Thesis on Floorplanning VLSI by Ramesh

7/28/2019 Thesis on Floorplanning VLSI by Ramesh

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A Recursive approach to Floorplanning

by

Renishkumar V. Ladani

A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of

Master of Technology

in

Information and Communication Technology

to

Dhirubhai Ambani Institute of Information and Communication Technology

May, 2005

DA-IICT


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Sponsored By: Bonrix Software Systems, Ahmedabad, India.www.bonrix.net, www.bonrix.co.in

Declaration

This is to certify that

(i) the thesis comprises my original work towards the degree of Master of Technology in

Information and Communication Technology at DA-IICT and has not been submitted

elsewhere for a degree,

(ii) due acknowledgement has been made in the text to all other material used.

Renishkumar V. Ladani

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Certificate

This is to certify that the thesis work entitled A Recursive Approach to Floorplanning

has been carried out byRenishkumar V. Ladani (200311014) for the degree of Master of

Technology in Information and Communication Technology at this Institute under mysupervision.

Prof. Ashok T. Amin

Thesis Supervisor

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Acknowledgements

I am thankful to my guide Prof. Ashok T. Amin for guiding me throughout my thesis work.

His suggestions and constant support during this research are motivating factor for me. I

consider myself very lucky to get an opportunity to work with him. I am thankful to my

co-guide Prof. Amit Bhatt for providing initial insight into field of floorplanning. I am

thankful to my evaluation-committee members, Prof. D. Nag Chaudhary and Prof.

Hemangi Kapoor for providing useful suggestions for my research work. I am thankful to

my colleagues and friends for motivating me for research and providing constant support

in difficult times. I am thankful to J. M. Lin and Y. W. Chang (@cc.ee.ntu.edu.tw) for

making available their floorplanning algorithm implementation and test cases on their homepage. I am thankful to Dr. Hirendu P. Vaishnav of Synapps Corp., USA for his suggestion

of floorplanning as a research topic. I am thankful to DA-IICT for providing me the

resources needed and a favourable environment to carry out my work. I am thankful to my

family for supporting me in all the ways.

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Contents

Page No.

DECLARATION ...................................................................................................... .........................I

CERTIFICATE ................................................................................................................................ .II

ACKNOWLEDGEMENTS ......................................................................................................... .....III

CONTENTS ......................................................................................................................... ..........IV

ABSTRACT .................................................................................................................. ................VII

LIST OF PRINCIPAL SYMBOLS AND ACRONYMS ............................................................ ......VIII

LIST OF TABLES ........................................................................................................................ ..IX

CHAPTER 1 ....................................................................................................................... ...........1

CHAPTER 2 ....................................................................................................................... .............7

FLOORPLANNING CONCEPTS AND APPROACHES TO PROBLEM ........................................7

CHAPTER 3 ..................................................................................................................... .............17

A RECURSIVE APPROACH ............................................................................................... ..........17

C ............................................................................................................................................. ....19

C ............................................................................................................................................. ....19

B .............................................................................................................................................. ......19

A ...................................................................................................................................... .............19

B .............................................................................................................................................. ......19

A ...................................................................................................................................... .............19

C ............................................................................................................................................. ....19

A ........................................................................................................................................... .........19

B .............................................................................................................................................. ......19

C ............................................................................................................................................. ....19

B .............................................................................................................................................. ......19

A ........................................................................................................................................... .........19

B .............................................................................................................................................. ......21

A ........................................................................................................................................... .........21

A ........................................................................................................................................... .........21

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B .............................................................................................................................................. ......21

D ............................................................................................................................................. ....30

C ............................................................................................................................................. ....30

D .............................................................................................................................................. ......30

C ............................................................................................................................................. ....30

B ............................................................................................................................................. ....30

B ............................................................................................................................................. ....30

A ...................................................................................................................................... ...........30

A ...................................................................................................................................... ...........30

D ............................................................................................................................................. ....32

C ............................................................................................................................................. ....32

B ............................................................................................................................................. ....32

A ...................................................................................................................................... ...........32

CHAPTER 4 ..................................................................................................................... .............49

CONCLUSION AND FUTURE WORK ....................................................................................... .49

REFERENCES ...................................................................................................................... ........54

