CSE 242A Integrated Circuit Layout Automation

Lecture: Partitioning

Winter 2009

Chung-Kuan Cheng

Outlines Motivations Formulations

2-way partitioning, multi-way partitioning, multilevel partitioning, replication cut, clustering.

Net Modeling Algorithms

Optimal Methods: Special cases, branch and bound,

Heuristic Methods: group migration, network flow, clustering, simulated annealing, genetic approaches

Motivation

Huge designs 100 Millions Trans Design Analysis Engineering Change Orders

Good partitioning makes difference on design quality

Motivation: Applications

Physical Hierarchy

Divide and Conquer

Project decomposition

Complexity reduction of each level

Emulation

Hardware & Software codesign

Management design resource

Formulations: Two Way Partitioning

Random two-way partition

N-pin net to be cut

a

a

a

ab

bb

b

2 2

2

n

np

Min-Cut

,,

min ( , ) ijs X t X

i X j X

C X X C

X

X

s t

{ } { }X s or X t

X

Theorem:

There exists an optimal linear placement

s.t. X and are separated.

Trend:

Min-Cut

Opt->Linear Placement

Trend: X=V

S

S

( , ) ( , )min

| |

C X X C X s

X

Bisection

X

X

1 1

2 2

1 2

min ( , )

| |

| |

| | | |

C X X

Const

L X U

L X U

Trend

X L or X L

Ratio Cut

( , )?

| || |

C X X

X X

Multi-Way Partitioning

K-Way Partitioning

X1 X2

Xk

1 2

,

( ) min ( , ,..., )

( ) ( )

( )

( ) min max ( )

i

i

i i

i i i

k

netnet is not contained in any subset

ii

i nn X n X

i i

L X U

i C X X X

C

ii E X

E X C

iii E X

Cluster Ratio Cut

1 2,

| | 15

| | 24

( , ,..., )min ( )

| || |k

k c ci j

i j

V

E

C X X XR k

X X

Multi-Level Partitioning

K-Level Partitioning

Li<= Xi<= Ui

Min Connection Cost E(Ti) <= Ci

External connection cost <= threshold at level i

Generic Binary Tree

L <= Xi <=U Level of Node = longest

path to leaves Connection cost of

node i at level L

,

,: min 2

l i nn gothrouth node i

ll i

l i

F C

Obj F

L4

L3

L2

L1

L0

Replication Cut

X Y

R

X

R

R

Y

C(X,Y)+C(Y,X)+C(Y,R)+C(X,R)

Performance-Driven Partitioning

Need an incremental timing analysis to reflect the performance fast

Replication helps

R

XR

Y

Retiming (Pipelined process)

Allocate one clock cycle for interpartition communication System performance is dominated by

Loops: max#

loop

loop

d

reg

1

1

2

2

2

#

#

l

l

l

l

d T

reg

d

reg

1 2 1 2

1 2 1 2

2, ,

# # #l l l l

l l l l

d d d d T

reg reg reg reg

Clustering

K-Way Partitioning K>>10 Obj strongly depends

on applications Performance driven

Obj min max # cuts between registers

Complexity reduction min i

i i

E

IEi external connection

Ii internal connection

Net Modeling

Shifting: For each shift, we update k, n-k. The cost of the net changes only when k = 0, 1 or n-k =0, 1

Two pin net clique 2/k total weight k-1 1/(k-1) S.K.

k n-k

Net Modeling: Loop Model

Suppose relative positions of pins are given, we can use a loop model

The model remains correct if any two adjacent (in order) pins swap

1/2

Net Modeling: Hypernet Model (Flow Approach)

star

n

Xin0

in ni

in nii i

x C

x x

Optimal Methods: Branch & Bound

1

2 3

L R

L LR R

X X

Prune the branches when size constraint is violated

Partial cost >= existing cost

For U=L=|V|/2

# combinations = |V| ! / (|V/2)! (|V|/2)!

With an elegant implementation |V| <= 60 is feasible

Optimal Methods: Serial & Parallel Graph

Dynamic Programming on Series-Parallel graph G(V, E, s, t)

C(a, i, j) a: s, t on different sides

C(b, i, j) b: s, t on the left side, left side has i nodes, right side has j nodes

1 2

, 1 2

, 1 2

1 2

, 1 2

//

( , , ) min ( , , ) ( , , )

( , , ) min ( , , ) ( , , )

( , , ) min(min ( , , ) ( , , )

k l

k l

k l

G G G

C a i j C a i k j l C a k l

C b i j C b i k j l C b k l

G G G

C a i j C a i k j l C b k l

Series s1

t1 s1

Parallel s1 s2

t1t2

Optimal Methods: Serial & Parallel Graph (Cont)

Dynamic Programming on Series-Parallel graph G(V, E, s, t)

