An Exact Algorithm for CouplingFree Routing

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An Exact Algorithm for Coupling-Free Routing

• Ryan Kastner, Elaheh Bozorgzadeh,Majid
  Sarrafzadeh
• kastner,elib,majid_at_cs.ucla.edu

Embedded and Reconfigurable Systems
Group Computer Science Department UCLA Los
Angeles, CA 90095
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Outline

• Basic definitions
• Coupling-Free Routing
• Coupling-Free Routing Decision Problem (CFRDP)
• Implication graph
• Maximum Coupling-Free Layout (MAX-CFL)
• Conclusion

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Coupling

• Definition - capacitance between adjacent wires
• Deep submicron trends
• Interconnect has more dominant role
• Scale wire height at slow rate compared to width

smaller device sizes

• Coupling can account for up to 70 of
  interconnect capacitance even in .25 micron
  designs

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Simplify definition of coupling

• Pair-wise boolean relationship between any 2 nets
• Easy to generalize to non-boolean, makes
  algorithm more complex

length gt l
distance lt d
Two wires couple if Cc(i,j) gt threshold
Two wires couple if distance lt d and length gt l
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Coupling-Free Routing (CFR)

• Given a set of nets SNi(x1i,y1i),(x2i,y2i)
  1 ? i ? n
• S is coupling-free if there is a single bend
  layout for every net such that no two routes
  couple

Coupled layout
Coupling-free layout
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Benefits of CFR

• Smaller interconnect delay
• One bend routing insures minimum wirelength
• Only one via
• Coupling between nets minimized
• Aids in wire planning
• Speeds up routing process
• See Predictable Routing, ICCAD00

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CFR in Global Routing

• Coupling at global routing is hard to determine
• Routes are not exact, makes it difficult to know
  adjacency relations of nets
• Detailed router will often make local changes
• Global routing allows global changes, it is very
  hard to make global changes at the detailed stage
• Coupling-avoidance in congested areas very
  important
• A coupling-free global layout will produce a
  coupling-free detailed layout

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CFRDP definition

• Given a set of two-terminal nets. Is there a
  single bend routing for every net such that no
  two routings couple?
• We solve via a transformation to the
  2-satisfiability problem (2SAT)

CFRDP algorithm overview
2SAT
CFRDP
transform to
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CFRDP to 2SAT

• Assign a boolean variable to every net
• A variable is true (false) if it is routed in
  upper-L (lower-L)
• The clauses are formed based on forcing
  interactions between the nets

Net A uA
Net A
Net B uB
Net B
Net C
Net C uC
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Forcing interactions

• In order to avoid coupling, a routing of a net
  forces another net into a particular route

Net 1
Net 1
To avoid coupling Net 2 must be lower-L
Net 2
Net 2
An lower-L routing of Net 1 forces a lower-L
routing of Net 2 ûA ?ûB
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Forcing interactions

• There are 10 possible forcing interactions
  between Net A (solid - uA) and Net B (dotted -
  uB)
• Each interaction can be encoded into clauses in
  2SAT

1) No interaction, no clauses 2) Unroutable,
return FALSE 3) (uA,,uB) 4) (uA,,ûB) 5)
(ûA,,uB) 6) (ûA,,ûB) 7) (uA,,ûB), (ûA,,uB) 8)
(uA,,ûB), (ûA,,ûB) 9) (uA,,uB), (ûA,,uB) 10)
(uA,,uB), (ûA,,ûB)
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Implication Graph

• Given a set of nets N
• Implication graph is a directed graph G(V,E)
• The vertices are the routes of the nets
• The edges are defined by the forcing
• If Route A forces Route B, then there is an edge
  from Vertex A to Vertex B
• Implication graph is constructed in time O(N2)

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Property - Direct Forcing

• Edges formed according to (direct) forcings
• Therefore, the out degree of the Vertex A is the
  number of routes forced by Route A

C has out degree 2 Every other vertex has out
degree 1
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Property - Indirect Forcing

• A routing may force other routes indirectly

Route 1
Route 1directly forces Route 2 Route 2 directly
forces Route 3 Route 3 directly forces Route 4
Route 2
Route 3
Route 4
Route 1 indirectly forces Routes 3 and 4
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Property - Indirect forcing

• The number of indirect forcings of Route A is the
  number of vertices reachable from vertex A minus
  the out degree of vertex A

A indirectly forces 2 routes (B, D)
Modified DFS algorithm can be used to
determine indirect forcings
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MAX-CFL Definition

• Given a set of two-terminal nets S and a positive
  integer K lt S. Is there a single bend routing
  for at least K nets such that no two routings
  couple?
• Additional routing constraints can easily be
  added
• Routed nets must be planar
• Routed nets must be routed on two layers
• MAX-CFL for planar layouts is NP-Complete
• General MAX-CFL NP-Complete?

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Algorithms

• We developed three algorithms
• Greedy
• Direct Forcing
• Implication
• Algorithms try to maximize number of nets routed
  and/or criticality of routed nets

Presented at ASIC00
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Implication Algorithm

• 1 Given a set of nets N
• 2 Create an implication graph G(V,E)
• 3 R ? NULL
• 4 for each vertex v ? V
• 5 do r.net ? route(v)
• 6 r.num_forcing ? Forcings(route(v))
• 7 R ? R U r
• 8 Sort R by function(direct forcings, indirect
  forcings)
• 9 for each routing r ? R
• 10 do if r.net is unrouted and r is routable
• 11 then route r

Running time is O(n3)
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Benchmarks
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Experimental Flow
greedy, direct forcing, implication algorithms
Benchmark fract,biomed, ibm01,etc.
Select x most critical nets
Perform MAX-CFL algorithm
Statistics criticality, routed
x 25,50,75, , 250
Results
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Conclusion

• Coupling-free routing useful for many routing
  algorithms
• Detailed routing
• Global routing
• Single layer routing
• Benefits of CFR
• Smaller interconnect delay
• Aides wire planning
• Speeds up routing process
• Implication algorithm maximizes routes placed
• Greedy algorithm maximizes criticality placed
• Open problems extending to consider weighted
  edges