May 2012 1

Connectivity Verification in VLSI Layout

Shmuel Wimer

Bar Ilan Univ., School of Engineering

May 2012 2

Outline

• Modeling connectivity

• Finding connected components in graphs

– Directed graphs by DFS

– Non directed graphs by UNION_FIND

• Intersection of rectangles

– Interval Tree

– Priority search tree

• Intersection of non rectilinear shapes

May 2012 3

Modeling Physical Connectivity

N-diff

P-diff

poly

contact

metal1

via1

metal2

Vcc

OUT

Xn

OUT

Xn

Vcc

May 2012 4

Vcc

OUT

Xn

Xn

Vcc

Polygon in layout and its corresponding graph vertex are labeled

with net name.

A short occurs when two polygons of different nets are physically

connected, hence connected component has more than one label.

May 2012 5

Vcc

OUT

Xn

OUT

An open occurs when a polygon is missing from layout, hence several connected component have same label.

Algorithm for physical connectivity check will:

1. First construct G(V,E) by reporting pair-wise polygon intersections.

2. Then find connected components and check labeling consistency

May 2012 6

• Layout polygons are labeled with net name

• A connectivity graph G(V,E) is defined where vertices correspond to

layout polygons.

• An arc e(u,v) in E is defined for vertex pair whose corresponding

polygons are intersecting and physically connected by process

technology.

• Consequently, every net has corresponding connected component

in G(V,E), labeled uniquely by net name.

• A short occurs when a connected component has more than one

label.

• An open occurs when two or more connected components have

same label.

May 2012 7

Intersection of Rectangles

1 2 1 2

1 1 2

1 1 2

Given two intervals , and , , iff one

of the following mutually exclusive conditions is satisfied:

or .

I y y I y y I I

y y y

y y y

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

Given two rectangles , , and , , ,

, iff , , and , , .

R x x y y R x x y y

R R x x x x y y y y

1 Given a set of isothetic rectangles , report all

their pairwise intersections.

n

i in R

Problem :

May 2012 8

1 1 2y y y

1 1 2y y y

2y

1y

1 2 1 2

Solution is implemented by scan-line algiorithm. Thecondition

, , is satisfied by definition for all rectangles

currently intersected by scan-line (active rectangles).

x x x x

1 2 1 2, , can be checked by data structures called

or .

y y y y

Interval Tree Priority Search Tree

May 2012 9

Interval Tree

Introduced by McCreight 1981, Edelsbrunner 1980. We study

a static version, although dynamic version is possible.

1 1

1 2 2

Given rectangles , , are the bottom and

top ordinates of . , , , are the sorted ordinates.

nn

i i ii i

i n

R I b e

R y y y

1 2 2

1

The interval tree has a primary skeletal structure defined

staticaly in memory for , , , , handling the set

of vertical edges.

n

n

i i

T

y y y I

R

May 2012 10

2

1

3

4

5

LLIST w

1

2

3

4

5

RLIST w 1 2n nw y y

1

is dynamically storing a subset , called active

subset. is defined by scan-line position in ordinary

manaer, by those of it intersects at abscissa .

active

active

n

i i

T I x I

I x

R x

May 2012 11

1The root of has a discriminator 2

and two dynamic secondary lists and .

n nw T w y y

LLIST w RLIST w

stores lower ends of those intervals

containing , sorted in ascending order.

activeLLIST w I

w

left1 2 active active

leftactive active

Left sub-tree of rooted at is an interval tree

of , , , and | ,

are intervals whose right end fall left to .

n

w LSON w

y y y I I e w

I I w

I I

stores upper ends of those intervals

containing , sorted in descending order.

activeRLIST w I

w

May 2012 12

is calassified as if (and

hence ) or trees rooted at

and both contain active nodes. Otherwise

is .

v T LLIST v

RLIST v LSON v

RSON v

v

active

inactive

right1 2 2 active active

rightactive active

Right sub-tree of rooted at is an interval tree

of , , , and | ,

are intervals whose left end fall right to .

n n n

w RSON w

y y y I I b w

I I w

I I

May 2012 13

1 2 13

0,2 , 1,3 , 2,3 , 4,7 , 5,13 , 6,9 , 8,10 , 11,12

, , , 0,1,2,3,4,5,6,7,8,9,10,11,12,13

I

y y y

4 5 76321 108 90 11

12 13

0 1 2 3 8 10

5 6 9 13 3.5 11.5

7 8 2 7.5

1.5 9.5 5.5

4 7

11 12

May 2012 14

Properties of Interval Tree

1 2 2 is balanced binary tree. Its leaves are the ordered sequence , , , . nT y y y

The lists and associated with an internal node must

support effective insertion and deletion, hence realized by balanced binary tree.