C(a, i, j) a: s, t on different sides

C(b, i, j) b: s, t on the left side, left side has i nodes, right side has j nodes

1 2

, 1 2

, 1 2

1 2

1 2

( , , ) min(min ( , , ) ( , , ),

min ( , , ) ( , . ))

( , , ) min(min ( , , ) ( , , ),

min ( , , ) ( , , ))

k l

k l

G G G

C a i j C a i k j l C b k l

C b i k j l C a k l

C b i j C a i k j l C a l k

C b i k j l C b l k

Series s1

t1 s1

Parallel s1 s2

t1t2

Heuristic Methods

Group Migration Kernighan & Lin Fiducccia-Matheyses

Programming Network Flow Replication Cut Clustering

Group MigrationKermighan & Lin Bisection Cost Ci: change of #cuts by moving node i to

the other side Heapsort nodes in each partition according to

CiRepeat

Repeat

Choose among the top k the best pair to swap

Update the cost, lock the moved nodes

Until all nodes are locked

Find the best sequence to swap

Until no more improvement

2

/30

,

(log ) #

(assuming # )

Probability of Opt:

( ) 2

ij i i j j ij

n

swap i j

S E I E I C

Complexity

n n k passes

pins n

P n

Cut

1 2 3

Group Migration

Hill climbing to jump over local optimal solutions. Locking mechanism to avoid repeated moves.

Fiduccia-Mattheyses

No swapping, move a single node each time Replace the heap with an array

max maxd b

max maxd b

max max max max

: degree

: cost of each net

i

d

b

d b C d b

max

max max max

max max max

Total increase

Total decrease 2

# top shifting = ( )

: #terminals

b p

b p b d

O b d pb

p

High Order Gains

, ,

,

,

( ) ( ) ( )

{ | , ( ) , ( ) 0

{ | , ( ) 0, ( ) 1

( ) 2

( ) 1

#nodes in A, B

pos ki

ke E e Eneg ki

pos ki A B

neg ki A B

A

B

D i C e C e

E e E i e e k e

E e E i e e e k

e

e

1 1 1 1

0 0 0

-1

B

i

eA

Probabilistic Model

( ,1), ( ,2)

( ) ( ( ) ( ))

( ) ( )

( ) ( ( ))

( ) 1 if S is an empty set

n x n i ii n i x i n

x n xn

x x

ii S

g V C p V p V

g V g V

p V f g V

p V

P

G(Vx)

Pmax

Pmin

For any i locked ( ) 0ip V

Move all nodes according to a single net

Adv: move more nodes a the same operation. Good for multi-way, hierarchical where cost function dominated by the way to handle the nets

Dis: Complicate

Simulated Annealing

21 2 1 2

crossing nets

( , ) ( ) ( ( ) ( ))

( ) size of i i

C V V C e S V S V

S V V

Simulated Annealings, nexts: configurations, T: real, count: integer Begin

S= random initial configuration T=T0

Repeat Count= 0 Repeat

Count= count+1 Nexts= generate(s) If c(nexts)<= c(s) or f(c(s),c(nexts),T)> random(0,1) Then s= nexts

Until equilibrium(count,s,T) T= update(T) Unitl frozen(T)

End

Programming

2

,

0 1 | | 1

0 0

1

For each node i if , 1

, 1

1 1then ( , ) ( )

4 4

,

1 0, | |

The eigenvalues of B ...

0, 1

The opt solution of programming

A l

i i

i i

Tij i j

i j

ij ij ii ijj

T T

V

n X x

n X x

C X X C x x X BX

b C b C

set X X X V

V

X V

1ower bound of bisection

Improving the lower bound

~2

1

1

Let C( , ) ( , )

The cost function won't change the opt.

Solution of bisection

max (nonlinear programming)

/ | |

2nd smallest eigenvalue of B+D , 0

Adv: global v

i

i ii

d

ii

ii i ij

X X C X X d x

d V

d d d

1

iew

Disadv: ... are near (numerical problem)

many points are near

no fixed can not be done recursively

k

i

i

x

x

Network Flow Approaches

t

fi

S

f

jXij

max f

0

0

ij ij ij

s js sjj j

t jt tjj j

i ji ijj j

ij

x C

x x f

x x f

x x

x

max f

0

0

0

0

ij ij

js sjj j

jt tjj j

ji ijj j

ij

x C

x x f

x x f

x x

x

,

min

0

1

0

ij iji j

ij ij i j

t s

ij

C

x

,

min

1

ij iji j

ij i j

t s

C

Multiple Commodity Flow

i

a

b

jijf

ijfijab

ij

x

max

| | 1

0

kij ij ij

k

k k kk jk jk

j j

k k ki jk ij

j j

kij

f

x C

x x f

fx x

V

x

max

0

0| | 1

0

kij ij

k

k kjk jk

j j

k kjk ij

j j

kij

f

x C

x x f

fx x

V

x

Xl Xj

Xi

ij

min

0

1| | 1

0

ij ijij

k kij i j

kkik

k i k k

ij

C

V

min

( ) | | 1

0

ij ijij

k kij i j

k ki k

k i k

ij

C

V

( ) | | 1

Triangular inequality

longest path the distance of the longest path from i to k

def: distance(i,j) =

k kij i jk ki k

k k i

ij jl il

k ki k

ij

A V

B

B constraints can be used to replace A

Weighted Cluster Ratio Cut

-

i i

,

,

Theorem:

If k 4, there exists an opt ratio cut R(X,X), s.t.

