stem from the fact that the very notion of high-frequency comes from the modeling scale First, different frequency-range Fourier components of the error are most efficiently targeted by relaxation at different scales By coarsening the modeling scale, we can target a coarser scale of the error in the system—simply apply the same relaxation algorithm at the coarser scale Recursion over all scales of relevance produces highly accurate solutions in extremely fast and scalable runtime Second, both the outcome of relaxation and the convergence rates of individual variables during relaxation can be used to guide the construction of interpolation or aggregation operators This connection underlies the basic principles of successful multiscale algorithms for systems of linear and nonlinear differential equations, where a robust convergence analysis has been obtained [2]
In this way, analysis and practice in both linear and nonlinear systems of equations have established a link between relaxation and interpolation for problems with local structure [9]
19.2.5.1 Error-Correcting Algorithm MG/Opt
Elementary local convergence properties of a general error-correcting multiscale optimization algo-rithm have been established by Lewis and Nash [17] For simplicity, their consideration of constraints
is omitted here The error being corrected (to first-order) is introduced by transferring approximations from finer level to coarser levels
Consider the unconstrained minimization of a smooth nonlinear function F (s) over variables
s ∈ Rn within a modeling system which generates multiple resolutions of F and s as specified by subscripts h (finer scale) and H (coarser scale) In placement, a resolution is primarily a selection of
bin-grid dimensions used to enforce density constraints
A convergent, continuous nonlinear relaxation algorithm R is assumed given, along with continuous interpolation operator I h and continuous coarsening operator I H = I h T
Trang 4
The following steps comprise on single iteration (at level h) of a multiscale optimization algorithm
called MG/Opt [17] The algorithm makes explicit use of first-order coarse-grid corrections Given
a resolution
min
with initial estimate s (0) h If this(h) is the coarsest resolution level, solve Equation 19.7 to the fullest possibly accuracy by means of relaxation R Otherwise, apply the following steps.
1 Apply N1 > 0 iterations of R directly to Equation 19.7, obtaining improved estimate s (1)
h
Compute corresponding coarse-level estimate s (1) H ≡ I H
h s (1) h
2 Compute coarse-level gradient correction v H = ∇F H (s H,1 ) − I H
h ∇F h (s h,1 ).
3 Using initial point s H,1, recursively apply this algorithm to the corrected coarse-level problem minsH F H (s H ) − vT
H s H to obtain next iterate s H,2
4 Interpolate the coarse-grid step s H,2 −s H,1 back to a finer-level search direction e h = I h
H (s H,2−
s H,1 ).
5 Perform line search (for scalarα) at level h to obtain next iterate s h,2 = s h,1 + αe h
6 Apply N2 > 0 iterations of relaxation R to Equation 19.7 with initial point s h,2 to obtain s h,3
7 Finally, discrete (noncontinuous) refinement steps may be used to transform s h,3 to next
iterate s (1) h
At least one of N1, N2must be strictly positive; the other may be positive or zero
The algorithm is designed to be easy to implement from a given flat nonlinear relaxation and the ability to (a) model the problem at multiple resolutions and (b) transfer approximations between those resolutions
It is shown by Lewis and Nash [17] that MG/Opt converges under the given assumptions In particular, they establish the following facts rigorously
1 Corrected coarse-level model approximates the fine-level model to first order ine h
2 Multiscale search direction e h is a descent direction at the fine level: eT
h ∇F h (s (1)
h ) < 0.
3 limk→∞∇F h (s (k)
h ) = 0, i.e., algorithm MG/Opt converges.
