1. Detailed Routing
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2. Multi-Stage Routing of Signal Nets
Detailed Routing
Routing
Multi-Stage Routing of Signal Nets
Timing-Driven Routing
Large Single- Net Routing
Global Routing
Detailed
Routing
Coarse-grain assignment of routes to routing regions
Fine-grain assignment of routes to routing tracks
considering
critical nets
P&G
routing
Geometric Techniques
Clock routing

3. Detailed Routing
The routing regions are divided into channels and switchboxes.
So only need to consider the channel routing problem and the switchbox routing problem. A B
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4. Channel Routing for Different Styles
Gate-Array:
Channel widths are fixed.
The goal is to finish routing of all the nets.
Standard-Cell and Full-Custom:
Channels are expandable.
The goal is to route all nets using the minimum channel width.
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5. Standard Cells vs. Gate Array
(a) Two tracks required and
all connections routed.
(b) Shorter wire length but
three tracks required.
In a Standard Cell design, an additional track could be added
while in a Gate-Array, the designer is faced with extra wire length or no connection.

6. Channel Routing in Standard Cell
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7. Channel Routing
Standard Cell Row
External Pad
Channel
Power Rail

8. Channel vs Switchbox
Channel may have exits at left and right sides, but exit positions are not fixed
We may map exits to either lower or upper edge of a channel
Terminal positions on all four sides of a switchbox are fixed
Switchbox routing is more difficult! 2 1 2 3 3 4 4 1 1 3 2 4 4 3 1 2 1 2 2 3 3 1
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9. Projects Schedule Encounter Tool: 31/3/93 (in person)
Placement Algorithm: 15/4/93 ( )
Global Routing Algorithm: 11/5/93 (in person)
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11. Channel vs Switchbox D B E A B C D

12. Channel Ordering A The width of A is not known until
A is routed, we must route A first. B A B C D
The routing order should be:
???
BCAD or BCDA
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13. Channel Ordering No feasible channel order! C D B A
1. Fix the terminals between A & B
2. Route B, C, then D (channel)
3. Route A (switchbox)
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14. Channel Routing Terminology
Terminals
Via
Upper boundary
Tracks
Dogleg
Lower boundary
Trunks
Branches
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15. Grid-Based vs. Gridless Model
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16. Routing Layer Models 1 Layer 2 Layers Layer 1 Layer 2 Layer 3 Via
VH Model
HV Model
2 Layers
Layer 1
Layer 2
Layer 3
Via
VHV Model
HVH Model
3 Layers
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17. Unreserved Layer Model
Routing Layer Models
HVH Model
VHV Model 1 2 3 3 2 1
Unreserved Layer Model
Layer 1
Layer 2
Layer 3
Via
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18. Channel Routing Problem
Input:
Two vectors of the same length to represent the pins on two sides of the channel.
Number of layers and layer model used.
Output:
Connect pins of the same net together.
Minimize the channel width.
(Minimize the number of vias.)
Example: ( )
( )
where 0 = no terminal 1 3 2 1 1 3 1 2 3
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19. Problem Instance and Its Solution
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20. Constraint Graphs 1 6 1 2 3 5 1 6 1 2 3 5 1 2 3 4 5 6 6 3 5 4 2 4 6 3 5 4 2 4
Vertical Constraint Graph:
Horizontal Constraint Graph: 6 5 2 1 1 5 4 3 6 3 4 2
Maximum clique = ???
Longest path = ???
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21. Vertical Constraint GraphsB D E B F G D A C E F H G
Note: Transitive edges are not included B D G E F C A H 21

22. Lower Bounds on Channel Width1 6 1 2 3 5 6 3 5 4 2 4
1. Length of the longest path in the Vertical Constraint Graph (i.t.o. no. of vertices)
2. Channel Density = Maximal clique 1 6 1 2 3 5 6 1 2 3 1 4 5 5 6 3 4 6 3 5 4 2 4
Local
Density 1 3 3 4 4 4 2 2
Lower bound = 4
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Lower bound = 3

23. Lower Bounds on Channel Width1 2 3 1 2 1 2 3 3
Lower bound = 3
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24. Cycles in Vertical Constraint Graph
If there is a cycle in the vertical constraint graph, the channel is not routable.
Dogleg can solve the problem. 1 1 2
Vertical
Constraint
Graph 2 1 2 1 2 2 1
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