Routing under Constraints - Department of Computer …-based decision strategy (emulates...

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Routing under Constraints

Alexander Nadel Intel, Israel

Mountain View CA, USA

October 4, 2016

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2"PhysicalDesign" by Linear77 - Own work. Licensed under CC BY 3.0 via

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Routing

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Outline

Unsolved crafted and industrial RUC instances are routed!

DRouter through SAT Solver Surgery

A*-based decision strategy (emulates constraints!)

Graph conflict analysisNet 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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A graph G(V,E)

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A graph G(V,E)

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A graph G(V,E)

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A graph G(V,E)

Terminals

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A graph G(V,E)

Disjoint Nets Ni V

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Routing: Output

Input Output

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Routing: Output

Input Output

1. Each net is spanned by a tree, called the net routing

2. Net routings can’t intersect3. Optimization: minimize the total

routing length

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Routing: Output

Input Output

1. Each net is spanned by a tree, called the net routing

2. Net routings can’t intersect3. 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

• Example “short” rule:

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

Short rule is violated for these edges

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Design Rules

• Example “short” rule:

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

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

Some violations still persist Time-to-market is impacted

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Potential Solution

Constraint Solving

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Potential Solution

Constraint Solving

Next: formalizing Routing under Constraints

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Routing Induces Assignment

Edge variables

Bool e: edge activity

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0 0 0 0 0 0 0 0 0

1 1 1 1 0 0 1 1 1

0 0 0 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0

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Routing Induces Assignment

Bool v: activity statusBit-vector n: net id

( for inactive vertices)

Vertex variables

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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 1,0 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)

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Routing under Constraints (RUC): Problem Formulation

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1. Graph G(V,E)

2. Disjoint Nets Ni V

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1. Graph G(V,E)

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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1. Graph G(V,E)

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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• For 2-terminal nets:

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- e=(v,u) is active

- nid(v) = nid(u)

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- e=(v,u) is active

- nid(v) = nid(u)

– A terminal has one active neighbor edge

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- e=(v,u) is active

- nid(v) = nid(u)

– An active non-terminal has two active neighbor edges

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- e=(v,u) is active

- nid(v) = nid(u)

• For n-terminal nets:

– Encode directed trees

– Using edge directions

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- e=(v,u) is active

- nid(v) = nid(u)

• 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)

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Boolean Constraint Propagation

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Conflict Analysis & Learning

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Backtracking

No conflict

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Backtracking

No conflict

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Backtracking

Restarts

Time-to-

restart?

No conflict

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SAT DRouter through

Surgery

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Backtracking

Restarts

Time-to-

restart?

No conflict

SAT DRouter

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Backtracking

Restarts

Time-to-

restart?

No conflict

SAT DRouter

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Backtracking

Restarts

Time-to-

restart?

No conflictA*-based Router

SAT DRouter

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Backtracking

Restarts

Time-to-

restart?

Graph-based Learning

SAT DRouter

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Backtracking

SAT DRouter

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Backtracking

Net Swapping

Net Restarting

Time-to-flip? Time-to-

restart?

SAT DRouter

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DRouter

Backtracking

Net Swapping

Net Restarting

restart?

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DRouter

Backtracking

Net Swapping

Net Restarting

restart?

Encoded constraints:

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DRouter

Backtracking

Net Swapping

Net Restarting

restart?

1. Edge consistency

• e=(v,u) is active

• v and u are active

• nid(v) = nid(u)

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DRouter

Backtracking

Net Swapping

Net Restarting

restart?

1. Edge consistency

• nid(v) = nid(u)

2. User-provided constraints modelling

design rules

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DRouter

Backtracking

Net Swapping

Net Restarting

restart?

1. Edge consistency

• 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

Backtracking

Net Swapping

Net Restarting

restart?

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1-Net Example

Backtracking

Net Swapping

Net Restarting

restart?

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1-Net Example

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

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

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.

Decision

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1-Net Example

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.)

σ-violation

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1-Net Example

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.)

σ-violation

A* search

for new σ

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Solutions68

1-Net Example

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.)

A* search

for new σ

Path found

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Solutions69

1-Net Example

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.)

A* search

for new σ

No -violationPath found

Target is part of path?

Repeat

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1-Net Example

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.)

A* search

for new σ

Repeat

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1-Net Example

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.)

A* search

for new σ

Repeat

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1-Net Example

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.)

A* search

for new σ

σ-violationNo -violation

Repeat

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1-Net Example

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.)

A* search

for new σ

σ-violation

Graph conflict (s and t can’t be connected)

No -violation

Repeat

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1-Net Example

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.)

A* search

for new σ

σ-violationAdd conflicting clause: vertex cut (2,0) (3,1)

No -violation

Repeat

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1-Net Example

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.)

