An%20Exact%20Algorithm%20for%20Difficult%20Detailed%20Routing%20Problems

1
An Exact Algorithm for Difficult Detailed
Routing Problems

• Kolja Sulimma
• Wolfgang Kunz
• J. W.-Goethe Universität Frankfurt

2
Motivationrouting for automated leaf cell
generation

• 1-D and 1.5-D cells
• channel routing seems more adequate than maze
  routing
• Scenario somewhat different from conventional
  inter cell channel routing

3
Motivationchannel routing for automated leaf
cell generation

• a typical routing channel in a standard cell
  layout looks like this
• long channel with many tracks

4
Motivationchannel routing for automated leaf
cell generation

• a typical routing channel in a standard cell
  layout looks like this
• long channel with many tracks
• fast algorithms like Greedy Router perform very
  well they need only one or two tracks more than
  the optimal solution

5
Motivationchannel routing for automated leaf
cell generation

• a typical routing channel in a standard cell
  layout looks like this
• long channel with many tracks
• fast algorithms like Greedy Router perform very
  well they need only one or two tracks more than
  the optimal solution
• some algorithms need a few extra columns

6
Motivationchannel routing for automated leaf
cell generation

• a channel in a leaf cell is much smaller

7
Motivationchannel routing for automated leaf
cell generation

• a channel in a leaf cell is much smaller
• the use of extra area is relatively expensive

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Motivationchannel routing for automated leaf
cell generation

• a channel in a leaf cell is much smaller
• the use of extra area is expensive
• or might simply be not feasible

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Motivationchannel routing for automated leaf
cell generation

• small problem instances
• small constant area overhead of heuristical
  algorithms becomes a large relative area overhead
• asymptotically slow algorithms can be practical
• difficult instances
• obstacles / keep-out regions
• dense terminal placement

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Problem Formulationrouting model
Grid-based over the cell channel routingin the
restricted two layer routing model

• grid-based
• all elements are aligned to a routing grid
• over-the-cell-routing
• terminals can be located anywhere in the channel,
  not only at the top or bottom boundary
• restricted two layer model
• one layer for horizontal connections (trunks)
• one layer for vertical connections (doglegs)
• arbitrary obstacles
• Each signal crosses each column at most once
• (no forks or detours)

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Problem Formulationexample channel segment
1
1
3
3
5
2
4
2
5
1
4

• Connect all terminals that belong to the same net
  in a channel of length n and a given height of t
  tracks

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Problem Formulationexample channel segment
1
1
1
5
3
3
3
5
2
4
2
1
4
4

• Connect all terminals that belong to the same net
  in a channel of length n and a given height of t
  tracks

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Algorithmic Approachdecision problem

• algorithm does not search for a minimum height
  routing
• tries to find a routing for a given channel
  height of t tracks
• this is all that is needed for constant height
  leaf cells
• minimization is done by iterating over t
• obviously increases the runtime
• in the paper we show that the overhead is only
  O(1)

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Algorithmic Approachdynamic programming

• Sweep over the channel from left to right column
  by column like the Greedy Router
• Use track assignments to record for each track
  the signal that leaves the current column on this
  track
• Greedy Router heuristically selects one track
  assignment for each column
• We enumerate all reachable track assignments for
  the column
• Dynamic programming prevents an exponential
  growth of the search space with the channel
  length

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Algorithmpseudo code

• for (each column)
• for (each possible track assignment m of the
  previous column)
• check if m valid in this column with respect
  to terminals and
• obstacles
• use m to generate new track assignments by
  adding all combinations of doglegs

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Algorithmexample

• Denote a track assignment by a t-tupel of
  integers in the range 0, ..., t, where 0 means
  an empty track
• How can can signals 1 and 2 leave column 5 and
  enter column 6?

1
1
1
1
5
3
3
3
5
2
4
2
2
1
4
4
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Algorithmexample

• Denote a track assignment by a t-tupel of
  integers in the range 0, ..., t, where 0 means
  an empty track
• How can can signals 1 and 2 leave column 5 and
  enter column 6?

1 0 2 0
1
1
1
1
1
5
3
3
3
5
2
4
2
2
1
4
4
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Algorithmexample

• Denote a track assignment by a t-tupel of
  integers in the range 0, ..., t, where 0 means
  an empty track
• How can can signals 1 and 2 leave column 5 and
  enter column 6?

1 0 2 0
0 1 2 0
1
1
1
1
5
3
3
1
3
5
2
4
2
2
1
4
4
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Algorithmexample

• Denote a track assignment by a t-tupel of
  integers in the range 0, ..., t, where 0 means
  an empty track
• How can can signals 1 and 2 leave column 5 and
  enter column 6?

1 0 2 0
0 1 2 0
0 0 2 1
1
1
1
1
5
3
3
3
5
2
4
2
2
1
4
1
4
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Algorithmexample

• Denote a track assignment by a t-tupel of
  integers in the range 0, ..., t, where 0 means
  an empty track
• How can can signals 1 and 2 leave column 5 and
  enter column 6?

1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1
1
1
1
1
5
3
3
2
3
5
2
4
2
1
4
4
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Algorithmexample

• Denote a track assignment by a t-tupel of
  integers in the range 0, ..., t, where 0 means
  an empty track
• How can can signals 1 and 2 leave column 5 and
  enter column 6?

1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
1
1
1
1
1
5
3
3
3
5
2
4
2
1
4
2
4
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Algorithmexample

• Denote a track assignment by a t-tupel of
  integers in the range 0, ..., t, where 0 means
  an empty track
• How can can signals 1 and 2 leave column 5 and
  enter column 6?

