Rectangle.
with text and connection points. Can be stretched to any dimensions.
Square.
Drag & drop to add an arrow with a 45-degree arrowhead.
Arrow with flexible head shape.
e the shape of the arrow's head.
Arrowhead Shape
Arrowhead Shape 2
Tail Shape
Arrow with flexible head and tail shape.
pe of the arrow's head and tail.
Tail Shape 2
Root Entry CompObj Pictures Current User SummaryInformation PowerPoint Document DocumentSummaryInformation
Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Times New Roman Symbol ew Roman
Blank Presentation Practical method: check / enforce density constraints
only in ww windows of fixed dissection with tiles
OlePres000 OlePres000
Root Entry Pictures PowerPoint Document SummaryInformation DocumentSummaryInformation Current User CompObj
OlePres000 OlePres000 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Times New Roman Symbol ew Roman Theorem: If all fixed dissection ww windows has density between U and L, then any ww window has density in the range between and
Theorem: If all fixed dissection tiles have density between L and U then any ww window has density in the range between
and
OlePres000 OlePres000 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Times New Roman
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OlePres000 OlePres000 Equation Equation.30 Microsoft Equation 3.0 Times New Roman Symbol ew Roman Filling problem for a fixed r-dissection:
Given r-dissection with tiles of size
Given Area and Slack of each tile (Tij)
The fill amount (Pij) in each tile should satisfy:
0  Pij  Slack(Tij)
 Pij  max {UW 2-area(W),0} for all wxw windows
Min variation formulation seeks to maximize the minimum window density: Maximize ( min (Area(Tij) + Pij) )
(r9@$c$99 * d#@ +\+99*
OlePres000 OlePres000 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Times New Roman Symbol ew Roman Maximize M subject to:
Pij Pij  Slack(Tij) i,j = 1, Area(Wij) + i,j= 1,…, - r +1
where:
Area(Wij) =
b,PGG6(Y5PP6G|
$ #W@ r * 2 W@ + V $ #W @ r * 2 W@
OlePres000 OlePres000 Equation Equation.30 Microsoft Equation 3.0 Times New Roman Symbol ew Roman Accuracy = ; r=1
SurvivingTiles = all tiles
while ( Accuracy > desiredAccuracy)
{
Find maximum standard window area using surviving tiles (Max)
Find maximum bloated window area using surviving tiles (BloatMax)
Delete tiles that do not belong to any bloated window with area > Max
Accuracy = BloatMax / Max
r = 2r // replace surviving tiles by 4 sub-tiles
}
Output maximum window area = (BloatMax + Max) / 2
Root Entry Pictures PowerPoint Document Times New Roman Symbol ew Roman SummaryInformation DocumentSummaryInformation Current User CompObj
OlePres000 OlePres000 gQE>&
OlePres000 OlePres000 Times New Roman Symbol ew Roman Given: a design rule-correct geometry with:
k disjoint rectangles in nn layout region
a minimum feature size c
area density lower and upper bounds
window size w < n
Goal: add fill geometries into layout while:
preserving circuit function & design rule-correctness
all ww windows satisfy lower/upper density bounds
Root Entry CompObj Pictures Current User SummaryInformation PowerPoint Document DocumentSummaryInformation f3fff
FQ=XG\ Times New Roman Blank Presentation
Root Entry Pictures PowerPoint Document mN@9@+9c SummaryInformation DocumentSummaryInformation Current User CompObj
OlePres000 OlePres000 Times New Roman Symbol ew Roman Theorem (Hanan analogue): there is extremal ww window with a corner at Hanan gridpoint
mN@9@+9c
Root Entry CompObj Pictures OlePres000 Current User SummaryInformation PowerPoint Document
Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Times New Roman Symbol ew Roman
-time algorithm for k rectangles:
For each pivot rectangle R do:
Find the density of ww window W that abuts R on top and right