APPENDIX ..................................................................................................................... ...............56

A.1 CIRCUIT LAYOUT GENERATED BY ALGORITHM-I FOR HARD BLOCKS .......................56

B.1 CIRCUIT LAYOUT GENERATED BY ALGORITHM-I FOR SOFT BLOCKS ........................62

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Abstract

Due to increase in number of components on a chip, floorplanning is important step in

Very Large Scale Integration physical design to ensure quality of design. Various iterative

approaches have been suggested to carryout floorplanning in Electronic Design

Automation tools. Iterative approaches can produce good results but they are slower. In

this thesis, we have taken bottom-up, recursive approach to floorplanning. We have also

suggested efficient exhaustive search procedure for placing two, three or four rectangular

blocks in a floorplan. A rectangular block can either be hard or soft and resultant floorplan

can either be slicing or non-slicing. Further more exhaustive search procedure can also be

extended for five or more rectangular blocks. We have developed two algorithms, which

fall in class of constructive approaches rather than class of iterative approaches. Thesealgorithms use exhaustive search procedure, works in bottom-up constructive manner and

they are recursive in nature. These algorithms are very fast compared to other search

algorithms and also producing promising results. Complexity of these algorithms is O(n).

Experiments results with MCNC circuits indicate that area utilization of about 85-99% can

be achieved in very less time then iterative algorithms.

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List of Principal Symbols and Acronyms

VLSI Very Large Scale Integration

EDA Electronic Design AutomationSA Simulated Annealing

GA Genetic Algorithm

SAGA Simulated Annealing and Genetic Algorithm

NPE Normalized Polish Expression

SP Sequence Pair

BSG Bounded Slicing Grid

TCG Transitive Closure Graph

CBL Corner Block List

GPE Generalized Polish Expression

Other minor symbols are defined at first occurrence; where necessary some symbols are

redefined in the text.

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List of Tables

Page No.

TABLE 2.5 PACKING COMPLEXITY FOR NON-SLICING FLOORPLAN, HERE N IS THENUMBER OF BLOCKS IN THE PLACEMENT.....................................................................................16

TABLE 3.4.1 UNIQUE PLACEMENT STRUCTURE AND ITS TWO COMPOSITIONS FOR TWOBLOCKS PLACEMENT.............................................................................................................................22

TABLE 3.4.1 TWO COMPOSITIONS OF TWO UNIQUE PLACEMENT STRUCTURES FORTHREE BLOCKS.........................................................................................................................................25

TABLE 3.4.1 TWO COMPOSITIONS OF SIX UNIQUE PLACEMENT STRUCTURES FORFOUR BLOCKS...........................................................................................................................................39

TABLE 3.6.1 AREA UTILIZATION AND RUNTIME FOR ALGORITHM-I............................... ......43

TABLE 3.6.2 AREA UTILIZATION AND RUNTIME FOR ALGORITHM-II....................... ......... ...43

TABLE 3.6.3 AREA UTILIZATION AND RUNTIME COMPARISON FOR ALGORITHM-I ANDALGORITHM-II..........................................................................................................................................43

TABLE 3.6.3A AREA UTILIZATION AND RUNTIME FOR SP AND O-TREE.................................44

TABLE 3.6.3B AREA UTILIZATION AND RUNTIME FOR B*-TREE AND ENHANCED O-TREE.........................................................................................................................................................................44

TABLE 3.6.3C AREA UTILIZATION AND RUNTIME FOR CBL AND TCG......................... ........ ..44

TABLE 3.6.3D AREA UTILIZATION AND RUNTIME FOR TCG-S AND FAST-SP.........................45

TABLE 3.6.3E AREA UTILIZATION AND RUNTIME FOR GPE.......................................................45

TABLE 3.6.4 PATTERN OF HARD AND SOFT BLOCKS IN TEST CASE -I........................ ........ ..46

TABLE 3.6.5 PATTERN OF HARD AND SOFT BLOCKS IN TEST CASE -II...................................46

TABLE 3.6.6 AREA UTILIZATION AND RUNTIME FOR ALGORITHM-I APPLIED ON CASE-I.........................................................................................................................................................................46

TABLE 3.6.7 AREA UTILIZATION AND RUNTIME FOR ALGORITHM-I APPLIED ON CASE-II......................................................................................................................................................................47

