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Routing under Constraints Alexander Nadel Intel, Israel FMCAD Mountain View CA, USA October 4, 2016 Design and Technology Solutions 1 Design and Technology 2 "PhysicalDesign" by Linear77 - Own work. Licensed under CC BY 3.0

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Design and Technology Solutions

Routing under Constraints

Alexander Nadel Intel, Israel

FMCAD Mountain View CA, USA October 4, 2016

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Design and Technology Solutions

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"PhysicalDesign" by Linear77 - Own work. Licensed under CC BY 3.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:PhysicalDesign.png#/media/File:PhysicalDesign.png

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Design and Technology Solutions

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"PhysicalDesign" by Linear77 - Own work. Licensed under CC BY 3.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:PhysicalDesign.png#/media/File:PhysicalDesign.png

Routing

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Outline

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Unsolved crafted and industrial RUC instances are routed! DRouter through SAT Solver Surgery

A*-based decision strategy (emulates constraints!) Graph conflict analysis Net restarting & net swapping

Bit-Vector / SAT Encoding

Doesn’t scale

Routing under Constraints (RUC): Problem Formalization Goal: Design a Scalable Design Rule-aware Router

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Abboud et al, OR Spectrum’08

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Routing: Input

(AKA Steiner Tree Packing Problem)

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Input

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Routing: Input

(AKA Steiner Tree Packing Problem)

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Input

A graph G(V,E)

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Routing: Input

(AKA Steiner Tree Packing Problem)

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Input

A graph G(V,E)

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Routing: Input

(AKA Steiner Tree Packing Problem)

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Input

A graph G(V,E)

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Routing: Input

(AKA Steiner Tree Packing Problem)

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Input

A graph G(V,E)

Terminals

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Routing: Input

(AKA Steiner Tree Packing Problem)

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Input

A graph G(V,E)

Disjoint Nets Ni  V

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Routing: Output

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Input Output

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Routing: Output

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Input Output

  • 1. Each net is spanned by a tree,

called the net routing

  • 2. Net routings can’t intersect
  • 3. Optimization: minimize the total

routing length

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Routing: Output

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Input Output

  • 1. Each net is spanned by a tree,

called the net routing

  • 2. Net routings can’t intersect
  • 3. Optimization: minimize the total

routing length

It is NP-hard to find:

  • 1. Shortest solution for one multi-terminal net (Steiner tree problem)
  • 2. Any solution for many multi-terminal nets
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Design Rules

  • Routing is to satisfy design rules

– Originating in the manufacturing requirement

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Design Rules

  • Routing is to satisfy design rules

– Originating in the manufacturing requirement

  • Example “short” rule:

– The 2 vertices of any edge can’t belong to two distinct net routings

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Short rule is violated for these edges

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Design Rules

  • Routing is to satisfy design rules

– Originating in the manufacturing requirement

  • Example “short” rule:

– The 2 vertices of any edge can’t belong to two distinct net routings

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Short rule is violated for these edges When the short rule is on, this example is UNSAT

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Industrial Approach: Rip-Up and Reroute

  • Nets are routed one-by-one

– Using A*

– s-t shortest-path given costs’ under-approximation – A*Dijkstra if no costs’ under-approximation is provided

– Trying to heuristically obey design rules

  • Violations are allowed, hence the initial solution

might be problematic

– Net routings might intersect – Design rules might be violated

  • Clean-up is applied

– Rip-up: problematic net routings are removed – Reroute: un-routed nets are attempted again

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The Problem with the Current Solution

  • Design rule violations persist

– Manual clean-up is carried out

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Some violations still persist Time-to-market is impacted

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Potential Solution

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Constraint Solving

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Potential Solution

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Constraint Solving

Next: formalizing Routing under Constraints

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Routing Induces Assignment

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Edge variables Bool e: edge activity

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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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Routing Induces Assignment

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Bool v: activity status Bit-vector n: net id ( for inactive vertices) Vertex variables

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0, 0, 0, 0, 0, 1,1 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,1 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,1 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,1 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,1 0, 0, 0, 0, 1,0 1,0 1,0 1,0 1,0 1,1 1,0 1,0 1,0 1,0 0, 0, 0, 0, 1,0 1,1 1,0 0, 0, 0, 0, 0, 0, 0, 1,0 1,1 1,0 0, 0, 0, 0, 0, 0, 0, 1,0 1,1 1,0 0, 0, 0, 0, 0, 0, 0, 1,0 1,0 1,0 0, 0, 0,