A* search

for new σ

σ-violationAdd conflicting clause: vertex cut (2,0) (3,1)1UIP conflict clause: (2,0) ¬(3,2)

No -violation

Repeat

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1-Net Example

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.)

A* search

for new σ

No -violation

Repeat

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1-Net Example

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.)

A* search

for new σ

(2,0) ¬(3,2)

Learn & Backtrack

No -violation

Repeat

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1-Net Example

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.)

A* search

for new σ

No σ-violation

Graph conflict

Learn & Backtrack

Repeat

(2,0) ¬(3,2)

σ-violation

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Solutions79

1-Net Example

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.)

A* search

for new σ

No σ-violation

Graph conflict

Learn & Backtrack

Repeat

(2,0) ¬(3,2)

σ-violation

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Solutions80

1-Net Example

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.)

A* search

for new σ

No σ-violation

Graph conflict

Learn & BacktrackDONE!

(yes!)

Repeat

(2,0) ¬(3,2)

σ-violation

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1-Net Example

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.)

A* search

for new σ

No σ-violationPath found

Graph conflict

Learn & BacktrackDONE!

(yes!)

Repeat

Result:

Path that

follows

constraints!

(2,0) ¬(3,2)

σ-violation

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Multiple Nets Handling

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• Route the nets one-by-one

– Order is critical!

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• Example Order 1:

– Red

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– Red

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– Red

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– Red

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– Red

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– Red

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– Red

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– Red

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– Red

- 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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– Red

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– Red

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– Red

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– Red

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– Red

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– Red

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– Red

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– Red

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– Red

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– Red

• Too slow! Solution: dynamic net reordering!

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DRouter

Backtracking

Net Swapping

Net Restarting

restart?

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DRouter

Backtracking

Net Swapping

Net Restarting

restart?

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Net Swapping

– Red

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Net Swapping

– Red

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Net Swapping

– Red

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Net Swapping

– Red

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Net Swapping

– Red

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Net Swapping

– Red

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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Net Swapping

– Red

• Flip:

– Red

Swapped

Net Swapping:

the blocking net j

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Net Swapping

– Red

• Flip:

– Red

Swapped

Net Swapping:

the blocking net j

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Net Swapping

– Red

• Flip:

– Red

Swapped

Net Swapping:

the blocking net j

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Net Swapping

– Red

• Flip:

– Red

Swapped

Net Swapping:

the blocking net j

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Net Restarting

– Red

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Net Restarting

– Red

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Net Restarting

– Red

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Net Restarting

– Red

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Net Restarting

– Red

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Net Restarting

– Red

Net Restarting

Restart and move the blocked net to the top

(after M conflicts for that net)

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Net Restarting

– Red

• Flip:

– Red

Moved to

the top

Net Restarting

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Net Restarting

– Red

• Flip:

– Red

Moved to

the top

Net Restarting

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Net Restarting

– Red

• Flip:

– Red

Moved to

the top

Net Restarting

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Net Restarting

– Red

• Flip:

– Red

Moved to

the top

Net Restarting

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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

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

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Clock Routing

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• Reduction to finding bounded-path in graph

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• SAT solver surgery: graph-aware decision strategy & graph conflict analysis

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• 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

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

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• Can reason about graph predicates & SAT/BV

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• Graph conflict analysis

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• Shortest-path decision heuristic can be optionally applied

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• Shortest-path decision heuristic can be optionally 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 Monosatlanguage

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Monosat vs. DRouter for Routing under Constraints

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• Monosat’s algorithms are not routing-aware

– No net re-ordering

– Graph conflict analysis for routing is inefficient

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Routing in DRouter (net

swapping&restarting are off)

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Conflict

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

Conflict

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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Routing in Monosat

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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}

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DROUTER

DROUTER-R

DROUTER-S

DROUTER-SR

MONOSAT

MONOSAT+D

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DROUTER

DROUTER-R

DROUTER-S

DROUTER-SR

MONOSAT

MONOSAT+D

- Full-fledged DRouter only can solves all the instances

- Both net restarting and net swapping are essential!

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Technology

Solutions174

DROUTER

DROUTER-R

DROUTER-S

DROUTER-SR

MONOSAT

MONOSAT+D

- 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

Design and

Technology

Solutions175

DRouter on Industrial Instances

• Run DRouter on difficult clips from Intel designs

– Couldn’t be routed cleanly by 2 industrial routers

Design and

Technology

Solutions176

DRouter on Industrial Instances

• Run DRouter on difficult clips from Intel designs

– Couldn’t be routed cleanly by 2 industrial routers

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

Design and

Technology

Solutions177

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

Routing under Constraints - Department of Computer …-based decision strategy (emulates...

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