X or X

Weighted cluster ratio cut

( , )

min| || |

: distance between and

: distance of the longest path from

ij i ji j

ij i ji j

ij i j

ij

X X

C X X

d X X

X X

d

to

Theorem: Multi-Commodity flow derives an opt

solution of weighted cluster ratio cut

i jX X

Replication Cut

( , ) ( , )

: min

0

0

0

1

1

0

0

, 0

ij ij ji iji j E j i E

ij i j

ij i j

i i

s

s

t

t

ij ji

Obj C W C U

W p p

U q q

q p

p

q

p

q

W U

R

S T

R R

S T

S T

Replication Graph

Replication Graph

' '

'n

'

(1) Edges or nets are directed

(2) Duplicate the graph, create , create net

but has reverse direction, equal weight C

(3) Link directed edge ( , ) with weight

(4) Min Cut separatin

i i V n n E

n

i i

'g S & t

derives X, R, Y

Heuristic Flow Approaches

-

-

MFMC (Max-Flow Min-Cut) for bisection

1 Find two seeds as s, t

2 Find Min Cut separating s & t

3 If |X|<|X |, find a seed merge with s

4 Else if |X|<|X |, find a seed merge with t

5 Repeat 2-4 until |X|=|X-

|

Adv: Use parametric flow approach

The whole process is equivalent to one cut

Dis: The choice of the seeds

Cluster Ratio

1 2,

0.5

( , ,..., )min ( )

| || |

Complexity log ,

(1) Randomly choose two nodes

find the shortest path

(2) Add weight on the path

(3) Repeat (1)-(2) r times

kR C C

i ji j

C V V VR k

V V

rn n r n

k

2

2

2

ij

ij

ij

Radomly choose two nodes i & j

Prof that i & j X

| |

| |

| || |

| |

If are small,

dNet weight

C

d : #pahts went through wire (i, j)

kk

k kk

k

XP

V

X V XP

V

X P P

Clustering (Performance-Driven)

1

1

1

1

1

1

1

1

1

11

1

2

2

2

2

3

Cap <= 4

(1) Min delay

(2) Min #clusters

(3) Min #clusters, T>=delay

(4) Min delay #clusters <= C

(5) Min max #inputs, #cluster<= C

(6) Min #clusters, #inputs <= K

Clustering Heuristics

Cluster during target operation Recursive

partitioning (ratio cut) to find clusters.

Then treat each cluster as a single node to perform partitioning.

Linear Placement Iteration

Do linear placement Cluster adjacent nodes Treat each cluster as a single node repeat linear

placement Adv: Work on the right target Dis: Need an efficient & effective target operation

# Crossing

a b

a

a b

b

L

ba is stronger than bL

Max Pair

Given pairing cost, find the best pair to cluster

Treat each cluster as a single node. Repeat the process

The cost function does not encourage the largest node pair with other nodes continuously.

Max Matching

Match the n/2 pairs simultaneously

Adv: max matching can be solved in polynomial operation optimally

Dis: enforce unnatural pair to merge

c

ba

f

ed

C and f are merged because their choices are taken by others

Variations of cost function

Similarity of signatures (Data Path)

Bit #, pin sequence

Control signal, function of gate

The conductivity between nodes i & j

ij

ij

ikk

Replacing nets with resistance 1/C

We can use random walk to derive ij between pairs of nodes

Random walk: at node i, the probability to goto j is

C

C

Expected # of hops between i & j is proportio

ij ji

nal to ij

h h 1/ ij 2|E|

Kth-connectivity

' ' '

2 2

2

[ ], 0,

: connectivity of i, j with one hop

[ ]

: connectivity of i, j with 2 hops

[ ]

: connectivity of i, j with k hops

ij ii ij ij

ij

ij

ij

k kij

kij

C c c c c

c

C c

c

C c

c

Research Directions

a b c

ed

a c

d,e

b

d e

a bc

Logic Hierarchy Layout Hierarchy

(1) Obj: min distortion, max performance

(2) Linkage between two: eg ECO revision

(3) Improvement of operations by exploring the given hierarchy