2.
Background
• At the past, chips were
continuously getting
smaller and smaller,
and hence less power
consumption.
• However, we’re fast
approaching the end of
the road where optical
lithography(光刻)
cannot take us where
we need to go next.
4.
Sub-wavelength Lithograph
• Feature size <<
lithograph wavelength
o 45nm vs. 193nm
• What you see in the
mask/layout is not what
you get in the chip:
o 图形失真
o 成品率下降
5.
What is Double Patterning?
• Instead of exposing the photo-resist layer once
under one mask, as in conventional optical
lithography, expose it twice, by splitting the mask
into two, each with features half as dense.
6.
Key Techniques
• Novel polygon cutting algorithm to reduce the
number of rectangles and the total cut-length.
• Novel dynamic priority search tree for plane-
sweeping.
• Decompose the underlying conflict graph into its
tri-connected components using SPQR-tree
• Graph-theoretical approach instead of ILP
o Recast the coloring problem as a T-join problem and
is then by solved by Hadlock’s algorithm
7.
New Polygon Cutting Algorithm
• Allow minimal overlapping to
reduce the number of
rectangles, and hence to reduce
the number of conflicts.
• Limited support of diagonal line
segments
8.
Dynamic Priority Search Tree
• In plane sweeping, events are frequently
“inserted” and “deleted” to the scan line.
• In our PST, all data are stored at the leaf
nodes of PST, making “insert” and “delete”
operations very fast (O(1) time for each tree
rotation). The payoff is that the “query”
operation will be little slower than the
traditional PST.
9.
Splitting and Stitching
• Additional rectangle splits for resolving conflicts
10.
Conflict Detection
b
• Two rectangles are NOT conflict if
their distance is > b.
• Conflict: (A,C), (A,E), (E,B), (B,D),
but not (A,B), (A,D) (B,C)! A C
• Define: a polygon is said to be
rectilinearly convex if it is both x-
monotone and y-monotone.
• Rule: F
o (A,D) are not conflict because A-F-D
reconstructs a rectilinearly convex
polygon. E B D
o (A,C) are conflict because A-F-C
reconstructs a rectilinearly concave
polygon
12.
Layout Splitting Problem
Formulation
• INSTANCE: Graph G = (V,E) and a weight function
w : E N
• SOLUTION: Disjoint vertex subsets V0 and V1
where V = V0 ∪ V1
• MINIMIZE: the total cost of edges whose end
vertices in same color.
• Note: the problem is linear-time solvable for bipartite
graphs, polynomial-time solvable for planar graphs,
but NP-hard in general.
• To reduce the problem size, graph partitioning
techniques could be used.
13.
Bi-connected Graph
• A vertex is called a cut-vertex of G if removing it will
disconnect G.
• If no cut-vertex can be found in G, then the graph is called a bi-
connected graph.
• For example below, a and b are cut-vertices.
b
a
14.
Bi-connected Components
• A connected graph can be decomposed into
its bi-connected components in linear-time.
• Each bi-connected component can be solved
independently without affecting the final sol’n.
• Question: Is it possible to further decompose
the graph?
15.
Tri-connected Graph
• A pair of vertices is called a separation pair of a bi-connected
graph G if removing it will disconnect G.
• If no separation pair can be found, then the graph is called a tri-
connected graph.
• Eg below, {a,e}, {b,e}, {c,d}, {e,f}, {g,h} are separation pairs.
a e g h
d
c
b f
17.
SPQR-Tree
S
S
R
S P R
P
S S
• A bi-connected graph can be decomposed into its
tri-connected components in linear-time using a data
structure named SPQR-tree
18.
Divide-and-Conquer Method
• Three basic steps:
o Divide a graph into its tri-connected components.
o Solve each tri-connected components in a
bottom-up fashion.
o Merge the solutions into a complete one in a top-
down fashion.
We calculate two possible solutions for each
components, namely {s, t} in same color and {s, t}
in opposite colors.
20.
More Technical Details
• In Hadlock’s algorithm, voronoi graph instead
of complete graph is used.
• A brute-force method is used for solving the
maximum weighted planar subgraph problem
(could be improved)
26.
Current Status of Our SW
• fft_all: 320K polygons1.3M rectangles
o Conflict graph construction within 1 minute
o Color assignment within 9 minutes
o Compare: 26 minutes for just displaying the result
using “eog”
o Note: Only g++ 3.4.5 was used, no advanced
compiler optimization has been done yet.
27.
Conclusions
• Experiment results show that our method can
achieve 3-10X speedup
• We believe that it is a key to the success of
22nm process
• Unfortunately we didn’t have chance to try a
realistic 32/22nm layout yet
• because nearly everything is confidential
under 90nm
• Foundries may move to EUV if DPL fails.
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