TABLE 3.6.8 AREA UTILIZATION AND RUNTIME FOR ALGORITHM-I APPLIED ON CASE-III....................................................................................................................................................................47

TABLE 3.6.9 AREA UTILIZATION AND RUNTIME FOR ALGORITHM-II APPLIED ON CASE-I.......................................................................................................................................................................47

TABLE 3.6.10 AREA UTILIZATION AND RUNTIME FOR ALGORITHM-II APPLIED ONCASE-II..........................................................................................................................................................47

TABLE 3.6.11 AREA UTILIZATION AND RUNTIME FOR ALGORITHM-II APPLIED ON

CASE-III........................................................................................................................................................48

TABLE 3.6.12 SUMMARY OF AREA UTILIZATION AND RUNTIME FOR ALGORITHM-I.. .. ..48

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TABLE 3.6.13 SUMMARY OF AREA UTILIZATION AND RUNTIME FOR ALGORITHM-II.....48

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List of Figures

Page No.

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Chapter 1

Introduction

1.1 Floorplanning in Context of VLSI Physical Design

VLSI physical design layout can be carried out in bottom up fashion. In this methodology

designer either uses cells from library or designs her/his cells and subsequently compose

the overall layout of the chip by means of placement and routing. But most of time this

leads to poor utilization of the chip area and excessive wiring.

Only a well-conceived design methodology can result in a final design of high quality; one

such methodology is FLOORPLAN-BASED DESIGN METHODOLOGY. It is top-down

design methodology. It advocates that layout aspects should be taken into account in all

design stages. Three design domains in which design stages are classified are behavioral

design domain, structural design domain and physical design domain.

The floorplan-based design methodology can be represented on GAJSKI Y-chart in fig.

1.1.

Fig. 1.1 GAJSKI Y-chart [1]

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Advantage of Floorplan-based methodology: Taking layout into account in all design

stages also gives early feed back, thus structural synthesis decision can immediately be

evaluated for their layout consequence and corrected if necessary. The presence of layout

information allows for an estimation of wire lengths. From these lengths one can derive

performance properties of the design such as timing and power consumption. They both

increase when the wire lengths grow.

1.2 Floorplanning Definition

It is easy to deal with layout when structural detail at lowest abstraction is available, one

knows the exact number of transistors in the circuit and the way they are interconnected.

When this type of structural information is not available, one can estimate the area to be

occupied by various sub blocks and together with a precise or estimated interconnection

pattern, try to allocate distinct regions of the integrated circuit to the specific sub blocks.

This process is call floorplanning.

It is important to note that functionally equivalent sub blocks have different shapes and

terminal positions. This is one of the main characteristics of floorplan-based design, one

chooses the shape and terminal positions such that they fit best with the original structure

and assumes that there is a way to design the module satisfying the chosen shape and

terminal position. Above type of blocks are known as flexible or soft blocks. When the

block is flexible one could say that the realization needs an area A. Whichever shape the

block will have its height h and its width w have to obey the constraint hw A. Other type

of blocks are hard blocks, it means that their shape and terminal positions (pins) are fixed.

It is also important to note that area required for interconnection wiring (Routing) can

either provided by incorporating them in the area estimations for the blocks or in the case

of N-layer metal with over the block routing (wiring), channel less block layouts are the

norm of design.

Example of a structural description of some circuit and possible floorplan and Floorplan

view of PowerPC 604 and Pentium 4 is provided in fig. 1.2.

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Fig. 1.2 Structural description of some circuit and its floorplan, and Floorplan view of

PowerPC 604 and Pentium 4 [2]

1.3 Floorplan Problem Description

Given a set of blocks B = {b1, b2,, bn}. Each blockbi is rectangular and has fixed width

and height. The outputs of algorithm are coordinates of blocks (the absolute coordinates

of the lower left corner of the block). The objectives of floorplan optimization problem are

to minimize the area ofB and reduce wire lengths of interconnects subject to the

constraints that no pair of blocks overlaps. There may be other objectives such as

maximize routability (minimize congestion), delay of critical path, noise, heat dissipation,

etc. But either they are not of much interest or in some way they are related to reduction

of wire lengths of interconnects [4].

