Fill Compression by Hierarchical Rectangle Covering

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Fill Compression by Hierarchical Rectangle
Covering
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Fill Compression by Hierarchical rectangle
Covering
AREF structure
(X1,Y1)
(X0,Y0)
(X2,Y2)
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Fill Compression by Hierarchical rectangle
Covering

• AREF description
• three coordinates
• number of rows
• number of columns
• degrees of rotation (optional)
• reflection (optional)

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Fill Compression by Hierarchical rectangle
Covering

• AREF (XY,COLROW,AREF(XY,COLROW(..)))
• hierarchical Fill definition

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Fill Compression by Hierarchical rectangle
Covering

• Objective
• to define a fill pattern in GDSII format
• to have maximum reduction of the size of the
  GDSII file

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Fill Compression by Hierarchical Rectangle
Covering

• A mathematical formulation
• Given
• A 0-1 matrix with each element corresponding to a
  filling site, where
• 1 site filled with "dummy" features
• 0 unfilled sites

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Fill Compression by Hierarchical Rectangle
Covering

• Definitions
• Region in a matrix is a single element
  (row-column intersection) of matrix
• OR a collection of h- or v-adjacent
  elements in it
• Block in a matrix is a single element of the
  matrix, surrounded by elements of different
  values OR a region with gt1 elements of identical
  values
• Rectangle is a rectangular region
• WFR is a largest set of h- or vertically
  equally-spaced rectangles of identical values, or
  WFRs.

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Fill Compression by Hierarchical Rectangle
Covering

• To find -
• a sequence of "well-formed" rectangles, covering
  all the
• "fill" geometries, in a "hierarchical bottom-up
  manner so
• as to achieve maximum compression of the GDSII
  file

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Fill Compression by Hierarchical Rectangle
Covering
Filled geometries
Rectangular block
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Fill Compression by Hierarchical Rectangle
Covering
Nested well-formed rectangle
Inner well-formed rectangle
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Fill Compression by Hierarchical Rectangle
Covering

• The Problem
• Given a binary (0,1) matrix B with size m x n
• find the minimum cover of all (possibly
  overlapping) blocks in B
• find all (possibly nested) well-formed
  rectangles in B

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Fill Compression by Hierarchical Rectangle
Covering

• The Proposed Methodology
• find all the maximal rectangles of adjacent 1s
  in B
• Recursively find all maximal rectangular
  patterns of blocks of 1s
• until no more such rectangle is obtained

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Fill Compression by Hierarchical Rectangle
Covering
recursive detection of maximal rectangular
patterns
x1,y1
x0,y0
Describing a block or wfr l x1 x0 w y2 y0
x2,y2
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Fill Compression by Hierarchical Rectangle
Covering
A
B
C
D
E
F
L
G
H
I
J
K
Given fill matrix
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Fill Compression by Hierarchical Rectangle
Covering
IJK
A
B
C
D
WFR hierarchy
E
F
J
K
L
I
K
G
H
I
J
K

• Information contents of a node of WFR hierarchy
• origin of parent WFR
• no. of component WFRs of parent
• inter-WFR distance between components
• orientation (V/H) of parent
• pointer to structure of component WFR

Fill matrix
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Fill Compression by Hierarchical Rectangle
Covering

• input a set of maximal rectangles of adjacent
  1s known origin of each maximal
  rectangle (rowno,colno) number of rows
  number of columns inter-row and
  inter-column separation (design specs) for each
  maximal rectangle of 1s form a node
  containing the above 4 items assume a grid of
  same dimension, row-columns as the input matrix
  assume a mapping of each of the above nodes to a
  cell of the grid at a position same as the
  origin coordinates apply find_max_chain output
  of find_max_chain a set of new nodes
  with the same information as above
  orientation of node a set of
  pointers to the constituent nodes

Constructing WFRs
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Fill Compression by Hierarchical Rectangle
Covering
Portion of a WFR Graph
A
B
C
ABC
ADE
D
E
B
D
C
A
E
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Fill Compression by Hierarchical Rectangle
Covering
A chain formed from WFRs
B
A
C
D
E
WFR symbol
L-R scan
D
Scanned WFR
Insert in D
D
Comparison
Output
(A,B)
d(A,B)
2
Nil
B
(A,B)
(A,C)
d(A,C)gtd(A,B)2
(A,B)
d(A,B)
C
(A,D)
d(A,D)ltd(A,C)2
d(A,C)
D
(A,C,E)
d(A,E)d(A,C)2
d(A,E)
E
d(A,E)ltd(A,D)2
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Fill Compression by Hierarchical Rectangle
Covering
Rectangles with holes

• 1-rectangles with interior 0-rectangles
• Steps
• find all isolated 0-rectangles in a preprocess
• preserve locations and sizes of these
  0-rectangles
• fill all such 0-rectangles with 1s
• find which 0-rectangles contained in which
  1-rectangles
• one 0-rectangle may be contained gt 1
  1-rectangles
• Questions
• what if a polygonal hole?
• how to integrate?

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Fill Compression by Hierarchical Rectangle
Covering
Isomorphic WFRs
A
A
B
C
B
C
Find a function that can map both WFRs to a
unique value ?
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Fill Compression by Hierarchical Rectangle
Covering
Time Complexity m number of rows n number
of columns Phase I O(mn) Finding the longest
chain O(mn2.d), where d is the depth of the WFR
graph Updating data structure for WFRs
O(mn) Chain to-rectangle constant
time Insertion of a rectangle in WFR graph - ?