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Modeling Routing under Constraints

  • Design rules can be easily expressed in BV logic

– Variables:

– Edge & vertex activities – Vertex nids – Any auxiliary variables

  • “Short” rule example

– For every edge e=(v,u): v  u  nid(v)=nid(u)

Short rule is violated for these edges

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Routing under Constraints (RUC): Problem Formulation

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Routing under Constraints (RUC): Problem Formulation

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Routing under Constraints (RUC): Problem Formulation

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Input

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Routing under Constraints (RUC): Problem Formulation

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  • 1. Graph G(V,E)
  • 2. Disjoint Nets Ni  V

Input

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Routing under Constraints (RUC): Problem Formulation

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  • 1. Graph G(V,E)
  • 2. Disjoint Nets Ni  V

Input A quantifier-free bit-vector formula F(V  E  N  A)

  • V : vertex activity
  • E : edge activity
  • N : vertex net id
  • A : any auxiliary variables

(represents the design rules)

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Routing under Constraints (RUC): Problem Formulation

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  • 1. Graph G(V,E)
  • 2. Disjoint Nets Ni  V

Input A quantifier-free bit-vector formula F(V  E  N  A)

  • V : vertex activity
  • E : edge activity
  • N : vertex net id
  • A : any auxiliary variables

(represents the design rules) Output: a model to F, which induces a routing:

  • e=(v,u) is active 
  • v and u are active, and
  • nid(v) = nid(u)
  • For each net i: active vertices with nid i and

active edges span the net’s terminals

  • Optional optimization requirement: the overall

weight of active edges is as small as possible

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Solving Attempt: Encoding into Bitvector Logic / SAT

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Solving Attempt: Encoding into Bitvector Logic / SAT

  • For 2-terminal nets:

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Solving Attempt: Encoding into Bitvector Logic / SAT

  • For 2-terminal nets:
  • e=(v,u) is active 
  • v and u are active, and
  • nid(v) = nid(u)

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Solving Attempt: Encoding into Bitvector Logic / SAT

  • For 2-terminal nets:
  • e=(v,u) is active 
  • v and u are active, and
  • nid(v) = nid(u)

– A terminal has one active neighbor edge

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Solving Attempt: Encoding into Bitvector Logic / SAT

  • For 2-terminal nets:
  • e=(v,u) is active 
  • v and u are active, and
  • nid(v) = nid(u)

– A terminal has one active neighbor edge – An active non-terminal has two active neighbor edges

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Solving Attempt: Encoding into Bitvector Logic / SAT

  • For 2-terminal nets:
  • e=(v,u) is active 
  • v and u are active, and
  • nid(v) = nid(u)

– A terminal has one active neighbor edge – An active non-terminal has two active neighbor edges

  • For n-terminal nets:

– Encode directed trees

– Using edge directions

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Solving Attempt: Encoding into Bitvector Logic / SAT

  • For 2-terminal nets:
  • e=(v,u) is active 
  • v and u are active, and
  • nid(v) = nid(u)

– A terminal has one active neighbor edge – An active non-terminal has two active neighbor edges

  • For n-terminal nets:

– Encode directed trees

– Using edge directions

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SAT Solver’s Internals

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Decision Strategy (Conflict-driven)

SAT Solver’s Internals

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation

SAT Solver’s Internals

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation

SAT Solver’s Internals

Conflict Analysis & Learning

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation

SAT Solver’s Internals

Conflict Analysis & Learning Backtracking No conflict

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation

SAT Solver’s Internals

Conflict Analysis & Learning Backtracking No conflict

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation

SAT Solver’s Internals

Conflict Analysis & Learning Backtracking Restarts Time-to- restart? No conflict

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SAT  DRouter through Surgery

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking Restarts Time-to- restart? No conflict

SAT  DRouter

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Boolean Constraint Propagation Conflict Analysis & Learning Backtracking Restarts Time-to- restart? No conflict

SAT  DRouter

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Boolean Constraint Propagation Conflict Analysis & Learning Backtracking Restarts Time-to- restart? No conflict A*-based Router

SAT  DRouter

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Boolean Constraint Propagation Conflict Analysis & Learning Backtracking Restarts Time-to- restart? No conflict A*-based Router Graph-based Learning

SAT  DRouter

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Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

SAT  DRouter

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Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart?