In addition to above problem description, other then rectangular block study of L-shaped

and U-shaped blocks has been carried out [14]. Also, Flexible blocks have been not

addressed in above problem description. There also exist some representations and

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algorithms, which addressed floorplan problem with flexible blocks, e.g., Normalized

Polish Expression (NPE) [5], SP [10], Fast-SP [13], O-tree [1] and B*-tree [4]. For such

type of representations and algorithms following problem formulation would provide more

insight.

Let B = {b1,b2, , bn} be a set ofn rectangular blocks. Each blockbi B is associated

with a three tuple (hi, wi, ai), where hi, wi, and ai denote the width, height, and aspect ratio

ofBi, respectively. The areaAi ofBi is given by hi * wi, and the aspect ratio ai ofBi is given

by hi/wi, Let ri,minand ri,max be the minimum and maximum aspect ratios, i.e., hi/wi [ri,min,

ri,max]. Here both soft(flexible) and hard blocks are being considered. A hard module is not

flexible in its shape, but free to rotate. A soft module is free to rotate and change its shape

within range [ri,min, ri,max]. Output of foorplanning is a placement (floorplan) P = {(xi, yi) |

bi B} is an assignment of rectangular blocks with the coordinates of their bottom-left

corners being assigned to (xi, yi)s so that no two blocks overlap (and Hi/wi [ri,min; ri,max],

i ). As previously describe in problem description, the objective of floorplanning is to

minimize a specified cost metric such as a combination of the area Atot and wire length Wtot

induced by the assignment ofbis, whereAtotis measured by the final enclosing rectangle

ofPand Wtot the summation of half the bounding box of pins for each net.

Cost = *Atot + *Wtot

Where,

Atot= Total area of the packing.

Wtot= Total wire length of packing.

and= User specified constant.

Here wire length estimation is to be done because exact wire length of each net is not

known until routing is done and also pin positions are not known yet. Two possible ways

of wire length estimation are center-to-center estimation and half-perimeter estimation.

Fig. 1.3. Center-to-center estimation and half-perimeter estimation.

1.3.1 Floorplan Sizing: A optimization problem in Floorplanning

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The availability of flexible blocks implies the possibility of having different shapes for the

same hardware units. Its therefore possible to choose a suitable shape for each flexible

block such that the resulting floorplan is optimal in some sense (e.g. minimal area).

1.3.2 Some Constraints in Floorplanning

In floorplanning, it is important to allow users to specify placement constraints. Three

common types of placement constraints are preplaced constraint, boundary constraint, and

range constraint. For preplaced constraint, we require a block to be placed exactly at a

certain position in the final packing. For boundary constraint, we require a block to be

placed along one particular side of the final floorplan: on the left, on the right, at the

bottom, or at the top. This is useful when users want to place some specific block along

the boundary for inputoutput connections. For range constraint, we require a module to

be placed within a given rectangular region in the final packing. This is indeed a more

general formulation of the placement constraint problem and any preplaced constraint can

be written as a range constraint by specifying the rectangular region such that it has the

same size as the module itself. Some representations and algorithms for floorplan are

extended for above given constraints.

1.4 Motivation

Due to the growth in design complexity, circuit sizes are getting larger. To cope with the

increasing design complexity, hierarchical design and IP modules are widely used. The

trend makes module floorplanning much more critical to the quality of a VLSI design. And

with current EDA tools with practice we can create good initial placement by

floorplanning hints and a pictorial display. This is one area where the human ability to

recognized patterns and spatial relations is currently superior to a computer programs

ability. Thus practically floorplanning is not fully automated till now date.

1.5 Organization of the Thesis

The rest of the report is organized as follows. The second chapter starts with different

approaches to floorplanning problem with different representation of floorplan as well as

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available algorithms. The second chapter end with previous work that has been done in this

particular direction and its comparison. The third chapter provides description of our

work. It includes introduction and description of our suggested recursive bottom-up

algorithms. It also includes how these algorithms use exhaustive search procedure for

placing two; three or four rectangular blocks in a floorplan. The chapter 3 ends with

experiments results and resultant floorplan view of MCNC benchmark suite. The forth

chapter contains conclusion to our thesis work and scope of future work.

