Double patterning layout decomposition for simultaneous conflict and stitch minimization.
ABSTRACT Double patterning lithography (DPL) is considered as a most likely solution for 32 nm/22 nm technology. In DPL, the layout patterns are decomposed into two masks (colors), and manufactured through two exposures and etch steps. If the spacing between two features (polygons) is less than certain minimum coloring distance, they have to be assigned opposite colors. However, a proper coloring is not always feasible because two neighboring patterns within the minimum distance may be in the same mask due to complex pattern configurations. In that case, a feature may need to be split into two parts to resolve the conflict, resulting in stitch insertion which causes yield loss due to overlay and lineend effect. While previous layout decomposition approaches perform coloring and splitting separately, in this paper, we propose a simultaneous conflict and stitch minimization algorithm with an integer linear programming (ILP) formulation. Since ILP is in class NPhard, the algorithm includes three speedup techniques: 1) grid merging; 2) independent component computation; and 3) layout partition. In addition, our algorithm can be extended to handle design rules such as overlap margin and minimum width for practical use as well as offgrid layout. Our approach can reduce 33% of stitches and remove conflicts by 87.6% compared with two phase greedy decomposition. Index Terms—Double patterning lithography, integer linear programming, layout decomposition.

Conference Paper: Dealing with IC manufacturability in extreme scaling
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ABSTRACT: As the CMOS feature enters the era of extreme scaling (14nm, 11nm and beyond), manufacturability challenges are exacerbated. The nanopatterning through the 193nm lithography is being pushed to its limit, through double/triple or more general multiple patterning, while nonconventional lithography technologies such as extreme ultraviolet (EUV), ebeam directwrite (EBDW), and so on, still have grand challenges to be solved for their adoption into IC volume production. This tutorial will provide an overview of key overarching issues in nanometer IC design for manufacturability (DFM) with these emerging lithography technologies, from modeling, mask synthesis, to physical design and beyond.Proceedings of the International Conference on ComputerAided Design; 11/2012  SourceAvailable from: libdspace.uwaterloo.ca

Conference Paper: WISDOM: wi re s preading enhanced d ecomposition o f m asks in double patterning lithography
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ABSTRACT: In Double Patterning Lithography (DPL), conflict and stitch minimization are two main challenges. Postrouting mask decomposition algorithms [14] may not be enough to achieve high quality solution for DPLunfriendly designs, due to complex metal patterns. In this paper, we propose an efficient framework of WISDOM to perform wire spreading and mask assignment simultaneously for enhanced decomposability. A set of Wire Spreading Candidates (WSC) are identified to eliminate coloring constraints or create additional splitting locations. Based on these candidates, an Integer Linear Programming (ILP) formulation is proposed to simultaneously minimize the number of conflicts and stitches, while introducing as less layout perturbation as possible. To improve scalability, we further propose three acceleration techniques without loss of solution quality: oddcycle union optimization, coloringindependent group computing, and suboptimal solution pruning. The experimental results show that, compared to a postrouting mask decomposition method [2], we are able to reduce the number of conflicts and stitches by 41% and 23% respectively, with only 0.43% wire length increase. Moreover, with proposed acceleration methods, we achieve 9x speedup.Proceedings of the International Conference on ComputerAided Design; 11/2010
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IEEE TRANSACTIONS ON COMPUTERAIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 29, NO. 2, FEBRUARY 2010185
Double Patterning Layout Decomposition for
Simultaneous Conflict and Stitch Minimization
Kun Yuan, JaeSeok Yang, and David Z. Pan, Senior Member, IEEE
Abstract—Double patterning lithography (DPL) is considered
as a most likely solution for 32nm/22nm technology. In DPL,
the layout patterns are decomposed into two masks (colors),
and manufactured through two exposures and etch steps. If
the spacing between two features (polygons) is less than certain
minimum coloring distance, they have to be assigned opposite
colors. However, a proper coloring is not always feasible because
two neighboring patterns within the minimum distance may be
in the same mask due to complex pattern configurations. In that
case, a feature may need to be split into two parts to resolve the
conflict, resulting in stitch insertion which causes yield loss due to
overlay and lineend effect. While previous layout decomposition
approaches perform coloring and splitting separately, in this
paper, we propose a simultaneous conflict and stitch minimization
algorithm with an integer linear programming (ILP) formulation.