4 Search directions e his well scaled; i.e., the natural stepα = 1 is likely to be accepted close
to a solution s∗ The latter property is necessary for fast convergence
19.3 MULTISCALE PLACEMENT IN PRACTICE
To design and implement a multiscale algorithm, one would like to know what requirements should
be imposed on its coarsening and interpolation, relaxation, and iteration flow, and what trade-offs among them can be reasonably expected In this section, we summarize characteristics of some leading multiscale algorithms and attempt to illustrate the trade-offs they create and the ways they manage them
Placement by multilevel optimization can be viewed as the natural recursive extension of clustering-based approaches considered in earlier work Schuler and Ulrich [18] compared top-down and bottom-up approaches to clustering for linear placement They observed large speedups compared
to flat algorithms They also observed that balancing cluster sizes (size incorporating both area and connectivity) was important Mallela and Grover [19] studied clustering as a means of accelerat-ing placement by simulated annealaccelerat-ing They maintained a priority-queue-like structure for cluster candidates Sechen and Sun [20] employed three levels of clustering in an annealing-based flow Hur and Lillis [21] used three levels of clustering in linear placement Cong and Xu [22] studied
Trang 5To our knowledge, Ultrafast VPR [24] is the first published work to recursively cluster a circuit model into a hierarchy of models for placement by multiscale optimization Ultrafast VPR is used
to accelerate the annealing-based VPR algorithm (versatile packing, placement and routing [25])
to reduce design times on field-programmable gate arrays (FPGAs) at some expense in placement quality (FPGA placement quality in Ultrafast VPR is measured by the area used.)
19.3.2 COARSENING
An aggregation strategy defines a recursive transformation of the data functions (objectives and con-straints) from finer-level to coarser-level representations These transformations reduce the numbers
of variables and constraints but sometimes increase their complexity as information is compressed Although some results exist for the accuracy of these transformations as approximations to the orig-inal problem in the graph context [2], formal results are, to our knowledge, not yet known in the hypergraph setting For this reason, contemporary convergence criteria in practice rely on heuristics and empirical observations
General clustering algorithms are described in Chapter 7 Typically, clustering algorithms for placement merge tightly connected cells in a way that eliminates as many nets at the adjacent coarser level as possible while satisfying some constraint on variation in cluster areas A class of coarsening algorithms more general than clustering is described briefly in Section 19.2.3
Important questions for coarsening in practice include the following:
1 How accurately do coarsened objectives and constraints approximate their corresponding finer-level counterparts? What error is incurred?
2 How much coarser than its finer-level source should a coarsened problem be?
3 How much variation in cluster areas should be allowed?
4 How coarse is too coarse? That is, when should recursive coarsening stop?
5 What trade-offs exist between (a) coarsening and relaxation and (b) coarsening and iteration flow? For example, how often can an algorithm afford to revise or completely reconstruct its coarsening hierarchy, and by what means?
6 Should the coarsening model the solution at the finer level, or the change in a given approximation to that solution? Why?
The questions are interdependent, and precise answers for placement are not yet known Leading academic multiscale placers model the full placement problem at coarser levels rather than the change
in a given placement as described in Sections 19.2.1 and 19.2.5 Only force directed placement (FDP)/line search directed placement (LSD) changes its coarsening/interpolation operators over its flow (Section 19.3.3)
The following classifications of coarsening algorithms for multiscale placement are sometimes used
1 Connectivity-based versus location-based: Although netlist connectivity is always used to some extent, location-based algorithms also require a given current placement as input and attempt to keep neighboring modules together (see Section 19.3.2.2)
2 Transient versus persistent [26]: Transient clusters appear as part of the inner steps of
an algorithm but are not associated with coarse-level variables For example, clusters are formed in multiscale partitioning algorithms used in top-down bisection-based placement, but they are not separately placed Persistent clusters, on the other hand, are computed a-priori and are actually placed
Trang 63 Score-based versus scoreless [27]: In scoreless algorithms, clusters are committed as soon