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Chapter 2

Floorplanning Concepts and Approaches to Problem

2.1 Background

The floorplan problem is known to be NP-complete [11]. Various heuristic approaches

have been taken to solve this problem. These approaches can be categorized in Simulated

Annealing (SA), Genetic Algorithm (GA) and Hybrid approach (SAGA: simulated

annealing and genetic algorithm). This type of algorithm searches through the feasible

solution space for floorplan. Evaluate each solution at each stage to know its cost or

fitness compare it with earlier available results. Keep it or discard it according to

strategies. Carry out different moves to obtain different feasible solutions from a available

feasible solution.

In Genetic algorithms [15] moves are crossover, mutation and inversion. Similar types of

moves exist for simulated annealing. Hence these algorithms depend on representation of

feasible solution space. Representation for floorplan can be categorized in slicing floorplan

representation and non-slicing floorplan representation.

2.1.1 Slicing StructureA rectangle dissection is a subdivision of a given rectangle by horizontal and vertical line

segments into a finite number of non-overlapping rectangles. The non-overlapping

rectangles are called basic rectangles. By slicing a rectangle, we mean to divide the

rectangle into two rectangles by a vertical or horizontal line. A slicing structure is a

rectangle dissection that can be obtained by recursively slicing rectangles into smaller

rectangles (see Fig. 2.1.1a).

The hierarchical structure of a slicing structure can be described by an oriented rooted

binary tree, called aslicing tree (see Fig. 2.1.1b). A Slicing tree is essentially a top down

description of a slicing structure. It specifies bow a given rectangle is cut into smaller

rectangles by horizontal and vertical slicing lines. Each internal node of the tree is labelled

either * or +, corresponding to either a vertical or a horizontal cut, respectively. Each leaf

corresponds to a basic rectangle and is labelled by a number between 1 and n when the

slicing structure has n basic rectangles. Wong and Liu proposed an algorithm for slicing

floorplan designs using a normalized polish expression [5] to represent a slicing structure.

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Fig. 2.1.1a A slicing floorplan Fig. 2.1.1b A slicing tree

2.1.2 Non-Slicing Structure

Not all floorplans are slicing. If the basic rectangles corresponding to leaf nodes in slicing

structures cant be obtained by recursive cutting rectangles into smaller rectangles then the

floorplan has non-slicing structure (See Fig. Fig. 2.1.2) and represented in different ways.

The representation are sequence pair [6], bounded slicing grid (BSG) [7], O-tree [1],

Transitive Closure Graph (TCG) [2], Corner Block List (CBL) [3] and B* Trees [4].

Fig. 2.1.2 A non-slicing structure

2.1.3 Normalized Polish Expression

A binary sequence b1,b2, , bm, is a balloting sequence iff for any k, 1


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A Polish expression 12 2n-1 is said to be normalized iff there is no consecutive *s

or +s in the sequence. (e.g. 1 2 + 4 3 * + is a normalized Polish expression.)

In general, there might be two or more Polish expressions (slicing trees) that correspond

to a given slicing structure (see Fig.3.2e). The number of Polish expressions corresponding

to a slicing structure can vary from slicing structure to slicing structure. This makes Polish

expressions an undesirable choice for representation of solutions in a simulated annealing

setting for the following reasons: 1. There is an unnecessary increase in the number of

states. 2. The set of slicing structures is unevenly distributed over the set of Polish

expressions, which might lead to unintentional and undesirable biases toward some slicing

structures. It is observation that given any slicing structure, it can be described by a unique

skewed slicing tree by performing the cuts always from right to left and from top to

bottom. Hence, the set of normalized Polish expressions as the solution space in our

simulated annealing algorithm. The Polish expression in fact is the Polish postfix notation

for this arithmetic expression.

Fig. 2.1.3 Two different slicing trees for the same slicing structure.

2.1.4 Neighbourhood Structures

We define three types of moves that can be used to modify a given normalized Polish

expression.

M1. Swap two adjacent operands.

M2. Complement some chain of nonzero length.

M3. Swap two adjacent operand and operator.

Two normalized Polish expressions are said to be neighbours if one can be obtained from

the other via one of these three moves. We also want to make sure that the move selected

will also produce a normalized Polish expression.

