048918 VLSI Backend CAD Konstantin Moiseev – Intel Corp. & Technion Shmuel Wimer – Bar Ilan...

Lecture 6Dynamic Programming: Slicing

Floorplans and Technology Mapping

048918VLSI Backend CAD

Konstantin Moiseev – Intel Corp. & Technion Shmuel Wimer – Bar Ilan Univ. & Technion

VLSI design flow overview Introduction to algorithms and optimization Backend CAD optimization problems:

◦ Design partitioning◦ Technology mapping◦ Floorplanning◦ Placement◦ Routing

Introduction to layout Layout optimization and verification

◦ Layout analysis◦ DRC and LVS checks◦ Finding objects in the layout

Course outline

Dynamic programming◦ Formal definition ◦ Simple examples

Slicing floorplans◦ General description◦ Dynamic programming algorithm

Technology mapping◦ Problem definition◦ Solution stages◦ Dynamic programming algorithm

Lecture outline

Dynamic Programming is a bottom-up technique that can be applied to solving optimization problems that are decomposable

A minimization problem is decomposable if each instance of the problem can be decomposed (perhaps in multiple ways) into “smaller” problem instances such that we can obtain a solution (not necessary an optimal one) of by combining optimal solutions of the

Each sub-problem has a positive decomposition size (d-size) that decreases in each decomposition step

Problem instances whose d-size is below some threshold are required to be directly solvable

In order to be solved by DP the problem has to exhibit optimal substructure

Dynamic Programming (DP)

p

1, , kp pps p

1, , kp p1, ,

kp ps s

Principle of optimality states that:1. There is a monotonically increasing function such that:

Reminder: is a cost of solution (not necessarily an optimal one) is monotonically increasing if for ,

2. There must be at least one way of breaking up into such that an optimal solution of results from combining arbitrary optimal solutions of the sub-problems and

3. We must be able to compute each such quickly by combining the The problem decomposable in this way exhibits optimal

substructure

Optimal substructure and principle of optimality

: kf

1, ,

kp p pc s sf c c s

pc s ps 1, , nf x x

1 1' , , 'n nx x x x 1 1', , ' , ,n nf x x f x x

1, , kp pps p

1, , kp p

1, ,

kp psc cf s c s

sips

Solution by DP can be seen as a sequential decision process◦ A stage is a set of all solutions to sub-problems with the same

d-size:

◦ Individual solutions in each stage are called states◦ The sequence of stages is built bottom-up, starting with stage

containing solutions to atomic sub-problems and ending with stage containing solutions to the whole problem

◦ Only the states in stage are used to create states in stage ◦ Most of newly created states in stage are pruned, leaving

only optimal (non-redundant) solutions◦ The optimal states are saved and re-used for next stage states

generation – this is called memoization◦ After stage n is reached, optimal solution itself is reconstructed

by top-down backtracking

Sequential decision process

1, , | ,1k ji p i jp kp d_size

i 1i 1i

7

Floorplan Graph representation

Back to floorplanning

B2B1

B3 B5

B4

B6

B12

B9

B8 B7

B10

B11

B2

B1

B1

0

B5

B12B6

B3

B9

B8

B7

B11B4

Vertices - vertical lines. Arcs - rectangular areas where blocks are embedded.

Floorplan is represented by a planar graph.

A dual graph is implied.

8

Actual layout is obtained by embedding real blocks into

floorplan cells.

◦ Blocks’ adjacency relations are maintained

◦ Blocks are not perfectly matched, thus white area (waste)

results

Layout width and height are obtained by assigning blocks’

dimensions to corresponding arcs.

◦ Width and height are derived from longest paths

Different block sizes yield different layout area, even if

block sizes are area invariant.

From Floorplan to Layout

9

Optimal Slicing Floorplan

hh

vv vv

B2B1

B3 B5

B4

B6B11B3 B4 B5 B6 B8 B9 B10

B1 B2 B7

hhhh

B12

B12

B9

B8 B7

B10

B11

Slicing tree. Leaf blocks are associated with areas.

v

Top block’s area is divided by vertical and horizontal cut-lines

10

Let block , 1 , have possible implementations , ,

1 , having fixed area .

In the most simplified case , 1 , have 2 implementations

corresponding to 2 orientations.

:

i i

i j j j

i i

i j j i

i

B i b h w

j n h w a

B i b

Problem Find among the 2 possible block orientations , 1 2 ,

the one of smallest area.

b b

i i

(L. Stockmeyer): Given slicing floorplan of blocks whose

slicing tree has depth , finding the orientation that yields the smallest

area takes time and storage.

b

d

O bd O bd

Theorem

Notation , is a cost function - non-decreasing in both arguments and computable

in constant time.

h w

slicing floorplan

corresponding binary slicing tree

F

T

and dimensions of floorplan with orientation F F Fh w

floorplan described by the subtree routed at node

set of leaves in that subtree

u uF

uL

The problem: ,F Fh w

min

By induction

Proof

1 1 2 2

1 1

For each node of the algorithm constructs a list of pairs

with the following properties:

.