as they are formed Examples described in Chapter 7 include edge coarsening, heavy-edge matching, and first choice In score-based algorithms, alternative candidates are assigned scores and iteratively refined As the refinement proceeds, the clustering scores are updated, eventually only candidates with sufficiently high scores are selected to serve as clus-ters Examples described below include Best choice [26], Fine-granularity [28], and Net cluster [27]
Leading multiscale algorithms limit variation of cluster areas at each level of hierarchy APlace [29] and NTUPlace3 [30] limit cluster areas to at most 1.5 times the target cluster area FastPlace3 limits its cluster areas to at most 5 times the target
Among connectivity-based algorithms, experiments to date suggest that local-connectivity-driven greedy strategies like first-choice vertex matching [31,32] and best choice [26] may be more effective than global-connectivity-driven approaches like edge-separability clustering (ESC) [33] How best to define coarse-level netlists without explosive growth in the number and degree
of coarsened hyperedges relative to coarsened vertices is particularly challenging [27,28,32]; Sections 19.3.2.3 and 19.3.2.4 describe recently proposed solutions
Although various forms of clustering dominate the recent multiscale placement literature, Rent’s rule and the importance of limiting cutsize also make recursive partitioning an attractive means
of generating the multiscale hierarchy [7] Hybrid algorithms for floorplanning or placement that combine clustering with netlist partioning in some form continue to be developed [7,12,34–38]
19.3.2.1 Best-Choice Clustering
As described in Chapter 7 and briefly summarized here, best-choice clustering [26] is the method used by the leading multiscale tools APlace [29], mPL [39], FastPlace 3.0 [40], and RQL [14] Best choice is a netlist-based, score-based algorithm typically used in placement for persistent or semipersistent clustering
A graph is defined on the netlist vertices (modules) with each edge weighted by the affinity of the given two vertices The affinity may represent some weighted combination of complex objectives, such as hypergraph connectivity, spatial proximity, timing delay, area balance, coarse-level hyperedge
elimination, etc The affinities s (i, j) between vertices i and j used by APlace, mPL, FastPlace3, RQL,
and NTUPlace3∗for connectivity-based clustering are all equal to or slightly modified from the basic affinity formula
s (i, j) =
e ∈Nij
w (e)
a i + a j
(19.8)
where
N ij is the set of nets containing both module i and module j
a i and a j are the respective areas of modules i and j
w (e) denotes the weight of hyperedge e, typically taken proportional to 1/(|e| − 1)
An affinity-ordered priority queue (heap) of the vertices (including clusters and partial clusters)
is formed; each vertex in the heap is associated with its nearest neighbor under the given affinity metric At each step, the pair of vertices(u, v) with the highest affinity is removed from the heap and
clustered, if its total area does not violate the maximum-area constraint For efficiency, a lazy-update
strategy is then employed: the affinities of the netlist neighbor vertices of u and v are marked as
invalid rather than being immediately updated Affinities of invalid vertices are updated only when they arrive at the top of the heap
∗ NTUPlace3 uses first choice rather than best choice.
Trang 7reported for APlace [41] and mPL [39].
19.3.2.2 Location-Based Clustering
Location-based clustering is also called layout-based clustering or physical clustering After an initial placement has been obtained as a starting configuration, spatial proximity can be incorporated into the vertex-affinity metric used for clustering [12,16] A simple three-level illustration is given in Figure 19.6
Earlier versions of mPL [16] incorporated distance as a reciprocal factor in a second V-cycle.∗ FDP [12] uses an (additive) convex combination of connectivity and spatial proximity for its
ver-tex affinity function in hybrid first-choice clustering Specifically, the affinity score s (i, j) between vertices i and j is defined for a given placement as
sFDP(i, j) = λ
⎛
e ∈Nij
1
1+ |x i − x j | + |y i − y j | − ζ
⎞
⎠ + (1 − λ)
e ∈Nij
1
|e| − 1
where
N ij is the set of nets containing both vertex i and vertex j
|e| is the number of vertices in e
ζ specifies the minimum displacement possible between nonoverlapping modules i and j
Proximity-based aggregation
Proximity-based aggregation
Defines level 1
Optimization
at level 1 Defines level 2
Initial placement
at level 0
(f) (e)
(d)
FIGURE 19.6 Location-based clustering in a three-level multiscale-placement flow Given an initial placement
at the finest level, clusters at each subsequent coarser level are determined from a combination of netlist connectivity and spatial proximity
∗ Later versions of mPL, however, abandoned the location-based approach in favor of a single V-cycle derived by purely connectivity-driven best-choice clustering [42].
