steiner routing

8/3/2019 Steiner Routing

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Steiner RoutingECE6133

Physical Design Automation of VLSI Systems

Prof. Sung Kyu LimSchool of Electrical and Computer Engineering

Georgia Institute of Technology


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Routingplacement

Generates a "loose" route for each net.

Assigns a list of routing regions to each net withoutspecifying the actual layout of wires.

global routing

compaction

Finds the actual geometric layout of each net withinthe assigned routing regions.

detailed routing

Global routing

Detailed routing


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Routing Constraints

100% routing completion + area minimization, under a set of constraints: Placement constraint: usually based on xed placement Number of routing layers Geometrical constraints: must satisfy design rules Timing constraints (performance-driven routing): must satisfy delay

constraints Crosstalk? Process variations?

Twolayer routing

ws

Geometrical constraint


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Graph Models for Global Routing: Grid Graph

Each cell is represented by a vertex.

Two vertices are joined by an edge if the corresponding cells are adjacentto each other.

The occupied cells are represented as lled circles, whereas the others

are as clear circles.

a b

c

d a b

c

d


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Graph Model: Channel Intersection Graph

Channels are represented as edges.

Channel intersections are represented as vertices.

Edge weight represents channel capacity.

Extended channel intersection graph: terminals are also represented asvertices.

channelintersectiongraph

extendedchannelintersectiongraph


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Global-Routing Problem

Given a netlist N= {N 1 , N 2 , . . . , N n }, a routing graph G = ( V, E ), nd aSteiner tree T i for each net N i , 1 i n , such that U ( e j ) c( e j ), e j E and ni=1 L ( T i ) is minimized,where

c( e j ): capacity of edge e j ;

x ij = 1 if e j is in T i ; x ij = 0 otherwise; U ( e j ) = ni=1 x ij : # of wires that pass through the channel corre-

sponding to edge e j ;

L ( T i ): total wirelength of Steiner tree T i .

For high-performance, the maximum wirelength (max ni=1 L ( T i )) is mini-mized (or the longest path between two points in T i is minimized).


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Classication of Global-Routing Algorithm

Sequential approach: Assigns priority to nets; routes one net at a timebased on its priority (net ordering?).

Concurrent approach: All nets are considered at the same time (com-plexity?)

globalrouting algorithm

sequential approach

twoterminal multiterminal

linesearch maze Steinertree based

Lee Hadlock Soukup

concurrent approach

hierarchical integer programming


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j

A l g o r i t h m s f o r V L S I P h y s i c a l D e s i g n A u t o m a t i o n

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D a t a S t r u c t u r e s a n d B a s i c A l g o r i t h m s

S p a n n i n g T r e e

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P r o b l e m F o r m u l a t i o n :

G i v e n a g r a p h G = ( V E ) , s e l e c t a s u b s e t V

0

V ,

s u c h t h a t V

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h a s p r o p e r t y P .

~

M i n i m u m S p a n n i n g T r e e

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P r o b l e m F o r m u l a t i o n :

G i v e n a n e d g e - w e i g h t e d g r a p h G = ( V E ) , s e l e c t a s u b s e t

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t o t a l c o s t o f e d g e s

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w t ( e

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{ U s e d i n r o u t i n g a p p l i c a t i o n s .


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j



3 . 1 3


S t e i n e r T r e e s

~

1 . P r o b l e m f o r m u l a t i o n :

G i v e n a n e d g e w e i g h t e d g r a p h G = ( V E ) a n d a s u b s e t D V ,

s e l e c t a s u b s e t V

0

V , s u c h t h a t D V

0

a n d V

0

i n d u c e s a t r e e o f m i n i m u m c o s t o v e r a l l s u c h t r e e s .

T h e s e t D i s r e f e r r e d t o a s t h e s e t o f d e m a n d p o i n t s a n d

t h e s e t V

0

; D i s r e f e r r e d t o a s S t e i n e r p o i n t s .