1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
0 1 0 2
1
1
1
1
5
3
3
1
3
5
2
4
2
1
4
2
4
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Algorithmdetails of column processing

• // takes as input a set Mi-1 of all track
  assignments of column i-1 and
• // produces a set Mi of all track assignments for
  column i
• next_column(Mi-1, i)
• Mi Ø
• for (each m ? Mi-1)
• m removeRightEdgeTerminals(m, i-1)
• m addLeftEdgeTerminals(m, i)
• m connectTerminals(m, i)
• m checkForObstacles(m, i)
• if (m ? 0)
• Mi Mi ? m
• Mi Mi ? doglegs(m)

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removeRightEdgeTerminals()addLeftEdgeTerminals()

• Remove all signals that had their last terminal
  on the previous column.
• Add signals to the assignment that have their
  first terminal on the current column.
• Delete assignment if above is impossible

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removeRightEdgeTerminals()addLeftEdgeTerminals()

• Remove all signals that had their last terminal
  on the previous column.
• Add signals to the assignment that have their
  first terminal on the current column.
• Delete assignment if above is impossible

1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
0 1 0 2
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removeRightEdgeTerminals()addLeftEdgeTerminals()

• Remove all signals that had their last terminal
  on the previous column.
• Add signals to the assignment that have their
  first terminal on the current column.
• Delete assignment if above is impossible

1 2 4 0
1 0 4 2
0 1 4 2
1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
0 1 0 2
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connectTerminals()

• Existing signals might have terminals in this
  column that are not connected by the incoming
  assignment.
• Add dogleg to connect the signal to the terminal.
• Delete the assignment if the track with the
  terminal is blocked by another signal.

1 2 4 0
1 0 4 2
0 1 4 2
1 2 4 0
1 0 4 2
0 1 4 2
1
1
1
1
4
4
4
4
2
2
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doglegs()

• Enumerate all the possible dogleg combinations
  for this assignment.
• Multiple layouts might generate the same track
  assignment.
• These layouts are equivalent under routability
  aspects
• only one is selected by dynamic programming
• selection can be used to minimize number of vias,
  wire length, ...

1 0 4 2
1 0 4 2
1 0 4 2
1 2 4 0
1
1
1
1
4
4
4
4
2
2
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Computational Complexityworst case

• Up to t! possible assignments per column
• dogleg() dominates all other subroutines
• less than 2t/2 doglegs per assignment
• worst case runtime per column O(t!2t)
• O(nt!2t) for a channel of length n
• fixed parameter tractable
• overall runtime linear for a fixed channel height
• runtime exponential in the channel height

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Data StructuresMDDs

• Represent the set of track assignments by its
  characteristic function f0, ..., tt ? 0, 1
• represent f by an MDD (Multi Valued Decision
  Diagram)
• reduces the size of the representation
• typically 1M assignments require 80K MDD-nodes
• allows efficient manipulation of the assignments
  in the set

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Data StructuresMDD example

• MDD representation of the example set in column 6
• connections to the 0-terminal are omitted for
  clarity.

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Data StructuresMDD example
1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
0 1 0 2

• Each path from the root to the 1-terminal
  represents a track assignment

1
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Data StructuresMDD example
1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
0 1 0 2

• Each path from the root to the 1-terminal
  represents a track assignment

1
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Data StructuresMDD example
1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
0 1 0 2

• Each path from the root to the 1-terminal
  represents a track assignment

1
35
Data StructuresMDD example
1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
0 1 0 2

• Each path from the root to the 1-terminal
  represents a track assignment

1
36
Data StructuresMDD example
1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
0 1 0 2

• Each path from the root to the 1-terminal
  represents a track assignment

1
37
Data StructuresMDD example
1 0 2 0
0 1 2 0
0 0 2 1
1 2 0 0
1 0 0 2
0 1 0 2

• Each path from the root to the 1-terminal
  represents a track assignment

1
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Data StructuresMDD example LeftEdgePorts()
1
4

• To be able to connect a new signal to track 2 the
  subset of assignments with an empty track 2 must
  be found
• calculate cofactor ftrack20

1
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Data StructuresMDD example LeftEdgePorts()
1
4

• To be able to connect a new signal to track 2 the
  subset of assignments with an empty track 2 must
  be found
• calculate cofactor ftrack20

1
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Data StructuresMDD example LeftEdgePorts()
1 2 0 0
1 0 0 2
0 1 0 2
1
4

• To be able to connect a new signal to track 2 the
  subset of assignments with an empty track 2 must
  be found
• calculate cofactor ftrack20

1
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Data StructuresMDD example LeftEdgePorts()
1 2 0 0
1 0 0 2
0 1 0 2
1
4

• To insert signal 4 in the empty track move track
  2 edges from 0 to 4

1
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Data StructuresMDD example LeftEdgePorts()
1 2 0 0
1 0 0 2
0 1 0 2
1
4

• To insert signal 4 in the empty track move track
  2 edges from 0 to 4

1
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Data StructuresMDD example LeftEdgePorts()
1 2 4 0
1 0 4 2
0 1 4 2
1
4

• To insert signal 4 in the empty track move track
  2 edges from 0 to 4

1
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Experimental results

• runtime per column in milliseconds
• tracks Hashset MDD
• 3 2 5
• 5 4 5
• 7 100 5
• 9 6000 20
• 11 300
• 13 8000

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Conclusion

• Intra cell channel routing can be solved exactly
  by exhaustive enumeration of reachable track
  assignments
• dynamic programming yields a runtime linear in
  the channel length
• Possible application to conventional channel
  routing
• less than exhaustive search
• more than one assignment per column as Greedy
  Router does.
• gt heuristically select a set of maybe a few
  hundred track assignments for each column