Trang 8The authors report best results forλ ≈ 0.25 The effect stabilizes iterations and supports multiple
V-cycles
Location-constrained clustering is used in a somewhat different way to define the coarsest level
of a three-level formulation used in FastPlace 3.0 [40] While the first level of fine-grain clustering (cf Section 19.3.2.3) relies solely on netlist affinities (Equation 19.8) and best choice, the second coarse-grain clustering imposes the added restriction that vertices must be closer than a certain parametrized limit in order to be clustered In FastPlace 3.0, that limit is set to 10 percent of the maximum chip dimension
A similar hybrid approach to clustering, using only netlist-connectivity at finer levels but both connectivity and proximity at the coarsest level, is used by mFar [11,43]
The work of Chen et al [44] describes a more radical approach.∗A cluster hierarchy is derived from four separate preliminary placements Several iterations of the successive-overrelaxation (SOR) variant of coordinate-wise relaxation [15] are applied to the linear equations for the optimality of flat, unconstrained quadratic HPWL perturbed by density-balancing forces computed by a Poisson solver [10] In each of four separate trials, the cells are initially placed all at the same point: one
of the four corners of the placement region Clusters are selected according to cells’ average final proximity over the results of all four trials Although this iterative, empirical approach to clustering requires significant runtime, it is a fixed initial cost that can be amortized over the cost of subsequent iterations Numerical results confirm the scalability of this approach
19.3.2.3 Mutual Contraction and Fine-Granularity Clustering
A central obstacle to multilevel placement of netlists is that, in the course of recursive clustering, modules tend to be eliminated far more rapidly than nets The result is often extremely densely interconnected coarse-level netlists with very different structure from the original finest-level netlists they are intended to approximate The problem has been partly addressed in the literature by clustering schemes that strive to eliminate as many nets as possible [32], in particular, nets of low degree To this end, the mutual contraction formulation [28] models the relative strength of a connection between
modules u and v relative to u’s total connectivity as
wr(u, v) = w (u, v)
x
w (u, x)
where the weight w (e) = 2/{[d(e) − 1]d(e)} comes from clique-model (graph-based)
approxima-tions of hyperedges in the netlist The contraction of two modules is defined as the symmetrized
product cp(x, y) = wr(x, y)wr(y, x); a pair of modules with a large contraction relative to other pairs
is a good candidate for clustering The notion is readily extended to arbitrary subsets of modules
In fine-granularity clustering [28], the contraction metric is used to order connections between modules in a priority queue The modules at the front of the queue are grouped together if their grouping does not violate area-balance or other constraints A target cluster limit is set a priori; clustering typically stops with most clusters containing two or three modules of average area
19.3.2.4 Net Cluster
The above strategies share a dependence on metrics derived from pairs of vertices but are largely oblivious to netlist structure involving more than two vertices The net–cluster study [27] shows that aggregation criteria defined over local subsets broader than simple pairs of vertices can be used
to improve quality and runtime of existing multiscale placers In particular, clustering the vertices
of a single multipin net is shown in many instances to improve quality of results over what can be
∗ The hierarchy is used by Chen et al to solve the linear system of equations for a flat, Poisson-based analytical placement formulation rather than to directly support multilevel placement.
Trang 93 7
FIGURE 19.7 Netlist with a natural clustering the vertex-pair-based algorithms typically fail to find The
natural clustering most appropriate for placement (or netlist partitioning) groups vertices 1–4 in one cluster and vertices 5–7 in another Vertex-pair-based algorithms overwhelmingly prefer to cluster vertices 4 and 5, precluding the natural clustering
obtained with pairwise strategy; cf Figure 19.7 Another advantage of net cluster is that it reduces the number of nets at approximately the same rate as the number of vertices
The net cluster algorithm consists of five distinct steps partitioned into two phases
Phase 1 Candidate cluster (a) identification, (b) refinement, and (c) scoring
Phase 2 Net-cluster (a) scoring and (b) final formation
Initial cluster candidates for phase 1 are simply the nets in the netlist; note that these are not disjoint Refinement of the candidates proceeds one by one on each of them For each one, several iterations
of FM bipartitioning (Chapter 7) are performed, the cluster candidate used as one subset of the netlist and the rest of the netlist as the other That is, this step attempts to determine whether each vertex
in a cluster candidate is more strongly connected to other vertices in the cluster or to the vertices
external to it After this refinement step, each candidate cluster C iis scored by
sc(C i ) = #nets inside the cluster
#modules inside the cluster× 1
cluster area That is, the score prefers candidate clusters that (1) absorb many nets, (2) aggregate many modules, and (3) have low area At the end of phase 1, candidate clusters still are not disjoint
In phase 2, each net N iis then also assigned a score equal to the sum of the scores of the candidate clusters containing it minus the sum of the scores of the candidate clusters cutting it (see Ref [27] for the precise form) Nets are then visited in descending score order, with one of the four following possible cases applied to each net
1 If clustering the cells violates cluster-area constraints, then this net is ignored, and the next net is considered
2 Otherwise, the cells in the net are clustered if none of them have already been clustered
3 Otherwise, if just one cell in the net has already been clustered, the net can be merged with
an existing cluster, if doing so does not violate cluster-area constraints
4 Otherwise, if at least two cells in the net have already been assigned to different clusters, a merge of all these overlapping clusters and the cells in the current net is made, if doing so does not violate cluster-area constraints
The net cluster study supports the view that considerable improvement in both runtime and QoR
of existing multilevel algorithms may be attainable through a more accurate modeling of local netlist structure during aggregation On ICCAD2004 test cases of up to approximately 200,000 objects,
a single pass of net cluster improves final HPWL of leading multiscale placers mPL6 [39] and NTUPlace3 [30] by 2–5 percent on average
Trang 10TABLE 19.1
Approximate Numbers of Clusters Ncoarsestat Coarsest Levels
of Leading Academic Multiscale Placers
Note: The finest-level netlist has Nfinest modules.