U s e d i n t h e g l o b a l r o u t i n g o f m u l t i - t e r m i n a l n e t s .

Demand Point

B

CD

FJ

H

7

7

54

9

5 6

12

8

6 5 5

2 3

5

6

66

A

I

E

G

(a) (b)


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3 . 1 4


U n d e r l y i n g G r i d G r a p h

~

T h e u n d e r l y i n g g r i d g r a p h i s d e n e d b y t h e i n t e r s e c t i o n s o f t h e

h o r i z o n t a l a n d v e r t i c a l l i n e s d r a w n t h r o u g h t h e d e m a n d p o i n t s .

Hanan's Thm (69'):There exists an optimal RST with all Steiner points (set S) chosen

from the intersection points of horizontal and vertical lines drawn from points of D.


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j



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D i e r e n t S t e i n e r t r e e s c o n s t r u c t e d f r o m a M S T

(a) (b)

(c) (d)

(e)

Hwang's Thm (76'):The ratio of the cost of a rectilinear MST to that of an optimal RST is no greater than 3/2.


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Routing Practical Problems in VLSI Physical CAD 1-Steiner Algorithms

The 1-Steiner ProblemDefinition


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Why 1-Steiner Insertion?Can Reduce Wirelength


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1-Steiner by Kahng/Robins

Iterative 1-Steiner Insertion AlgorithmKeep adding 1-Steiner point one-by-one until no more gain

Nave implementation: O(n 2 n log n n)Sophisticated implementation: O(n 3)


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1-Steiner by Borah/Owens/Irwin

Interesting Observation


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Gain Computation

Things to do

Thus, the gain is


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Overall Algorithm

Multi-pass HeuristicEntire algorithm can be repeated


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Comparison

Kahng/Robins vs Borah/Owens/IrwinKahng/Robins tends to give better results

Borah/Owens/Irwin runs much faster: O(n 4 log n) vs O(n 2)


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Practical Problems in VLSI Physical Design 1-Steiner Algorithm (1/17)

1-Steiner Routing by Kahng/RobinsPerform 1-Steiner Routing by Kahng/Robins

Need an initial MST: wirelength is 2016 locations for Steiner points


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First 1-Steiner Point InsertionThere are six 1-Steiner points

Two best solutions: we choose (c) randomly

beforeinsertion


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First 1-Steiner Point Insertion (cont)

beforeinsertion


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Second 1-Steiner Point InsertionNeed to break tie again

Note that (a) and (b) do not contain any more 1-Steiner point: sowe choose (c)

beforeinsertion


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Third 1-Steiner Point InsertionTree completed: all edges are rectilinearized

Overall wirelength reduction = 20 16 = 4

beforeinsertion


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1-Steiner Routing by Borah/Owens/IrwinPerform a single pass of Borah/Owens/Irwin

Initial MST has 5 edges with wirelength of 20Need to compute the max-gain (node, edge) pair for each edge inthis MST


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Best Pair for ( a,c )


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Best Pair for ( b,c )Three nodes can pair up with ( b,c )

l(a,c ) l( p,a ) = 4 2


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Best Pair for ( b,c ) (cont)All three pairs have the same gain

Break ties randomly

l(b,d ) l( p,d ) = 5 4

l(c,e ) l( p,e) = 4 3


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Best Pair for ( b,d )Two nodes can pair up with ( b,d )

both pairs have the same gain

l(b,c ) l( p,c) = 4 3

l(b,c ) l( p,e) = 4 3


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Best Pair for ( c,e )Three nodes can pair up with ( c,e )

l(b,c ) l( p,b) = 4 3

l(b,d ) l( p,d ) = 5 4


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Best Pair for ( c,e ) (cont)

l(e,f ) l( p,f ) = 3 2


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Best Pair for ( e,f )Can merge with c only

l(c,e ) l( p,c) = 4 3


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SummaryMax-gain pair table

Sort based on gain value


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First 1-Steiner Point InsertionChoose { b, (a,c )} (max-gain pair)