Since ILP is in class NPhard, the algorithm includes three
speedup techniques: 1) grid merging; 2) independent component
computation; and 3) layout partition. In addition, our algorithm
can be extended to handle design rules such as overlap margin
and minimum width for practical use as well as offgrid layout.
Our approach can reduce 33% of stitches and remove conflicts
by 87.6% compared with two phase greedy decomposition.
Index Terms—Double patterning lithography, integer linear
programming, layout decomposition.
I. Introduction
A
due to the delay of the next generation lithography equipment
such as extreme ultraviolet [1]. Double patterning lithogra
phy (DPL) [2]–[5] emerges almost the only alternative for
32nm/22nm nodes and it is already used for NANDflash
production. In DPL, a single layout is decomposed into two
masks and manufactured through two exposure/etching steps.
As a benefit, the pitch size is doubled, which enhances the
resolution as illustrated in Fig. 1. Although DPL requires two
masks and increases the design cost, it is widely considered
as a most likely solution for 32nm, 22nm, and even 16nm.
Double patterning layout decomposition [6]–[8] is a process
that assigns two features within the given minimum space
S the minimum feature size decreases, semiconductor
industry is facing the limitation of patterning sub32nm
Manuscript received June 9, 2009; revised August 17, 2009. Current version
published January 22, 2010. This paper was supported in part by the National
Science Foundation, Semiconductor Research Corporation, Sun, Qualcomm,
and equipment donations from Intel. This paper was recommended by
Associate Editor P. Saxena.
The authors are with the Department of Electrical and Computer
Engineering, University of Texas, Austin, TX 78731 USA (email:
kyuan@cerc.utexas.edu; jsyang@cerc.utexas.edu; dpan@cerc.utexas.edu).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCAD.2009.2035577
Fig. 1.
is increased effectively in DPL.
One single design is decomposed into two masks and the pitch size
to different masks. A layout may contain a pattern which
is unable to assign a color. In this case, a feature may be
split into two parts and colored differently to resolve the
conflict, which generates stitches. Stitches will cause yield
loss and increase manufacturing cost due to overlay errors,
which is 5nm or 6nm under current 32nm double patterning
lithography. Some mask misalignment direction [4] could be
actually beneficial for printability. However, on the presence of
various process uncertainties, such as dose, focus, and mask
errors, the printed stitch width could be easily smaller than
25nm and result in design failure. Pushing overlay below
3nm [9] is very challenging. Moreover, the additional line
ends may cause more pattern degradation and reduce yield
in case of defocus and dose variation. After splitting, a few
unresolved or even unresolvable conflicts may remain and will
be corrected by time consuming layout redesign. Therefore, it
is important to produce high quality decomposition solution
with less conflicts and stitches.
There are a few works focusing on standalone layout
decomposition. A heuristic approach is proposed in [7] to
cut troublesome patterns after finding the coloring conflicts.
The patterns are prefragmented into smaller pieces in [8] to
perform coloring. All these works do not have a systematical
way to minimize the number of conflicts and stitches. Coloring
and splitting are considered in separate steps while they are
highly correlated tasks. Pattern matching technique is proposed
in [10] to decompose the layout. However, it might not be
able to work on large scale problem, hence limits the solution
quality. Recently, a practical layout decomposition flow is
proposed in [11] to address design needs for double patterning.
They first detect the features associated with unresolvable
conflict cycles for layout modification. The remaining design
is then decomposed to minimize the number of stitches based
on an ILP formulation. However, in their work, the number
of unresolvable conflict cycles and splitting stitches are not
optimized together, and conflict elimination technique is quite
greedy.
02780070/$26.00 c ? 2010 IEEE
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186IEEE TRANSACTIONS ON COMPUTERAIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 29, NO. 2, FEBRUARY 2010
Fig. 2. Concept of conflict and stitch.
In this paper, we propose an algorithm to decompose layout
for minimizing conflicts and stitches simultaneously. The
proposed approach reduces the conflicts by 87.6% with 33%
less stitches compared to a greedy two phase decomposition
flow. When compared to a methodology based on [11], we are
also able to achieve averagely 87.2% and 10% reduction on
conflicts and stitches, respectively. Although our approach is
comparatively slower, we can obtain coloring solutions for all
the test cases within a few minutes. The runtime shows linear
complexity with respect to problem size.