2.1.5 The Cost Function

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Cost = *Atot + *Wtot

Where,

Atot= Total area of the packing.

Wtot= Total wire length of packing.and= User specified constant.

2.1.6 Comparisons between slicing and non-slicing approach

Slicing representation has some advantages such as smaller encoding cost and solution

space bringing faster runtime for packing. Furthermore it is flexible to deal with hard,

preplaced, soft and rectilinear blocks. However in real designs optimal solution might not

be in the solution space of slicing structure. While with non-slicing representation optimalsolution might be achieved but it needs more evaluating runtime for packing then slicing

approach.

2.2 Algorithmic Approaches

2.2.1 Simulated annealing

Simulated annealing is a well-known high performance optimization technique for

combinatorial problems. The simulated annealing algorithm is presented below:

01 Temperature = Initial Temperature;02 Current placement = Random initial placement;

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03 Current score = Score (Current placement);

04 While equilibrium at temperature not reached Do

05 Selected component = Select (at random);

06 Trail placement = Move (selected component);07 Trail score = Score (trail placement);

08 If trail score < current score then

09 Current score = trial score;10 Current placement = trail placement;

11 else

12 if uniform random(0,1) < e-(trail score current score)/temperature then13 Current score = trial score;

14 Current placement = trail placement;15 temperature = temperature * Alpha; // alpha ~ 0.95

The temperature in initialised to a relatively high value and its slowly decrease until a

freezing point is reached. At each temperature, components are selected for possible

movement until equilibrium is reached. If movement of the selected components results inan improved placement, the movement is performed. Otherwise the movement is

performed with a probability that decrease exponentially with temperature. Components

are typically selected randomly for pair wise exchange.

2.2.2 Genetic Algorithm

The original GA and its many variants collectively known as genetic algorithms are

computational procedure that mimics the natural process of evolution. Darwin observedthat as variations are introduced into a population with each new generation the less fit

individuals are tend to die off in the competition for food and this survival of the fittest

principle leads to improvement in the species.

GAs has also applied to optimisation problems, and the applications like floorplanning in

EDA tools falls into this category. The objective of the GA is then to find an optimal

solution to a problem. Since Gas are heuristic procedure, they are not guaranteed to find

the optimum but experience has shown that they are able to find very good solutions for

wide range of problems.

GAs work by evolving a population of individual in the population where the fitness

computation depend s on the application. For each generation individuals are selected from

the population for reproduction, the individuals are crossed to generate new individuals

and the new individuals are muted with some low mutation probability. The new individual

may completely replace the old individuals in the population with distinct generation

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evolved; alternatively the new individuals may be combined with old individuals in the

population.

Since selection is biased towards more highly fit individuals, the average fitness of the

population tends to improve from one generation to the next. The fitness of the best

individuals is also expected to improve overtime, and the best individual may be chosen as

a solution after several generations.

Simple GA: Also referred to as total replacement algorithm. Flowchart of this simple

genetic algorithm is available in fig. 2.2.2. [15]

Stopping Criteria: The GA may be limited to a fixed number of generations or it may be

terminated when all individuals in the population coverage to the same string or no

improvements in fitness values are found after given number of generation.

Since selection is biased towards more highly fit individuals the fitness of the overall

population is expected to increase in successive generations. However, the best individual

may appear in any generation.

Evaluate each individual

Generate initial population

Select Np individuals with repetition, such that

the probability of selection of each individual isproportional to its fitness

Perform inversion on the offspring with

probabilityp1 if the algorithm calls for itPair the individuals randomly to form parentsMutate the offspring with a small probability, PmReplace all individuals of the previous

generation with the Np offspring

With a high probability, Pc, perform crossover on

the pair s to generate two offspring. If crossoveris not performed, then the parents are copied

unchanged to the offspring.

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Fig. 2.2.2 Flowchart of the simple genetic algorithm

2.2.3 SAGA

Rather than simply using a GA for floorplanning, its better to use a new stochastic

optimization algorithm called SAGA, Which is combination of genetic algorithm and

simulated annealing algorithm applied to floorplanning. The aim of this idea is to improve

the typical convergence rate of the pure GA by combining it with simulated annealing.

The typical GA convergence curve is shown in fig Fig. 2.2.3.