Mark e1 = (a,c ), e2 = (b,c )Skip { a , (b,c )}, { c, (b,d )}, { b, (c,e )} since their e1 / e2 are alreadymarkedWirelength reduces from 20 to 18


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Second 1-Steiner Point InsertionChoose { c, (e,f )} (last one remaining)

Wirelength reduces from 18 to 17


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ComparisonKahng/Robins vs Borah/Owens/Irwin

Khang/Robins has better wirelength (16 vs 17) but is slower


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Routing Practical Problems in VLSI Physical CAD BRBC Algorithm

Bounded Radius Routing

Why Radius?Longest source-sink path length among all sinks

Smaller path resistance: better performanceBoth Radius and Cost?

Cost = wirelength

Radius (= R) and wirelength (= C) are both important for RC-delay reduction

Bounded PRIM vs Bounded Radius/Cost J. Cong, A. B. Kahng, G. Robins, M. Sarrafzadeh, and C. K. Wong, "Provably good performance-driven global routing", TCAD, 1992.


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Radius vs Wirelength


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Bounded PRIM Algorithm

Variation of PRIMs MST algorithm


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Bounded PRIM Algorithm

Comparison (e = 0, 0.5, infinity)Radius bound/value increase

Wirelength decreases


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Practical Problems in VLSI Physical Design Bounded Radius Routing (1/16)

Bounded Radius RoutingPerform bounded PRIM algorithm

Under = 0, = 0.5, and =

Compare radius and wirelengthRadius = 12 for this net


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BPRIM Under = 0Example

Edges connecting to nearest neighbors = ( c,d ) and ( c,e)We choose ( c,d ) based on lexicographical order

s-to- d path length along T = 12+5 > 12 (= radius bound)First appropriate edge found = ( s,d )


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BPRIM Under = 0 (cont)Radius bound = 12

edges connecting tonearest neighbors

s-to- y path lengthalong T

ties brokenlexicographically

should be 12;otherwise

appropriate used

first feasibleappr-edge


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BPRIM Under = 0 (cont)


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BPRIM Under = 0 (cont)


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BPRIM Under = 0.5Radius bound = 18

edges connecting tonearest neighbors

s-to- y path lengthalong T

ties brokenlexicographically

should be 18;otherwise

appropriate used

first feasibleappr-edge

should be 12


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BPRIM Under = 0.5 (cont)


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BPRIM Under = 0.5 (cont)


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BPRIM Under =

Radius bound = = regular PRIM


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BPRIM Under = (cont)


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ComparisonAs the bound increases (12 18 )

Radius value increases (12 17 22)Wirelength decreases (56 49 36)


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Bounded Radius Spanning Tree

Shallow Light Algorithm


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Bounded Radius Spanning Tree

Bounded-radius Spanning Tree Construction Augmentation of Q: added edges shown in dotted lines

Final BR-MST is SPT on Q

h C k ?


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Why BRBC Works?

Wh BRBC W k ?


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Why BRBC Works?

B d d R di B d d C t


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Bounded Radius Bounded CostPerform BRBC under = 0.5

defines both radius and wirelength boundPerform DFS on rooted-MST

Node ordering L = { s, a, b, c, e, f, e, g, e, c, d, h, d, c, b, a, s }We start with Q = MST

MST A g t ti


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MST AugmentationExample: visit a via ( s,a )

Running total of the length of visited edges, S = 5Rectilinear distance between source and a, dist ( s,a ) = 5We see that dist ( s,a ) = 0.5 5 < S Thus, we reset S and add ( s,a ) to Q (note ( s,a ) is already in Q)

MST Augmentation (cont)


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MST Augmentation (cont)

dotted edges are added

visit nodes based on L

Last Step: SPT Computation


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Last Step: SPT ComputationCompute rooted shortest path tree on augmented Q