. and for with

. For each with th

, , , ,

ere is

1

3

,

11

,

2

1

m m

i i i i

h w h w h w

m L u

h h w w i i m

i i m

u T

( ) ( )

( ) ( )

an orientation of the cells in such

that:

. For each orientation of the cells in there is a pair in the

list with:

,

and

,

4 ,

i i F u F u

i i

i F u i F u

L u

h w h w

L u h w

h h w w

ProofThe algorithm constructs lists bottom up.

For each of with dimensions and w ,l ith the list is e ,af ,T a b a b a b b aThe conditions 1-4 are satisfied lfor eaf.

1 1

1 1

'

, , , ,

', ' , ,

nonLet be a vertical slice node of with children and and let

be the lists constructed for and respectively, where

', '

' 1

-l

eaf

1

and

'

k k

m m

u T v v

h w h w

h w h w

v v k L v

m L v

Let us assume by induction that properties 2-4 are satisfied for these lists.

14

v1 v2 v3 v4

v1’ v2’ v3’ + =

+ =

+ =

u

''

max ,u v v

u v v

h h h

w w w

ProofThe pairs in list for are created by procedure:u maxplus merge

1 1// vertical cut-line

// lists are sorted in descending

HorizontalMerging ( , , , ) {

1; 1;

while (( ) && ( )) {

max ' , ;

'

order of he

;

ight

ts

i i j ji j

u i j

u i j

w h w h

i j

i s j t

h h h

w w w

if ( ' ) { }

else if ( ' ) { }

else { ; // ' }

}

}

i j

i j

i j

h h i

h h j

h hi j

u

Size of new width-height list equals sum of lengths of children lists, rather than their product (redundant solutions are pruned).

Proof

Understanding pruning

w

h

Pareto frontier (irredundant solutions) - saved

Redundant solutions - pruned

For continuous problems Pareto frontier is always convex

Proof. The new list created for satisfies properties 1-4.uClaim

. The maximum number of loop iterations is .

Therefore:

1

1 1

Property 1

' 1 1 1

k m

k m L v L

wh

v

il

L u

e

. Assume last created pair in list is .

For any and

max , ' , '

' ' , , ' '

max , ' max , ' w ' '

, , therefo

Property

re

and

2 i j i j

k i l j k i l j

k l i j k l i j

u h h w w

k i l j h h h h w w w w

h h h h w w w

. Satisfied by construction - each pair in the list represents some

ori

Pro

ent

perty 3

ation.

. Satisfied by construction. Only non-redundant pairs are saved in Pro theperty 4

list.

Since is non-decreasing, the optimal solution can be obtained by minimizing

over all pairs in the list constructed for the roo, t, i i i ih w h w T

Proof

: there are in total pairs in all lists at each level

of the tree and there are levels. Each pair is processed only once at each

level. Therefore, run-t

Run-

ime

time complex

complexity i

ity

s

O n

d

O nd

: the same. Space to keep atStorage each oncomplexity e of levels.O n d

: each partial solution holds two pointers to its

"parents". Top-down backtracking is used to find orie

Optimal solution reconstr

ntations of individual bl

u

o

ction

cks.

A question for self-check how algorithm and proof would change if each basic

block had implementations? What would be run-time and space complex

:

ity?p

19

Another example - Optimal Tree Covering

A problem occurring in mapping logic circuit into new cell library

Given:◦ Rooted binary tree T(V,E) called subject tree (cone of logic

circuit), whose leaves are inputs, root is an output and internal nodes are logic gates with their I/O pins.

◦ A family of rooted pattern trees (logic cells of library), each associated with a non-negative cost (area, power, delay). Root is cell’s output and leaves are its inputs.

A cover of the subject tree is a partitioning where every part is matching an element of library and every edge of the subject tree is covered exactly once.

Find:◦ A cover of subject tree whose total sum of costs is minimal.

Optimal Tree Covering

The problem: ◦ Find covering of subject graph (tree) with minimal area

?Pattern treesSubject tree

21

r

s t

u

t1 (2) t2 (3) t5 (5)t4 (4)t3 (3)

t2

t1 t1

t3

3+2+2+3=10

t4t1

t3

4+2+3=9

t2

t5

3+5=8

Tree coveringBrute force approach Brute force solution, recursive exploration:

Start from root, try all possible patternsFor each root pattern, try all possible patterns for its

descendants, etc. Run-time:

◦ For each node try all possible patterns◦ For tree with nodes and patterns in library run time is ~

◦ Should look for another approach

n p

pO n

Tree coveringDynamic programming Revealing optimal substructure:

◦ If pattern P is min cost match at some node of subject tree…◦ then it must be that each leaf of pattern tree is also the root of some min cost

matching pattern Assume three different patterns match at root of subject tree

◦ Pattern P1 has 2 leaf nodes: a and b◦ Pattern P2 has 3 leaf nodes: x, y and z◦ Pattern P3 has 4 leaf nodes: j, k, l and m◦ Which is the cheapest pattern if we know cost of each pattern?