Our main contributions are as follows.
1) We propose a new grid model to enable bigger solution
space than previous works [7], [8] and perform simul
taneous conflict and stitch optimization.
2) We develop an ILP algorithm to minimize the number
of conflicts and stitches for a high quality solution.
3) We propose three speedup techniques (grid merging,
independent component computing and layout partition)
to improve the runtime and scalability of our algorithm.
For layout partition, we identify and solve a coloring
flip optimization problem to minimize the conflicts and
stitches across the boundary of different partitions.
4) We discuss how to extend our proposed grid model
to handle various splitting rules and design patterns in
practice.
The rest of the paper is organized as follows. Section II
provides the preliminaries and motivates. In Section III, we
discuss the problem formulation with related model and defi
nitions. The basic ILP formulation is described in Section IV
with three speedup techniques. The extensive discussion
on grid model for practical design issues is presented in
Section V. Section VI presents the experiment results and
Section VII concludes this paper.
II. Preliminaries and Motivation
A. Double Patterning Layout Decomposition Considerations
As explained in Section I, in DPL, the original design will
be assigned into two masks. There are two critical issues with
this layout decomposition: coloring conflict and splitting stitch.
1) Coloring Conflict: If the distance between two separate
features is less than minimum coloring spacing mincs, they
Fig. 3.
works [7], [8]. An unplanned coloring will need much extra effort during
splitting.
Shortcoming of two phase layout decomposition flow in previous
should be assigned to different masks (colors). Otherwise,
there will be a coloring conflict.
Fig. 2(a) shows a layout with three features, and any two
of them are required to have different colors because of the
insufficient spacing. A coloring conflict will be unavoidable
as in Fig. 2(b). Sometimes, such a violation can be eliminated
by appropriately splitting the features like Fig. 2(c). There
are also unresolvable conflicts, as Fig. 2(d) indicates, which
requires modifying the design.
2)
Splitting Stitch: The stitch exists when two touched
features are assigned to different masks. The stitch can be
inserted to split some features to resolve the conflict as shown
in Fig. 2(c). However, stitch insertion can have negative effects
on yield due to overlay error between the two masks as
Fig. 2(e) illustrates. In addition, the lineend will cause pattern
degradation.
There are several practical guidelines for splitting. As
Fig. 2(f) shows, in order to control the overlay, there is a
minimum overlap length, minol, requirement for stitch inser
tion. The segments h1 and h2 on different masks should be
overlapped to certain amount ensuring better manufacturabil
ity. Moreover, we do not want to have any minimum width,
minwi, rule violation during splitting, as marked by the circle
in Fig. 2(f).
Without altering layout in the scope, the general objective
of layout decomposition can be stated as minimizing the
unresolved conflicts by introducing as few as possible stitches.
B. Simultaneous Optimization
The previous works insert stitches after coloring to resolve
conflicts. Without planning possible splitting during coloring,
it is hard to eliminate the conflict. Considering a layout in
Fig. 3(a), we have a coloring solution in Fig. 3(b). During
the splitting, the U feature should be cut into two parts to
remove the conflict but we have to further check whether
the splitting will result in another conflict like Fig. 3(c). In
such case, the coloring of the neighborhood features needs to
be reconsidered to avoid unnecessary stitches like Fig. 3(d)
and enable optimal solution in Fig. 3(e) or (f). This is a
simple example, but as we can see, given the complexity
of modern design, the twophase approach will have extreme
difficulty handling the exploding consideration and producing
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YUAN ET AL.: DOUBLE PATTERNING LAYOUT DECOMPOSITION FOR SIMULTANEOUS CONFLICT AND STITCH MINIMIZATION187
Fig. 4. Different stitch candidates can lead to different solution qualities.
Fig. 5. Difficulty of predicting where the splitting is needed.
high quality solution. This motivates us simultaneous conflict
and stitch minimization during layout decomposition.
III. Problem Formulation
In this section, we will first motivate and introduce our grid
model in Section IIIA. The basic terms will be defined in the
following Section IIIB. The formal problem definition will be
described in the end.