Stopping

criteria

Yes

No

Runtime

Cost

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Fig. 2.2.3. Typical convergence of a GA

Initially the solution cost improves very rapidly, however obtaining further improvement

soon becomes difficult and the majority of runtime is spent in the later phase of the process

in which small improvements are obtained very slowly, while in case of simulated annealing

algorithm. The typical convergence curve of SA is very different from that of the GA.

Initially SA converges much slower but in the late phase of the process, SA may be able to

obtain improvement faster than the GA. The unified algorithm called SAGA (an acronym

for simulated annealing and genetic algorithm) is designed in such a way that the initial fast

convergence of the GA is combined with the faster convergence of SA in the late phase.

The SAGA algorithm is application independent and highly adaptive. When applied to the

floorplanning SAGA perform better than a pure GA.

2.2.4 Comparisons between SA and GA

Both simulated annealing and the genetic algorithm are computation intensive. One

difference is that simulated annealing operates on only one solution at a time while genetic

algorithm maintains a large population of solutions which are optimized simultaneously.

Thus the genetic algorithm takes advantages of the experience gained in the past

exploration of the solution space. Both simulated annealing and the genetic algorithm have

mechanisms for avoiding entrapment at local optima. In simulated annealing this is

accomplished by occasionally discarding a superior solution and accepting and inferior

one. The genetic algorithm also relies on inferior individuals as a means of avoiding false

optima, but, since it has whole population of individuals, the genetic algorithm can keep

and process inferior individuals without losing the best one.

Simulated annealing is an inherently serial algorithm while genetic algorithm can be

parallelized on such loosely coupled distributed computer network with 100% processor

utilization.

2.5 State-of-art in floorplan representations

VLSI floorplans are often grouped into two categories, the slicing structure [5] and the

non-slicing structure [1, 2, 3, 4, 6, 7]. A binary tree whose leaves denote blocks can

represent a slicing structure, and internal nodes specify horizontal or vertical cut lines.

Wong and Liu proposed an algorithm for slicing floorplan designs [5]. They presented a

normalized Polish expression to represent a slicing structure, enabling the speed-up of itssearch procedure. However, this representation cannot handle non-slicing floorplans. It

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takes only O(n) time to derive a floorplan from a representation. Recently, proposed

several representations such as sequence pair [6], bounded slicing grid (BSG) [7], O-tree

[1], Transitive Closure Graph (TCG) [2], Corner Block List (CBL) [3] and B* Trees [4]

can handle non-slicing floorplans. Table 2.2.6 shows packing complexity for non-slicing

floorplan.

GPE Recently, a new representation for VLSI floorplan problem has been published [11].

They proposed a new and easy representation for VLSI floorplan and building block

problem. The representation effectively inherits the useful property of normalized polish

expression [5] and is able to present non-slicing floorplan. The test using MCNC

benchmarks and the experiments give promising results. The time complexity to transform

a GPE to a corresponding placement is also O(n). Results of GPE suggest that it achieves

better area utilization compared to previous non-slicing representation Fast-SP and

Enhance O-tree.

Flooplan sizing (shaping) as defined previously can be done optimally and efficiently for

slicing floorplans. It can also be done optimally for some non-slicing floorplans, but its

very time consuming. Shape Curve Computation is used for Shaping in slicing floorplans

[14] and the sizing algorithm runs in polynomial time for slicing floorpalns. LangrangianRelaxation method used for shaping in non-slicing floorplan. But it is not efficient and

applicable to only non-slicing floorplans, which are using Constraints graphs for packing

such as SP [10], Fast-SP [13], O-tree [1] and B*-tree [4].

Representation Runtime for packing

SP O(n2)

Fast-SP O(n lg n lg n)

BSG O(n2)

O-tree O(n)

B*-tree O(n)

CBL O(n)

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TCG O(n2)

GPE O(n)

Table 2.5 Packing complexity for non-slicing floorplan, here n is the number of blocks in

the placement

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Chapter 3

A Recursive Approach

3.1 Introduction

Algorithms for floorplanning are classified in two classes of approaches, iterative ap-

proaches and constructive approaches. Iterative approaches produce floorplan with better

areas utilization but they are slower then constructive algorithms. An iterative approach

starts with one initial solution, evaluate it and then generate more such solutions from

available solution. At each stage, an iterative approach evaluates new solution and com-

pares it with earlier available results and keeps only promising solutions. In these ap-

proaches, an algorithm run up to either reaching timeout or based on some criteria such as

no more improvement in results. While in case of a constructive approach a feasible solu-

tion is generated gradually from available inputs using some techniques and principles. We

propose and investigate two constructive algorithms based on the notion that grouping

blocks having nearly same area in a floorplan produce better results than placing blocks

having wide difference in area.