BPRIM vs BRBC


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BPRIM vs BRBCUnder the same = 0.5

BPRIM: radius = 18, wirelength = 49BRBC: radius = 12, wirelength = 52

BRBC: significantly shorter radius at slight wirelength increase

A tree Algorithm


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Practical Problems in VLSI Physical Design

A-tree AlgorithmMinimum Shortest Path Steiner Arborescence (MSP-SA)

Given: edge-weighted undirected graph G(V , E ), sink set N (=subset of V ), and root node r

MSP-RA of N : Steiner tree T rooted at r that spans N Every source-sink path in T is shortest in GTotal wirelength is minimized

Minimum Rectilinear Steiner Arborescence (MRSA)Rectilinear version of MSP-SACan use Hanan or grid graph as underlying graph GBoth MSP-SA and MRSA are NP-hardMRSA vs BR-MRT: similarity and difference?

A-tree algorithm [Cong et al, 1993]: computes MRSA

Overview of A-tree Algorithm


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Overview of A-tree AlgorithmOverall flow

Starts with initial forest F 0, where each sink becomes a rootGoal: make a sequence of moves

Move: either grow an existing tree or merge two treesContinue making moves until only one tree remains

Notation

F k : forest after k -th moveR(F k ): set of root nodes in F k

Types of moves

Safe (3 types): keep the forest wirelength-optimalHeuristic (2 types): use [Rao et at, 1992], non-WL-optimal

dx/dy/df Computation


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dx/dy/df ComputationThree types of neighboring nodes of p in F x

NW ( p): set of nodes with their x-coordinate < x( p) and y-coordinate > y( p)

SE ( p): set of nodes with their x-coordinate > x( p) and y-coordinate < y( p)D( p , F k ): set of nodes with both x/ y-coordiates p. we call thesenodes are dominated by p

Node Blockage


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Node BlockageTwo ways that q is blocked from p

If q is in NW ( p): draw vertical line from q to p until y( p)If q is in SE ( p): draw horizontal line from q to p until x( p)

Computing dx


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Computing dx

Computing dy


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Computing dy

Computing df


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Computing df

Example


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a p e

Safe Move Type 1


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yp

Safe Move Type 2


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yp

Safe Move Type 3


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yp

Heuristic Moves


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From [Rao et al, 1992]Used only when safe (= optimal) moves are not possible

Overall Algorithm


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A-tree Routing Algorithm


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Practical Problems in VLSI Physical Design A-tree Algorithm (1/13)

Compute dx(c, F 0), dy(c, F 0), df (c, F 0)We begin with root set R(F 0) = { a,b,c,d,e,f } for initial forest F 0

Recall that


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mx = a, dx = 3

my = d, dx = 2

Recall that (cont)


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MF = { f, i } , df = 4 , mf w = i, mf s = f

Computing dx/dy/df for Node c


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Computing dx/dy/df Values


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Compute dx/dy/df for all other nodes

Safe Move Computation


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What kind of safe moves does node a contain?We have dx(a , F 0) = , dy(a , F 0) = 1, df (a , F 0) = 5

Type 1: dx df and dy df

Type 2: dx df and dy < df Type 3: dx < df and dy df

So a has type-2 safe move

Safe Move Computation (cont)


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Compute safe moves for all nodes in F 0Type 1: dx df and dy df Type 2: dx df and dy < df

Type 3: dx < df and dy df All moves are safe

No heuristic moves necessary

Recall that


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Recall that (cont)


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Safe Move for Node a


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Perform safe move for node a (type 2)

Safe Move for Node a (cont)


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Updating dx/dy/df values and safe moves

Performing Remaining Safe Moves


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Choose the nodes in alphabetical order

Performing Remaining Moves


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Final rectilinear Steiner arborescenceAll source-sink paths are shortestTotal wirelength = 18

3 Steiner nodes (white square) usedAll moves performed were safe

steiner routing

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