SAT  DRouter

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DRouter

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart?

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DRouter

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart? Encoded constraints:

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DRouter

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart? Encoded constraints:

  • 1. Edge consistency
  • e=(v,u) is active 
  • v and u are active
  • nid(v) = nid(u)
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DRouter

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart? Encoded constraints:

  • 1. Edge consistency
  • e=(v,u) is active 
  • v and u are active
  • nid(v) = nid(u)
  • 2. User-provided constraints modelling

design rules

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DRouter

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart? Encoded constraints:

  • 1. Edge consistency
  • e=(v,u) is active 
  • v and u are active
  • nid(v) = nid(u)
  • 2. User-provided constraints modelling

design rules That’s it! What about disconnected terminals??? Routing correctness is guaranteed by the decision strategy!

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1-Net Example

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart?

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1-Net Example

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Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart?

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Design rules

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t σ (sugg.) Path Suggestion (not an actual SAT decision)

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path σ (sugg.) Activate edge in sugg. SAT Decision

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) σ-violation Activate edge in sugg.

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) σ-violation Activate edge in sugg. A* search for new σ

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ Path found Activate edge in sugg.

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ No -violation Path found Target is part of path? no Repeat Activate edge in sugg.

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ No -violation Path found Target is part of path? no Repeat Activate edge in sugg. BCP

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ No -violation Path found Target is part of path? no Repeat Activate edge in sugg. BCP

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ Activate edge in sugg.

x

σ-violation No -violation Target is part of path? no Repeat

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ Activate edge in sugg.

x

σ-violation Graph conflict (s and t can’t be connected) No -violation Target is part of path? no Repeat

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ Activate edge in sugg.

x

σ-violation Add conflicting clause: vertex cut (2,0)  (3,1) Graph conflict (s and t can’t be connected) No -violation Target is part of path? no Repeat

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ Activate edge in sugg.

x

σ-violation Add conflicting clause: vertex cut (2,0)  (3,1) 1UIP conflict clause: (2,0)  ¬(3,2) Graph conflict (s and t can’t be connected) No -violation Target is part of path? no Repeat

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ Activate edge in sugg.

x

σ-violation Add conflicting clause: vertex cut (2,0)  (3,1) 1UIP conflict clause: (2,0)  ¬(3,2) Graph conflict (s and t can’t be connected) No -violation Target is part of path? no Repeat

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ Activate edge in sugg. σ-violation Add conflicting clause: vertex cut (2,0)  (3,1) 1UIP conflict clause: (2,0)  ¬(3,2) (2,0)  ¬(3,2)

x

Graph conflict (s and t can’t be connected) Learn & Backtrack No -violation Target is part of path? no Repeat

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ No σ-violation Graph conflict Learn & Backtrack Target is part of path? no Repeat Activate edge in sugg.

x

(2,0)  ¬(3,2) σ-violation

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ No σ-violation Graph conflict Learn & Backtrack Target is part of path? no Repeat Activate edge in sugg.

x

(2,0)  ¬(3,2) σ-violation

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ No σ-violation Graph conflict Learn & Backtrack

DONE!

Target is part of path? (yes!) no Repeat Activate edge in sugg.

x

(2,0)  ¬(3,2) σ-violation

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1-Net Example

1 2 1 2 3 s: (0, 0) t: (3, 0) ¬(1,0)  ¬(2,0) ¬(1,0)  ¬(1,1) ¬(3,2)  ¬(3,1) Initial path: A* from s->t Real path

x x

σ (sugg.) A* search for new σ No σ-violation Path found Graph conflict Learn & Backtrack

DONE!

Target is part of path? (yes!) no Repeat Activate edge in sugg.

x

Result: Path that follows constraints!