LLIST v RLIST v v T

The active nodes of are connected in a binary tree imposed on as a

super structure. Thus, a primary node has two pointers and

. If is inactive and . if

the s

T T

v T LPTR v

LPTR v v LPTR v RPTR v LPTR v

T

ub-tree rooted at has an active node. Similarly,

if the sub-tree rooted at has an active node.

LSON v RPTR v

RSON v

May 2012 15

4 5 76321 108 90 11

12 13

0 1 2 3 8 10

3.5 5 6 9 13

11.5

7 8 2 7.5

1.5 9.5 5.5

4 7

11 12

May 2012 16

4 5 76321 108 90 11

12 13

0 1 2 3 8 10

5 6 9 13 3.5 11.5

7 8 2 7.5

1.5 9.5 5.5 11 12

By definition of active nodes, all leaves of have non-empty lists. It stems

from being binary tree that of its nodes have non-empty lists.at least half

T

T

May 2012 17

Insertion and Deletion in Interval Tree

2

Both and all and

are binary trees of

depth log .

T LLIST

RLIST

O n

*

*

Insertion of interval ,

starts at root and traverses

or nodes

until first satisfying

is found. A

path is thus defined.IN

b e

LSON RSON

v T

b v e

P

May 2012 18

*

*

The lower ordinate of ,

is inserted into and

the upper ordinate of ,

is inserted into .

b b e

LLIST v

e b e

RLIST v

*If was active is unchanged.

If it was inactive then by going

upward to first active node, with

constant work at node, can be

updated appropriately.

v T

T

*Two paths and are issued from , ending at leaves and , respectively.L R v b eP P

May 2012 19

Intersections Report

*

Intersection reports can occur with intervals allocated to nodes along

and intervals allocated to nodes in the sub-tree rooted at and bounded

between and .

IN

L R

P

v

P P

ib v

ie

b e

1

For either or . Let . , may intersect

with intervals of , . Since is ascending sorted, starting at

intersection is reported as long as . raversal stops at

f

IN

i i i

i

v P e v b v e v b e

b e LLIST v

b LLIST v b e T

irst time . Total work is linear in number of intersections. is

similar.

ib e b v

May 2012 20

1

Let (similarly ). There exist two cases. If , , may

intersect with intervals of , . Since is descending sorted,

starting at , intersection is reported as long as .

L R

i i i

i

v P v P v b b e

b e RLIST v

e RLIST v b e

raversal

stops at first time . Total work is linear in number of intersections.i

T

b e

ib v

ie

b e

ib v

ie

b e

Let .b v

May 2012 21

* *

Consider now any right sub-tree along (terminating at a leaf ). Since such

sub-trees are contained in left sub-tree of , there exists .

Hence, all intervals allocated to right sub-tr

LP b

v b v v e

ees along must be reported. This

is done efficiently by traversing only nodes of , at least half nodes of it have

non-empty lists. Consequrntly, amount of time is linear in report size.

LP

T

, intersects with all intervals of since all share (and with

). Intersections report is obtained by lists traversal without overhead.

b e LLIST v v

RLIST v

*

* *

Deletion of , finds first same as insertion did and deletes ,

from and . Intersections reports are not required since

insertion did it already. Update of is in order.

b e v b e

LLIST v RLIST v

T

May 2012 22

Performance of Algorithm

Storage is since there are 2 end points of

intervals and 4 1 nodes in , and 2 end points

at most are simultaneously stored in and

.

O n n

n T n

LLIST

RLIST

2Construction of takes log time at least

since sorting of the 2 end points is required.

T O n n

n

May 2012 23

2

2

Intervals are inserted or deleted in time log ,

including treatment of and and the

maintenance of active nodes connected in . and

traversal takes log .L

R

O n

LLIST RLIST

P

P O n

T

Less than half of visited nodes for intersection

report with and are empty, and

intersections of intervals are reported from

and with no overhead.