While W intersects R do:
slide W right untill intersection with next rectangle edge
record changes in density as you go
Repeat for R, this time sliding up
Repeat for all other orientations of W
Speedup via fixed preprocessing:
Partition layout into squares of size ww each
Record all rectangles intersecting each ww square
Proceed as above
Expected runtime: DocumentSummaryInformation OlePres000
OlePres000 OlePres000 Document Word.Document.80 Microsoft Word Document Document Word.Document.80 Microsoft Word Document Document Word.Document.80 Microsoft Word Document Times New Roman
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0&KKKK 9&KKKK 5&KKKK 1&KKKK dCKCKtC2K n\S&lCKC2B**KK UdSl \S&SK*KC UdSl l\K*C* UdSl KK&K KK&K 2&KKKK 2&KKKK 9&KKKK KK&K
KK&K KK&K 6&KKKK 4&KKKK 1&KKKK KK&K KK&K KK&K KK&K 0&KKKK 6&KKKK 3&KKKK KK&K KK&K KK&K
kdCKCKtC2K lCCK2CBM CK&lCK:*)M dSl&\*tC &KKKK &KKKK 0&KKKK 9&KKKK 5&KKKK 1&KKKK kdCKCKtC2K n\S&lCKC2B**KK UdSl \S&SK*KC UdSl UdSl
KK&K KK&K 2&KKKK 2&KKKK 9&KKKK KK&K KK&K KK&K 6&KKKK 4&KKKK 1&KKKK KK&K KK&K KK&K KK&K 0&KKKK 6&KKKK 3&KKKK KK&K KK&K KK&K
Root Entry Pictures PowerPoint Document PPPPGP(Y-,- SummaryInformation DocumentSummaryInformation Current User CompObj
OlePres000 OlePres000 Times New Roman
OlePres000 OlePres000 Document Word.Document.80 Microsoft Word Document Document Word.Document.80 Microsoft Word Document Times New Roman
dCKCKtC2K lU*BLKK)&:*CC slU&K&1CC)CKK*C: U&l*KKKl&:*CC KK&KKK KK&KKK KK&KKK KK&KKK KK&KKK KK&KKK KK&KKK 1KKK&KKK KK&KKK dCKCKtC2K CK&lCK:*)M dSl&\*tC 5&KKKK 0&KKKK 1&KKKK 1&KKKK KK&K
5&KKKK 0&KKKK KK&K kdCKCKtC2K elU*BLKK)&:*CC slU&K&1CC)CKK*C: U&l*KKKl&:*CC KK&KKK KK&KKK KK&KKK KK&KKK KK&KKK KK&KKK KK&KKK 1KKK&KKK KK&KKK
kdCKCKtC2K CK&lCK:*)M dSl&\*tC 5&KKKK 0&KKKK 1&KKKK 1&KKKK KK&K 5&KKKK 0&KKKK KK&K
OlePres000 OlePres000 Times New Roman Symbol ew Roman
Times New Roman Symbol ew Roman Blank Presentation
Times New Roman Blank Presentation
OlePres000 OlePres000 Equation Equation.30 Microsoft Equation 3.0 Times New Roman Symbol ew Roman Given: k rectangles in nn layout
Find: extremal-density ww window
Inclusion-exclusion -based -time algorithm:
initialize boolean nn array B with 0’s and 1’s
create integer nn array S[i, j] = # 1’s in B[1..i,1..j]
density ww window at (i, j) = S[i+w, j+w] - S[i, j+w] - S[i+w, j] + S[i, j]
Page-1
Black fill
White fill
Blue fill
Red fill
Green fill
Cyan fill
Yellow fill
Magenta fill
Gray fill
10% Gray fill
White line
30% Gray fill
Long dashed line
Blue dark fill
Cyan dark fill
Green dark fill
Magenta dark fill
Red dark fill
Yellow dark fill
50% Gray fill
70% Gray fill
90% Gray fill
1pxl line
3pxl line
9pxl line
Hairline
Short dashed line
Arial centered
Arial top left
Arial top
Times centered
Times top
Times top left
Arial 8pt centered
Rectangle
Rectangle.3
Rectangle.43
Rectangle.30
Rectangle.18
Flexi-arrow 1
Rectangle.25
Rectangle.26
Rectangle.27
Rectangle.28
Rectangle.31
Rectangle.15
Rectangle.33
Rectangle.24
Row_1
Rectangle.36
Rectangle.44
Rectangle.39
Rectangle.40
Rectangle.41
Rectangle.42
Rectangle.45
Rectangle.46
Flexi-arrow 3
Rectangle.49
Square
45 degree single
Rectangle.50
Rectangle.51
Rectangle.54
Square.53
Rectangle.56
Rectangle.57
Rectangle.58
Rectangle.59
Rectangle.60
Square.61
Square.64
Rectangle.65
Rectangle.66
Rectangle.67
Rectangle.68
Rectangle.69
Square.70
Flexi-arrow 1.47
Square.74
Rectangle.75
Rectangle.76
Rectangle.77
Rectangle.78
Rectangle.79
Square.80
Rectangle.14
Flexi-arrow 1.13
Flexi-arrow 1.5
Flexi-arrow 1.6
Row_2
Row_3
Row_4
Guide
Gesture Format
Arial Unicode MS
Symbol
Wingdings
Arial
SimSun
PMingLiU
MS PGothic
Dotum
Sylfaen
Estrangelo Edessa
Vrinda
Shruti
Mangal
Tunga
Sendnya
Raavi
Dhenu
Latha
Gautami
Cordia New
MS Farsi
Gulim
Times New Roman