A. Grid Layout Model
Considering splitting during coloring is a challenging prob
lem. First of all, the stitch configurations are highly correlated
and all the potential locations need to be considered for global
optimality. Fig. 4(a) is a case with two conflicts. As we can see,
two possible splitting choices on feature A lead to two different
solutions, Fig. 4(b) and (c). The first one has two stitches,
where the latter one associates with only one. Moreover, we
can even hardly predict where we could have a splitting due
to some chain effect. For example, the right most feature D is
not expected to be cut in Fig. 5(a) because it is only adjacent
to one single feature A. However, given a coloring assignment
as shown in Fig. 5(b), feature A will be split to resolve the
conflict between A and B like Fig. 5(c). As a result, feature
D also needs to be broken into two segments as shown in
Fig. 5(d).
In order to overcome these issues, we will map the whole
layout into grids with its size to be half the pitch of the
original design. Each grid is either empty or fully occupied
by the pattern, and each occupied grid will be assigned one
color. Therefore, any boundary between grids is a potential
splitting location. This is shown in the Fig. 6. Essentially, we
provide fine resolution for splitting options. This model is able
to offer sufficient stitch candidates for all the features across
the design in practice and the solution space is much bigger
than previous works [7], [8]. The discretization is reasonable
because a design usually follows underlying regular pitches
Fig. 6. Proposed grid layout model.
Fig. 7.
box.
Concept of blocking path. The solid rectangle marks the bounding
in modern layout. Minimum coloring spacing mincsis taken
as twogrid size to double the spacing for each mask in this
paper and also subject to change according to given mincs.
B. Terms and Problem Formulation
Before formulating our problem, we will first define the
terms in the grid layout model.
Definition 1 (Occupied Grid, OG): The grid filled by the
layout.
The OG must be assigned one of the two colors: gray and
black.
Definition 2 (Blocking Path, BP): Given two occupied grids
OG1and OG2, a blocking path is a path when:
1) it is fully composed of OGs and connects OG1 and
OG2;
2) OG1 and OG2 are touching its two ending grids, re
spectively, but not belonging to this path;
3) this path is within the bounding box of OG1and OG2.
The main usage of blocking path is to identify neighboring
but locally isolated layout grids. These grids, even belonging
to the same connection, need to be considered as different
features, and could form a coloring conflict.
As shown in Fig. 7(a), C–D is a blocking path for grid A
and B. In another example Fig. 7(b), C–F is not a BP for A–B,
because not all of them are in the bounding box of A–B as the
third rule defines. Some part of it (C–E) is beyond the box,
and hence locally A–B can be considered as isolated.
Definition 3 (Potential Conflict Grid Pair, PCGP, and
Potential Stitch Grid Pair, PSGP): Given two occupied grids
OG1and OG2.
1) If the distance between OG1and OG2is less than mincs
and the two grids are not touching, they form a potential
conflict grid pair.
2) If OG1 and OG2 are touching, they form a potential
stitch grid pair.
The distance between a pair of OGs is the minimum
distance between any two points from the OGs. For example
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188IEEE TRANSACTIONS ON COMPUTERAIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 29, NO. 2, FEBRUARY 2010
Fig. 8.
the bounding box of A and B.
Stitch grid pair and conflict grid pair. Dashed box in (c) and (d) is
in Fig. 7(b), the distance for untouched B and C is
size due to two closest corners, which is smaller than mincs.
Therefore, they form a PCGP.
Definition 4 (Stitch Grid Pair, SGP): If the grids of a PSGP
are assigned different colors, it is a stitch grid pair.
Definition 5 (Conflict Grid Pair, CGP):
the identical color, and there is no blocking path connecting
them in the same mask, it is a conflict grid pair.
The definition of SGP is straightforward as grids A and B
shown in Fig. 8(a). Fig. 8(b) shows the normal CGP cases,
where a PCGP is colored identically and unconnected. B–F
and A are within the minimum coloring spacing. There are
even no paths connecting them, not to mention blocking path.
The rule one of Definition 2 is violated. As a result, any of
B–F and A are a CGP.