LLIST RLIST

LLIST

RLIST

May 2012 24

2 2

Given isothetic rectangles having

intersecting pairs, interval tree algorithm reportes

intersections in log time, log

pre processing time and space .

n s

O n n s O n n

n

:Theorem

2

2

Is it possible to perform better than

log ?

Evidently, is mandatory (cannot perform

better than report size). So can we do better than

log ?

O s n n

O s

O n n

: Question

May 2012 25

1 2Given a set , , , , the problem called

ELEMENT_UNIQENESS is aiming at answering

whether the set has two identical elements.

nz z z R

2log is a lower bound of it. Define

rectangles , , , and apply

intersection report algorithm, which now will

solve the ELEMENT_UNIQENESS problem.

i i i

n n n

R a b z z

2

Consequently, no faster algorithm is possible,

and interval intersection runs in log .n n s

May 2012 26

Priority Search Tree

• Introduced by McCreight 1981 / 1985.

• Transform reporting rectangle intersections into point

inclusion problem.

• Paradigm is scan-line, abscissa of vertical edges are

sorted in ascending order {x1, x2,…, x2n}.

• Vertical edges [y1, y2] are associated with opening and

closing rectangle property.

May 2012 27

2y

1y

2y

1y

1 2 1 2

Intersection occurs if

and .y y y y

Unlike Interval Tree where intersection conditions were

checked explicitely, intersection problem is transformed

into a point containment problem as follows.

May 2012 28

p

q

2 1,y y

21 2

Every living rectangle (i.e., it is already opened and exists in memory)

is transformed into a point in 2D plane , , .y y p q R

1 2

22 1

A newly inserted left edge , is transformed into infinite

quadrant of 2D plane, , , .

y y

Q p q p y y q

R

1 2

1 2

1 2

New rectangle ,

intersects with a living

rectangle , iff

, .

y y

y y

y y Q

1 2,y y

May 2012 29

Priority Search Tree is a data structure supporting queries of point

containment very efficiently. maintains coordinates as a search

tree and it is a heap for maintaining coordinates.

T

T p

q

coordinate must be unique for rectangles in order to maintain

consistent insertion and deletion. This can be done by attaching to

the identifier (e.g., memory pointer) of its originating rectangle.

p

p

* *,p q

p

q * *The pair , with maximal

is attached to the root of .

p q

q T

range is then divided by a

vertical line .

p

p const

May 2012 30

The above steps are applied recursively for the

resulting left and right sub ranges. Resulting

trees are attached to and nodes

of . How to decide on the

vertical line ?

p

LSON RSON

root T

p const

Question :

* *If , is used, a data structure maned

results in. It has good average

performance but can be a linear list in worst

case.

p q

Cartesian Tree

May 2012 31

Since the entire data is known apriori, the

range of , is known too. Since

all the values of are distinct, bisection at

median is possible.

min maxp p p

p

2For , pairs the depth of is log .

successive bisections result in a stripe in

having one unit width in .

n p q T O n

O n

q p

May 2012 32

jhdb

niea

of

m

k p

c g l

da

b eh

g

cj

l

mf

i

k

o

p

n

p

q

A priority search tree is a heap in q and a radix search tree in p.

May 2012 33

Implementing Priority Search TreeAs other scan-line algorithms, vertical edges are first sorted in ascending

order of abscissa. Abutments are considered as intersection and therefore

opening edges preceed closing ones having same abscissa.

Along scan-line progression a

vertical edge , in

plane is translated into a point

, in plane.

b e xy

p q pq

( p , q ) = ( b , e )

e

b

Denote by , = , the key coresponding to a currently inserted

edge , and by , the key of an edge stored in a node

(recall that is unique).

E E

v v

p q b e

E b e p q v T

p

May 2012 34

A node stores a range , of values in plane. It has

two sub trees rooted at and .

v T B v E v p pq

LSON v RSON v

We'll present a code for edge insertion. Deletion is more complicated and

discussed briefly. See details in E. M. McCreight, "Priority search trees",

SIAM J. Comput., Vol. 14, pp. 257 - 276, 1985.

Unlike segment and interval trees which are static, priority search tree is

dynmic, storing only currently opened rectangles. This is an advantage

for typical VLSI data (why?).

Most of VLSI layout objects are relatively very small compared to chip / block

size. It means that at each insertion or deletion of an edge, rectangles

are opened, consuming storage.