Based on R. Rutenbar slides

Tree coveringDynamic programming

Min cost tree cover◦ Cheapest cover of root of subject tree is mincost(root) =

min(patterncost(P1) + + ,patterncost(P2) + + + ,patterncost(P3) + + + + )

- Each rectangle means recursive call to mincost (subtree) This shows optimal substructure of covering problem


mincost(a) mincost(b)

mincost(x) mincost(y) mincost(z)

mincost(j) mincost(k) mincost(l) mincost(m)

Tree coveringDynamic programming Revealing overlapping sub-problems

◦ Assume we calculate tree cost top-down◦ For picture below: node “y” in the subject tree

Will get its mincost cover computed (mincost(y)) when we put P2 at the root of the subject tree

… and again, when we put P3 at the root

◦ Instead, calculate tree cost bottom-up Will have to calculate mincost(y) only once and store – memoization

◦ This reminds Fibonacci numbers example …


Tree coveringDynamic programming Assume table[node] = ∞ for all nodes at the beginning The algorithm:


Tree coveringDynamic programming The algorithm works bottom up For each node, checks all possible patterns that can

be rooted at this node and combines each pattern’s cost with optimal solutions of sub-trees rooted at leafs of the pattern

Only optimal solution is saved in each node Complexity:

◦ In each one of nodes patterns are checked: The run-time complexity is Space complexity is

n p

O pn O n

Tree coveringDynamic programming - Example Cover following circuit using DP approach:

Tree coveringDynamic programming - Example

- NAND2 is only match for node a


- NAND2 is only match for node c- INV is only match for node b


- NAND2 is only match for node e- INV is only match for node d


-INV is possible match for node f- AOI21 is possible match for node f- NAND2 is only match for node g


-INV is only match for node h- NAND2 is possible match for node i- NAND3 is only match for node i


-NAND2 is only match for node j- NAND3 is possible match for node j

Tree coveringDynamic programming - Example Now backtrack to reveal optimal cover

Backup

38

Representing slicing floorplans using shape functions

Legal shapes Legal shapes

w

h

w

h

Block with minimum width and height restrictions

ha*aw A

39

Shape functions

w

h

Hard library block

w

Discrete (h,w) values

h

40

Corner points

52

2

5

2 5

2

5

w

h

41

Algorithm

This algorithm finds the minimum floorplan area for a given slicing floorplan in polynomial time. For non-slicing floorplans, the problem is NP-hard.

Construct the shape functions of all individual blocks

Bottom up: Determine the shape function of the top-level floorplan from the shape functions of the individual blocks

Top down: From the corner point that corresponds to the minimum top-level floorplan area, trace back to each block’s shape function to find that block’s dimensions and location.

42

4

2

2

4

Block B:

Block A:

5

5

3

3

Step 1: Construct the shape functions of the blocks

43

4

2

2

4

Block B:

Block A:

5

5

3

3


2

4

h

6

w2 64

5

3

44

4

2

2

4

Block B:

Block A:

5

5

3

3


2

4

h

w2 64

6

3

5

45

4

2

2

4

Block B:

Block A:

5

5

3

3

w2 6

2

4

h

4

6

hA(w)


46

4

2

2

4

Block B:

Block A:

5

5

3

3

hB(w)

w2 6

2

4

h

4

6

hA(w)


47

w2 6

2

4

h

4

6

hB(w)hA(w)

8

w2 6

2

4

h

4

6

hB(w)hA(w)

hC(w)

8

Step 2: Determine the shape function of the top-level floorplan (vertical)

48

w2 6

2

4

h

4

6

w2 6

2

4

h

4

6

hB(w)hA(w)

hB(w)hA(w)

hC(w)

3 x 9

4 x 7

5 x 5

88

Minimimum top-level floorplanwith vertical composition

Step 2: Determine the shape function of the top-level floorplan (vertical)

49

2 x 4 3 x 5

5 x 5

Step 3: Find the individual blocks’ dimensions and locations

w2 6

2

4

h

4

6

(1) Minimum area floorplan: 5 x 5

(2) Derived block dimensions : 2 x 4 and 3 x 5

8

Horizontal composition

048918 VLSI Backend CAD Konstantin Moiseev – Intel Corp. & Technion Shmuel Wimer – Bar Ilan...

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