There are also some special CGP cases that we need
to further consider blocking path in order to avoid false
recognition of lithography friendly pattern. If two nontouching
grids are electrically connected through a blocking path, we
should not consider them belonging to different features. The
printability will not be an issue. As shown in Fig. 8(c), grid
A and B have a BP C–D in the same mask between them,
so they do not form a CGP. It is indeed a normal jog, and
can be printed well. In contrast, although there is a path C–F
connecting A and B in Fig. 8(d), C–E is out of their bounding
box. In consequence, the path is not a blocking path. This
violates the third rule of Definition 2, so grid A and B form
a CGP. In this case, A and B are in fact locally isolated
but neighboring within the bounding box. This configuration
is a typical U shape pattern, and would have weak printa
bility.
√2 grid
If a PCGP is in
C. Problem Description
In our work, we use the number of SGPs and CGPs
as the cost, which assigns higher weight to the grids that
are associated with more conflicts/stitches. Formally, we for
mulate the layout decomposition optimization problem as
follows:
Problem Formulation: Given a grid layout, color it into two
parts (gray and black). The primary objective is to minimize
the number of CGPs and the second objective is to minimize
the number of SGPs.
We prefer a solution with less CGPs than one with smaller
number of SGPs but more CGPs, because a layout with
nonzero CGPs is essentially not manufacturable and a solution
with less CGPs reduces expensive redesign effort.
Fig. 9.Overall layout decomposition flow.
TABLE I
Notation for Basic ILP Formulation
ogi,j
xi,j
Occupied grid of which i and j are coordinates.
Binary variable that denotes the color of ogi,j.
xi,j= 1 if the color is gray, otherwise, it is black.
Binary variable sij,mn= 1 if ogi,jand
ogm,nis a SGP.
Binary variable cpq,uv= 1 if ogp,qand ogu,v
is a CGP.
Set of PSGPs.
Set of PCGPs.
Set of BPs connecting ogp,qand ogu,v.
kthBP connecting ogp,qand ogu,v.
Number of grids in pk
Binary variable gpq,uv= 1 if pk
is a gray BP.
Binary variable bpq,uv= 1 if pk
is a black BP.
sij,mn
cpq,uv
SP
CP
Ppq,uv
pk
pq,uv
nk
pq,uv
gk
pq,uv
pq,uv.
pq,uv
bk
pq,uvpq,uv
IV. Algorithm
In this section, we will present our ILPbased layout de
composition algorithm. The entire flow is shown in Fig. 9.
After mapping the design to grid model, we will process
the grids and formulate the basic ILP formulation. Since the
timing complexity for ILP is very high, we will then propose
three speedup techniques by either eliminating unnecessary
variables or dividing the whole problem into several smaller
ones. Finally, the layout decomposition for the entire design
can be obtained by merging the subproblem solutions. For
better solution reunion, we formulate a problem of coloring
flipping optimization through ILP.
A. Basic ILP Formulation
To better present our method, we first describe the notation
in Table I. The simultaneous coloring and splitting optimiza
tion can be formulated as follows:
⎛
sij,mn∈SP
subject to
min
⎝
?
sij,mn+ α
?
cpq,uv∈CP
cpq,uv
⎞
⎠
(1)
xi,j+ (1 − xm,n) ≤ 1 + sij,mn
∀sij,mn∈ SP
(2)
(1 − xi,j) + xm,n≤ 1 + sij,mn
∀sij,mn∈ SP
(3)
Page 5
YUAN ET AL.: DOUBLE PATTERNING LAYOUT DECOMPOSITION FOR SIMULTANEOUS CONFLICT AND STITCH MINIMIZATION 189
?
?
?
xe,f∈pk
pq,uv
xe,f≤ (nk
pq,uv− 1) + gk
pq,uv
∀pk
pq,uv∈ Ppq,uv
(4)
xe,f∈pk
pq,uv
(1 − xe,f) ≤ nk
pq,uv(1 − gk
pq,uv)
∀pk
pq,uv∈ Ppq,uv
(5)
xe,f∈pk
pq,uv
(1 − xe,f) ≤ (nk
?
xp,q+ xu,v≤ 1 + cpq,uv+
pq,uv− 1) + bk
pq,uv
∀pk
pq,uv∈ Ppq,uv
(6)
xe,f∈pk
pq,uv
xe,f≤ nk
pq,uv(1 − bk
pq,uv)
∀pk
pq,uv∈ Ppq,uv
(7)
?
k
gk
pq,uv
∀cpq,uv∈ CP
(8)
(1 − xp,q) + (1 − xu,v) ≤ 1 + cpq,uv+
?
k
bk
pq,uv
∀cpq,uv∈ CP. (9)
The objective function (1) is to minimize the weighted
summation of SGPs and CGPs. Parameter α is used to tune
the relative importance between SGP and CGP, and can be set
to ensure the priority of CGP elimination. All the PCGPs and
PSGPs are predetermined by examining the neighboring grids
for each OG.
