Rectangle.
with text and connection points. Can be stretched to any dimensions.
Use to create either a straight or curved connector.
Move control handle to change shape of curve.
Eccentricity
Rectangle with variable corner rounding.
f corners on rectangle.
Corner Rounding
Drag & drop to add an arrow with a 60-degree arrowhead.
Drag & drop to add an arrow with a 45-degree arrowhead.
Root Entry CompObj Pictures Current User SummaryInformation PowerPoint Document DocumentSummaryInformation Times New Roman Symbol ew Roman Monotype Sorts
1 port per terminal is involved
Interconnection is a forest blank.pot
OlePres000
Root Entry CompObj Pictures Current User SummaryInformation PowerPoint Document DocumentSummaryInformation
Times New Roman Symbol ew Roman blank.pot OlePres000
Times New Roman Symbol ew Roman Monotype Sorts blank.pot
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Times New Roman Symbol ew Roman MS LineDraw Tahoma Draw Monotype Sorts Proof requires concept of “edge reuse”
Green edge cost  corresponding path
Middle edge “reused” 4 times 
Green tree cost  4 x (black tree cost) blank.pot
Times New Roman Monotype Sorts Courier New blank.pot
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Root Entry CompObj Pictures Current User SummaryInformation PowerPoint Document DocumentSummaryInformation Times New Roman Monotype Sorts blank.pot
OlePres000 VG9@99j+
blank.pot In the greedy framework, we need to establish a measure
for the quality of partial d-stars.
With our heuristics, we measure the quality of a d-star
by its improvement over the cost of the corresponding
one-star which has the same root and spans the same groups.
We call this measure of quality the "norm” of a partial d-star
Our goal is to find low-norm partial d-stars. OlePres000
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Times New Roman Tahoma ew Roman Symbol ew Roman Monotype Sorts Strong Connectivity Version
Assign 0-cost to intra-group
edges  Steiner Tree Problem
The Steiner Tree Problem
Given: weighted graph G=(V,E) and node subset M Í V
Find: min-cost tree spanning M
NP-hard
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Root Entry CompObj Pictures Current User SummaryInformation PowerPoint Document DocumentSummaryInformation Times New Roman Monotype Sorts
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The graph shown here depicts experimental data from our
heuristic as well as that of Reich and Widmayer. Each point
represents the average tree cost over five hundred runs, given
as a percent improvement over Reich and Widmayer’s tree
cost (shown in green).
The blue line represents our basic heuristic, while the red line
represents our heuristic, further improved using a minimum
spanning tree -based post-processing step.
(BREAK TO OTHER SLIDE)
This is the result as the area of each group varies between 10%
and 100% of the
otal routing area and as the number of ports
per group varies from 3 to 8.
From this graph, we conclude that our algorithm works well in
practice as well as in theory. Our heuristic improves as the number
of ports per terminal increases and as the group area increases.
Our implementation may be found on the World Wide Web. OlePres000
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Left-most and right-most blocks not completely contained in subtree 
edge reused 2 times in such paths
Total 2-star cost  3 times original blank.pot
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Equation Equation.30 Microsoft Equation 3.0 Equation Equation.30 Microsoft Equation 3.0 Times New Roman Symbol ew Roman Monotype Sorts The Group Steiner Tree Problem:
Given: weighted graph G=(V,E) and k disjoint groups N1, …, Nk Í V
Find: min-cost tree containing at least one node from each group Ni
Even O(ln k) approx. is NP-hard
Best previous bound: O(k)
Our heuristics:
sub-linear performance bounds:
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Times New Roman Symbol ew Roman Monotype Sorts Modules can be rotated / flipped
Each terminal Þ 8 “virtual” ports blank.pot
Even if we have already determined the placement of
modules, we may wish to rotate them or flip them.
This means that every port may have up to eight “virtual"
positions, as we can see here. So, we can consider these
eight virtual positions as a group of ports, where only one
of the ports may be used in the actual interconnection
topology.
This suggests two variants of the basic problem OlePres000
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Root Entry CompObj Pictures Current User SummaryInformation PowerPoint Document DocumentSummaryInformation Times New Roman Times New Roman
OlePres000 sN9+c$@9#2 @@k N@#9$ U@@$#@@ G$99
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
Connector
Line connector
45 degree single.22
Rounded rectangle
45 degree single.9
45 degree single.16
Rectangle.25
60 degree single
45 degree single
Rectangle.10
Rectangle.42
Rectangle.6
Rectangle.44
Rectangle.45
Row_1
45 degree single.31
Guide
45 degree single.37
Gesture Format
Rectangle.40
Arial Unicode MS
Symbol
Wingdings
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SimSun
PMingLiU
MS PGothic
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Estrangelo Edessa
Vrinda
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Mangal
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Sendnya
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Cordia New
MS Farsi
Gulim
Times New Roman