(2,0)  ¬(3,2) σ-violation

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Multiple Nets Handling

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Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

30

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SLIDE 84

Design and Technology Solutions

84

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

30

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SLIDE 85

Design and Technology Solutions

85

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

30

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SLIDE 86

Design and Technology Solutions

86

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

30

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SLIDE 87

Design and Technology Solutions

87

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

30

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SLIDE 88

Design and Technology Solutions

88

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

30

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SLIDE 89

Design and Technology Solutions

89

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

30

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SLIDE 90

Design and Technology Solutions

90

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

30

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SLIDE 91

Design and Technology Solutions

91

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 92

Design and Technology Solutions

92

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 93

Design and Technology Solutions

93

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 94

Design and Technology Solutions

94

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

  • Graph conflict
  • black is blocked
  • Early conflict detection
  • Check for graph conflicts

after routing each terminal

  • Learn a conflict clause &

re-route

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SLIDE 95

Design and Technology Solutions

95

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 96

Design and Technology Solutions

96

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 97

Design and Technology Solutions

97

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 98

Design and Technology Solutions

98

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 99

Design and Technology Solutions

99

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 100

Design and Technology Solutions

100

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 101

Design and Technology Solutions

101

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 102

Design and Technology Solutions

102

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 103

Design and Technology Solutions

103

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

31

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SLIDE 104

Design and Technology Solutions

104

Multiple Nets Handling

  • Route the nets one-by-one

– Order is critical!

  • Example Order 1:

– Red

  • Example Order 2:

– Red

  • Too slow! Solution: dynamic

net reordering!

31

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SLIDE 105

Design and Technology Solutions

105

DRouter

32

Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart?

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SLIDE 106

Design and Technology Solutions

106

DRouter

32

Decision Strategy (Conflict-driven) Boolean Constraint Propagation Conflict Analysis & Learning Backtracking No conflict A*-based Router Graph-based Learning

Net Swapping Net Restarting

Time-to-flip? Time-to- restart?

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SLIDE 107

Design and Technology Solutions

107

Net Swapping

  • Example Order 2:

– Red

33

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SLIDE 108

Design and Technology Solutions

108

Net Swapping

  • Example Order 2:

– Red

33

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SLIDE 109

Design and Technology Solutions

109

Net Swapping

  • Example Order 2:

– Red

33

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SLIDE 110

Design and Technology Solutions

110

Net Swapping

  • Example Order 2:

– Red

33

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SLIDE 111

Design and Technology Solutions

111

Net Swapping

  • Example Order 2:

– Red

33

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SLIDE 112

Design and Technology Solutions

112

Net Swapping

  • Example Order 2:

– Red

33

Net Swapping: After N conflicts, swap the order between: the first blocked net i the blocking net j {A,j,B,i,C}  {A,i,j,B,C}

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SLIDE 113

Design and Technology Solutions

113

Net Swapping

  • Example Order 2:

– Red

  • Flip:

– Red

33

Swapped Net Swapping: After N conflicts, swap the order between: the first blocked net i the blocking net j {A,j,B,i,C}  {A,i,j,B,C}

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SLIDE 114

Design and Technology Solutions

114

Net Swapping

  • Example Order 2:

– Red

  • Flip:

– Red

33

Swapped Net Swapping: After N conflicts, swap the order between: the first blocked net i the blocking net j {A,j,B,i,C}  {A,i,j,B,C}

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SLIDE 115

Design and Technology Solutions

115

Net Swapping

  • Example Order 2:

– Red

  • Flip:

– Red

33

Swapped Net Swapping: After N conflicts, swap the order between: the first blocked net i the blocking net j {A,j,B,i,C}  {A,i,j,B,C}

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SLIDE 116

Design and Technology Solutions

116

Net Swapping

  • Example Order 2:

– Red

  • Flip:

– Red

33

Swapped Net Swapping: After N conflicts, swap the order between: the first blocked net i the blocking net j {A,j,B,i,C}  {A,i,j,B,C}

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SLIDE 117

Design and Technology Solutions

117

Net Restarting

  • Example Order 2:

– Red

34

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SLIDE 118

Design and Technology Solutions

118

Net Restarting

  • Example Order 2:

– Red

34

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SLIDE 119

Design and Technology Solutions

119

Net Restarting

  • Example Order 2:

– Red

34

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SLIDE 120

Design and Technology Solutions

120

Net Restarting

  • Example Order 2:

– Red

34

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SLIDE 121

Design and Technology Solutions

121

Net Restarting

  • Example Order 2:

– Red

34

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SLIDE 122

Design and Technology Solutions

122

Net Restarting

  • Example Order 2:

– Red

34

Net Restarting Restart and move the blocked net to the top (after M conflicts for that net)

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SLIDE 123

Design and Technology Solutions

123

Net Restarting

  • Example Order 2:

– Red

  • Flip:

– Red

34

Moved to the top Net Restarting Restart and move the blocked net to the top (after M conflicts for that net)

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SLIDE 124

Design and Technology Solutions

124

Net Restarting

  • Example Order 2:

– Red

  • Flip:

– Red

34

Moved to the top Net Restarting Restart and move the blocked net to the top (after M conflicts for that net)

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SLIDE 125

Design and Technology Solutions

125

Net Restarting

  • Example Order 2:

– Red

  • Flip:

– Red

34

Moved to the top Net Restarting Restart and move the blocked net to the top (after M conflicts for that net)

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SLIDE 126

Design and Technology Solutions

126

Net Restarting

  • Example Order 2:

– Red

  • Flip:

– Red

34

Moved to the top Net Restarting Restart and move the blocked net to the top (after M conflicts for that net)

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SLIDE 127

Design and Technology Solutions

127

Net Swapping vs. Net Restarting

  • Swapping is local
  • Restarting is global
  • In practice both techniques are crucial
  • Strategy:

– Swap for some time – If it doesn’t work, restart

35

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SLIDE 128

Design and Technology Solutions

128

Related Work 1: Clock Routing Erez & Nadel, CAV’15

36

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SLIDE 129

Design and Technology Solutions

129

Related Work 1: Clock Routing Erez & Nadel, CAV’15

36

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SLIDE 130

Design and Technology Solutions

130

Related Work 1: Clock Routing Erez & Nadel, CAV’15

36

Clock Routing

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SLIDE 131

Design and Technology Solutions

131

Related Work 1: Clock Routing Erez & Nadel, CAV’15

36

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SLIDE 132

Design and Technology Solutions

132

Related Work 1: Clock Routing Erez & Nadel, CAV’15

  • Reduction to finding bounded-path in graph

36

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SLIDE 133

Design and Technology Solutions

133

Related Work 1: Clock Routing Erez & Nadel, CAV’15

  • Reduction to finding bounded-path in graph
  • SAT solver surgery: graph-aware decision

strategy & graph conflict analysis

36

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SLIDE 134

Design and Technology Solutions

134

Related Work 1: Clock Routing Erez & Nadel, CAV’15

  • Reduction to finding bounded-path in graph
  • SAT solver surgery: graph-aware decision

strategy & graph conflict analysis

  • The decision strategy:

– Emulates constraints! – Guides the solver towards the solution – Considers additional optimization requirements

36

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SLIDE 135

Design and Technology Solutions

135

Related Work 2: Monosat Solver Bayless & Bayless & Hoos & Hu, AAAI’15

37

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SLIDE 136

Design and Technology Solutions

136

Related Work 2: Monosat Solver Bayless & Bayless & Hoos & Hu, AAAI’15

  • Can reason about graph predicates & SAT/BV

37

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SLIDE 137

Design and Technology Solutions

137

Related Work 2: Monosat Solver Bayless & Bayless & Hoos & Hu, AAAI’15

  • Can reason about graph predicates & SAT/BV
  • Graph conflict analysis

37

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SLIDE 138

Design and Technology Solutions

138

Related Work 2: Monosat Solver Bayless & Bayless & Hoos & Hu, AAAI’15

  • Can reason about graph predicates & SAT/BV
  • Graph conflict analysis
  • Shortest-path decision heuristic can be
  • ptionally applied

37

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SLIDE 139

Design and Technology Solutions

139

Related Work 2: Monosat Solver Bayless & Bayless & Hoos & Hu, AAAI’15

  • Can reason about graph predicates & SAT/BV
  • Graph conflict analysis
  • Shortest-path decision heuristic can be
  • ptionally applied
  • Path-finding (routing for one 2-terminal net) is

conceptually similar in Monosat and DRouter

– Main difference: – Lazy A* in DRouter vs. – Eager incremental Ramalingam-Reps in Monosat

  • RUC can be easily expressed in Monosat

language

37

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SLIDE 140

Design and Technology Solutions

140

Monosat vs. DRouter for Routing under Constraints

38

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SLIDE 141

Design and Technology Solutions

141

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

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SLIDE 142

Design and Technology Solutions

142

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 143

Design and Technology Solutions

143

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 144

Design and Technology Solutions

144

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 145

Design and Technology Solutions

145

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Conflict

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SLIDE 146

Design and Technology Solutions

146

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 147

Design and Technology Solutions

147

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 148

Design and Technology Solutions

148

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 149

Design and Technology Solutions

149

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 150

Design and Technology Solutions

150

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 151

Design and Technology Solutions

151

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 152

Design and Technology Solutions

152

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 153

Design and Technology Solutions

153

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off)