O n

O n

May 2012 35

INSERT ( , , ) {

( ) {

// heap maintenance

// r

( ) { , ; , , ; }

2;

( ) { ;

adix bisection

E E

E v E E v v E E

E

void key p q int b, int e, node v

while v

if q q p q v p q p q

m b e

if p m e m v LSON v

; }

{ ; ; }

}

allocate son node

// left search

// right search

// leaf re to ; , ;

ached

;

}

E E

else b m v RSON v

u v p q v

LSON u RSON u

May 2012 36

Deletion operates in two phases. In the first finds in the key , of

the deleted edge by ordinary search of a radix search tree. is compared

with 2. If search takes left to and if v

T p q

p

m b e p p LSON v

search takes right to . If then , is found.

v

v

p p

RSON v p p p q

A second phase starts at the node where , was found. "Knock out

tournament" starts, at which two sons compete on the vacant key at their

parent. The key of son having larger wins. Competition proc

p q

q eeds with

the position of the victor son, Tournament stops when a node with less

than two sons is reached. This maintains the heap property of . (Only

lowest internal nodes may have single son.)

T

May 2012 37

Reporting Intersections Report takes place prior to edge insertion. If , is the edge aimed

at insertion, an infinite quadrant is defined as follows:

E b e

( p , q ) = ( e , b )

e

b

By definition, all the rectangles corresponding to nodes whose , keys

are contained in the quadrant are intersecting with the rectangle of edge .

p q

E

Exploration of these points is done by first checking whether .

If test fails, it follows from heap property of that entire subtree can

be discarded ( for all its nodes).

v E

v E

q q

T

q q

May 2012 38

1

For a node satisfying , a radix search in the subtree rooted at

takes place, aiming at finding all points satisfying , . Given

an edge : , , define ( ), min and ( )

v E

v E

r E l i n i E

v q q v

p p

E b e p p e p b q q b

.

REPORT( , , , ) {

( ) {

( , ) { report intersection }

2 ;

// check heap propery

// apply radix

(( ) ( )) { R

search

EPO

l r

v

v l r

l r

r

void int p int p int q node v

if q q

if p p p

m p p

if m p and RSON v

RT( , , , ) }

( ) { REPORT( , , , ) }

}

}

r

l

m p q RSON v

if LSON v p m q LSON v

May 2012 39

2

REPORT( , , , ) correctly reports all points in

quadrant , , in log time, where is report size.

l r

r

p p q root T

p q O n k k

Lemma :

A reported point is correst. REPORT first verifies , ,

as otherwise it returns. Satisfaction of , is also explicitely

checked before reporting.

v

v r

q q

p p

Proof :

Conversely, if , , , it must first be approached

by ( ) query in REPORT which implements a search in a heap,

hence finding all nodes satisfying the ( ) query. Then the query

, must

v v r

v

v

v r

p q p q

q q

q q

p p

be positively answered.

May 2012 40

2

2

is a heap in and binary search tree in , whose depth is bounded by

log . REPORT implements a radix search in , . Without

( ) query, this was an ordinary search of log , where

is the

r

v p p

T q p

O n p

q q O n k k

2

number of nodes satisfying , . ( ) query prevents

overheads of quering wrong nodes, yielding log time.

v r vp p q q

O n k

Q.E.D

2

2

A priority search tree for a set of points uses storage

and can be built in log time. Reporting of all points in quadrant

, , takes log time, where is report size.r

n O n

O n n

p q O n k k

Theorem :

May 2012 41

1

1 2 2

1

FIND_ALL_INTERSECTIONS ( ) {

sort vertical edges of retngles in ascending order , , , ;

set min ; initialize root of priority search tree ;

( 1 ; 2

n

i i

n

l i n i

void rectangles R

x x x

p b T

for i i n

; ) {

( is openning ) {

REPORT_INTERSECTIO

// report

N ( , , , ) ;

INSERT ( , , ) ;

intersections only for openning ed

}

{ DEL

gei

l i i

i i i i

i

if x

p e b root T

b e b , e , root T

else

ETE ( ) ; // clo}

}

}

sing edgei ib , e , root T

May 2012 42

Layout Connectivity Graph

• Geometric intersection algorithms detected pair-wise

physical connectivity of polygons (labeled by net name),

yielding a connectivity graph G(V,E).

– V: polygons

– E: physical connections

• Connected components (CC) of G(V,E) represent parts

of layout having same electric potential.