19.3.2.5 Coarsest Level
Because the initial placement at the coarsest level may have a large influence at subsequent iterations, and because the coarsest-level problem is relatively small, the placement at this level is typically performed with great care, to the highest quality possible For example, mPL [39] uses nonlinear programming while mPG [5] uses simulated annealing How to judge the coarse-level placement quality is not necesssarily obvious, however, as the coarse-level objective may not correlate strictly with the ultimate fine-level objectives Under such circumstances, multiple iterations over the entire hierarchical flow may sometimes be important [6,16], evaluation of objectives and constraints at the finest level providing feedback to the representation and optimization at all coarser levels
Coarsest-level problem sizes of some leading multiscale algorithms are shown in Table 19.1 From the table it is evident that an enormous range of coarsest-level problem sizes exists, from
just four movable objects in Dragon [7] to N /4 in FastPlace3, where N is the number of modules
in the original netlist The observation that flat (non-multiscale) algorithms, e.g., [13,45] remain competitive suggests that there is no inherent upper limit to best coarsest-level problem size Similarly, Dragon’s results [7] and results from other problem domains [46,47] strongly suggest the absence
of any hard lower bound The complexity of integer nonconvex nonlinear programming, to which placement belongs, is high enough that multiscaling can sometimes produce superior solutions to flat algorithms even when fewer than 100 variables are present
19.3.2.6 Numbers of Levels
Netlist-driven clustering by variations of best choice or first choice seems to allow a rapid reduction
in the number of modeling levels, by allowing a large cluster-size target, without loss of final place-ment quality In APlace, each coarse cluster level has about 1/10 the number of movable objects
of its adjacent finer level; from Section 19.3.2.5, this rule produces just four levels of hierarchy for APlace on a 1M-module test case In NTUPlace3, this ratio is approximately 1/5, and in mPL6 it is approximately 1/4
In contrast, in the location-based clustering used by FDP/LSD [12,48], at most 35 percent of modules are aggregated in one pass of clustering, leaving a much larger number of levels From Section 19.3.2.5, this rule produces 20 levels of hierarchy for FDP/LSD on a 1M-module test case
Of all the leading multiscale placers cited in this chapter, FDP/LSD [48] is the only one known to rely
on more than a single pass of successive refinement from coarsest to finest level (cf Figure 19.3) After each such pass of successive refinement in FDP/LSD, the entire hierarchy is reconstructed, using location-based clustering (Section 19.3.2.2), with one less level of clustering than used in the preceding pass On a 1M-module test case, this flow constructs a total of 20+ 19 + · · · + 1 ≈ 200 distinct problem formulations over its history
... final formationInitial cluster candidates for phase are simply the nets in the netlist; note that these are not disjoint Refinement of the candidates proceeds one by one on each of them For. .. For each one, several iterations
of FM bipartitioning (Chapter 7) are performed, the cluster candidate used as one subset of the netlist and the rest of the netlist as the other That is,... containing it minus the sum of the scores of the candidate clusters cutting it (see Ref [27] for the precise form) Nets are then visited in descending score order, with one of the four following possible