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SLIDE 154

Design and Technology Solutions

154

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

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SLIDE 155

Design and Technology Solutions

155

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

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SLIDE 156

Design and Technology Solutions

156

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

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SLIDE 157

Design and Technology Solutions

157

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat Conflict

slide-158
SLIDE 158

Design and Technology Solutions

158

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

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SLIDE 159

Design and Technology Solutions

159

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

slide-160
SLIDE 160

Design and Technology Solutions

160

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

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SLIDE 161

Design and Technology Solutions

161

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

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SLIDE 162

Design and Technology Solutions

162

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

slide-163
SLIDE 163

Design and Technology Solutions

163

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

slide-164
SLIDE 164

Design and Technology Solutions

164

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

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SLIDE 165

Design and Technology Solutions

165

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

slide-166
SLIDE 166

Design and Technology Solutions

166

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

slide-167
SLIDE 167

Design and Technology Solutions

167

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

slide-168
SLIDE 168

Design and Technology Solutions

168

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

slide-169
SLIDE 169

Design and Technology Solutions

169

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

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SLIDE 170

Design and Technology Solutions

170

Monosat vs. DRouter for Routing under Constraints

  • Monosat’s algorithms are not routing-aware

– No net re-ordering – Graph conflict analysis for routing is inefficient

38

Routing in DRouter (net swapping&restarting are off) Routing in Monosat

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SLIDE 171

Design and Technology Solutions

171

Experimental Results on Crafted Instances

  • Solvers:

– Drouter (default) – Drouter – R: no net restarting – Drouter – S: no net swapping – Drouter – SR: no net swapping, no net restarting – Monosat (default) – Monosat + D: shortest-path decision strategy is on – BV: reduction to BV

  • Instances:

– 120 solid grid graphs of size M  20

– M  {3,5,7}

– 20 random 2-terminal nets – Generate C * |V| random binary clauses v  u

– v,u  V – C  {0,0.1,0.2,0.3}

39

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SLIDE 172

Design and Technology Solutions

172 40

5 10 15 20 25 30 10 20 30 # SOLVED

DROUTER DROUTER-R DROUTER-S DROUTER-SR MONOSAT MONOSAT+D BV

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SLIDE 173

Design and Technology Solutions

173 40

5 10 15 20 25 30 10 20 30 # SOLVED

DROUTER DROUTER-R DROUTER-S DROUTER-SR MONOSAT MONOSAT+D BV

  • Full-fledged DRouter only can solves all the instances
  • Both net restarting and net swapping are essential!
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SLIDE 174

Design and Technology Solutions

174 40

5 10 15 20 25 30 10 20 30 # SOLVED

DROUTER DROUTER-R DROUTER-S DROUTER-SR MONOSAT MONOSAT+D BV

  • Full-fledged DRouter only can solves all the instances
  • Both net restarting and net swapping are essential!
  • Monosat and BV can’t solve a single instance
slide-175
SLIDE 175

Design and Technology Solutions

175

DRouter on Industrial Instances

  • Run DRouter on difficult clips from Intel designs

– Couldn’t be routed cleanly by 2 industrial routers

41

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SLIDE 176

Design and Technology Solutions

176

DRouter on Industrial Instances

  • Run DRouter on difficult clips from Intel designs

– Couldn’t be routed cleanly by 2 industrial routers

41

Area in m2 Nets Vertices Constraints Time in sec. Memory in Gb. 24 110 42,456 484,008 25 0.7 24 230 42,456 484,008 391 1.0 32 352 63,740 667,764 705 2.2 129 788 127,480 2,669,056 14,733 6.5 129 891 127,480 2,669,056 92,950 6.5

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SLIDE 177

Design and Technology Solutions

177

Conclusion

  • DRouter: design-rule-aware router

– SAT solver surgery:

– Decision heuristic  A*-based router – Conflict analysis enhanced with graph reasoning – Restarts  net swapping & net restarting

  • Solves instances which can’t be solved by

existing tools

– Including clips from real Intel designs

42