– Short occurs when CC has more than one label

– Open occurs when multiple CCs have same label

May 2012 43

Disjoint Sets and Connected Components

MAKE_SET creates a set of single element .x x

UNION , applies . The new set is represented by

an element of either or , which are then destroyed from .

x y

x y

x y S S

S S

S

1 2 1

Invariants of connected component algorithm is a collection of

, , , of dynamic set , , ,

Every set is represented (identified) by one of its objects.

is supplamented by

kk i i i i j jS S S s S S S S S

S operations defined for every .x S

FIND_SET returns a pointer to where belongs to.xx S x

May 2012 44

What is the maximal number of UNION operations?

We measure performance by , the number of

MAKE_SET operations and , the total number

of MAKE_SET, UNION and FIND_SET.

n

m

CONNECTED_COMPONENTS( , ) {

( ) { MAKE_SET( ) }

( , ) {

(FIND_SET( ) FIND_SET( )) { UNION( , ) }

}

}

void graph G V E

foreach v V v

foreach e u v E

if v u u v

May 2012 45

UNION ( c , f )

Linked-List Data Structure of Disjoint Sets

a dcb

e gf

a cb gf

What is run-time complexity ?

MAKE_SET is O(1) FIND_SET is O(1)

What about UNION ?

May 2012 46

x1 x2 x3 Xn-1 xn

Let CONNECTED_COMPONENT do the following sequence

MAKE_SET(X1)

MAKE_SET(Xn)

UNION(X1,X2)

UNION(Xn-1,Xn)

n

2n

2n

Average time per operation is θ(n)

Question: How to improve run-time ?

May 2012 47

Time is consumed in UNION for updating the pointers to set’s

representative

Use weighted-UNION heuristics

• maintain size of set

• Always merge smaller set to larger one

Theorem: Given G(V,E) with |V|=n and |E|=m, finding connected

components by weighted-UNION heuristics takes O(m+nlog2n) time.

Proof: Pointer of vertex to the representative of its set may change log2n times at most !

Therefore, the time consumed by UNION is O(nlog2n).

There are O(m) operations of MAKE_SET and FIND_SET. Q.E.D

May 2012 48

UNION ( c , f )

Disjoint-Set Forests

a

d

c

b

e

g

f

a

d

c

b g

f

e

Sets are represented by rooted trees instead of linked lists.

This by itself doesn’t improve run-time. A sequence of n-1

UNION operations may result a tree that is just a linear

list. Two heuristics improve performance significantly: Union

by Rank and Path Compression.

May 2012 49

Similar to weighted UNION heuristics. Make the root of the deeper

tree the root of the new tree (hang the shallower tree on the root of

the deeper one).

It requires to maintain the depth of the tree, called rank, at the root.

Rank approximates the log2 of tree size.

Path compression: Implemented as a part of FIND_SET. It

consumes time proportional to the distance from root. It maintains

set trees as shallow as possible.

ba

cd

e

ba

c d

eFIND_SET ( b )

May 2012 50

Disjoint-Set Forests Implementation

Associate with every node an integer , which

is the longets path (edge count) to leaf in subtree rooted

at . points to parent node.

x rank x

x parent x

MAKE_SET( ) {

; 0 ;

}

void node x

parent x x rank x

UNION( , ) {

LINK(FIND_SET( ) , FIND_SET( ))

}

void node x node y

x y

May 2012 51

Disjoint-Set Forests Implementation

// and are roots, hence only their ra

LINK( , , ) {

( ) {

= ;

}

{

nk may chang

=

e

;

( ) {

++ }

}

}

void node x node y

if rank x rank y

parent y x

else parent x y

if rank x rank y

ra k y

x

n

y

May 2012 52

// Check whether is the root. If so, return it.

// Otherise, it is connected directly to tree

// representative (root), thus implementing path

F

/

IND_SET( ) {

/ compressio

(

n.

node node

if

x

x

) {

FIND_SET( );

}

;

}

x parent x

parent x parent x

return parent x

May 2012 53

FIND_SET is a two-pass process. It first goes

upward to root along find path to find set

representative. It then goes downward (exit

of recursion) and connects to root each node

along the path.

Run time is almost linear in . For proof see:

.

m

Introduction to Algorithms,

T.H. Cormen, C.H. Leiserson and R.L. Rivest

May 2012 54

Implementation Comments

• Connectivity graph edges are not required explicitly.

Instead, connected component finding algorithm is

invoked into the various intersection report algorithms.