Constraints (2) and (3) are used to identify SGP from
PSGP. According to the definition of SGP, we need to know
whether the PSGP grids have opposite colors. Whenever xi,j
and xm,nhave opposite values, the left hand side of one of the
constraints will be two. As a result, sij,mn must be assigned
one to satisfy the constraints, which detects a SGP.
The usage of Constraints (4)–(9) is to determine whether
a PCGP forms a CGP. Identifying CGP takes more effort.
Besides checking the colors of PCGP, we need to know
whether there is a blocking path in the same mask. All the
possible BPs Ppq,uv can be easily enumerated by depth first
search on the occupied grids within the bounding box. We
can investigate their coloring using Constraints (4)–(7). The
corresponding binary variable gk
if the grids of some blocking path are in the same mask.
Constraints (8) and (9) evaluate the conditions for CGP. A
conflict will be reported only if PCGP grids are assigned same
color and the possible BPs gk
Let nogbe the number of occupied grids, the basic formu
lation contains at most O(nog) variables. The constraints are
specified for detecting either PSGPs or PCGPs. Suppose there
are nspPSGPs and ncpPCGPs, the complexity of nspis O(nog).
ncpis linearly related to nog, but quadratically proportional to
mincs. The complexity of constraints due to PSGPs is O(nsp).
The constraint number for PCGPs is linear proportional to ncp.
It is also exponentially related to mincs, which results from
the enumeration of blocking paths. Although this formulation
shows exponential complexity in terms of mincs, when we fix
the value of mincsas the presetting for layout decomposition,
the number of variables and constraints is quadratic with
respect to nog.
The proposed integer linear formulation can minimize the
number of conflicts and stitches simultaneously. However,
because ILP is nondeterministic polynomial timecomplete, it
is not affordable to directly apply a basic ILP formulation for
large modern designs.
pq,uv/bk
pq,uvwill be true only
pq,uv/bk
pq,uvdo not exist.
Fig. 10. Main idea of grid merging.
Fig. 11.
having no interacted PSGPs/PCGPs and marked by the dashed circle.
Example of breaking big layout into two independent components,
B. SpeedUp Techniques
In this section, we will discuss three speedup techniques.
The clustering methodology is applied in grid merging to
reduce the number of variables and constraints. In contrast,
the key idea of the other two techniques is to use a divide
and conquer algorithm to convert the problem into smaller
subproblems.
1) Grid Merging: In the proposed grid model, we aim to
provide very fine resolution for stitch candidates. This may be
over skilled under certain situations.
Consider the layout segment L in Fig. 10(a) with unit grids
A–B–C–D. Only the two ending grids A and D may have
coloring interaction with other layout objects besides L. B and
C can be considered as isolated to some extent, because there
are no occupied grids outside A–B–C–D which are touching
them or within mincs of their boundary. Therefore, it is not
possible for B or C to form a stitch or conflict with other
layout apart from the grids of segment L.
We can make advantage of above property to reduce prob
lem size by combining this type of connected grids into a big
super grid. As graphically shown in Fig. 10(b), B and C can
be treated as a united grid T. This is equivalent to enforce B
and C the same color. It will not deteriorate the conflict and
stitch optimization. For this super grid, it does not have any
chance to form a conflict or stitch with surrounding grids other
than its two adjacent grids A and D.
Generally speaking, the elimination of internal splitting
candidates is not a problem for solution quality. For any
optimized solution obtained under original grid model with
internal stitches, it can be mapped to one solution in the
merged model with the stitch propagated to its ending grids,
such as from (c) to (d) in Fig. 10.
2)
Independent Component Computation: We propose
independent component computation for reducing the ILP
problem size without losing optimality. In real layout, we
observe many isolated occupied grid clusters, i.e., there are
no PSGPs or PCGPs formed between them. Therefore, we can