In both algorithms, exhaustive search procedure is carried out at each step to place four or

less blocks at a time to get a floorplan having best area utilization. This exhaustive searchprocedure is repeated in bottom up to construct a floorplan. .

Objectives of floorplanning problem is either area optimisation; wire length optimisation or

both. Although wire length optimisation is also critical to VLSI physical design but we will

focus on only area optimisation.

3.2 Problem Definition

Suppose, we are given a set of n blocks or rectangular objects b1, b2, , bn. A block can be

of a fixed type or a flexible type. A fixed block has fixed height and width. A flexible block

has constant area but can have height and width ratio, called aspect ratio, from a given set

of possible values.

These blocks are to be placed in a rectangular area in non-overlapping manner. A block

may be rotated by + 90os. The problem is to arrange n blocks inside a rectangle of

minimum possible area.

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With n blocks b1, b2, , bn, we are given a list ofn quadruplets of numbers (A1, r1, s1, d1),

(A2, r2, s2, d2), , (An, rn, sn, dn). This quadruplet of number (Ai, ri, si, di), with ri si,

specifies the area and the shape constrains for module i. In fact, if we let wi be the width of

module i and hibe the height of module i, we must have wi * hi = Ai and ri hi/wi si.Thus ri andsi are our lower and upper limit of aspect ratio. Blocki is a rigid (hard) block if

ri = si, otherwise its is a soft (flexible) block. If a block is hard then di has no meaning to it

and its just dont care value. But if a block is soft, di specifies all possible shapes for a

flexible block, having aspect ratios as ri, ri + di, ri + 2*di, , si.

A solution of the floorplan design problem consist of an enveloping rectangle R which

contains blocks b1, b2, , bn in non overlapping manner and floorplan F = {(x1i, y1i, x2i,

y2i) | 1 i n}, indicating that placement of blockbi with its bottom-left corner being at

(x1i, y1i) and top-right corner being at (x2i, y2i).

3.3 Terminology And Concepts

Minimum Area: Minimum Area (MA) is a summation of area of n blocks b1, b2, , bn.

Floorplan Area: Floorplan Area (FA) is area of minimum possible of rectangle which

accommodates n blocks b1, b2, , bn in non-overlapping manner. Clearly,FA MA.

Dead Area: A minimum possible rectangle which can accommodate n blocks in non-

overlapping manner has some area not occupied by any blocks. It is known as Dead Area

(DA) and measured in percentage of FA, namelyDA = (FA-MA)/FA*100. Area utilization

factor is defined to be 100-DA.

L-compact: A floorplan L-compact if and only if there is no block that can shift left from

its original position with other components fixed.

B-compact: A floorplan is B-compact if and only if there is no block that can shift bottom

from its original position with other components fixed.

LB-compact: A floorplan is LB-Compact if and only if its both L-compact and B-

compact.

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These types of floorplans are illustrated in fig 3.3.1.

Fig. 3.3.1 (a) A Floorplan Fig. 3.3.1 (b) B-Compact Floorplan

Fig. 3.3.1 (c) L-Compact Floorplan Fig. 3.3.1 (d) LB-Compact Floorplan

3.4 Exhaustive Search Procedure

Suppose, we are given a set of 2, 3 or 4 blocks say {A, B}, {A, B, C} and {A, B, C, D},

Where A, B, CandD are hard blocks. Here, (Ah, Aw), (Bh, Bw), (Ch, Cw) are (Dh , Dw) are

height and width of A, B, C, and D respectively. Let F is a floorplan generated after

placing either{A, B}, {A, B, C} or {A, B, C, D}. Here, (Fh, Fw) is height and width of

floorplan and FA = Fh * Fw , represent floorplan area. In this section, we present an

efficient way of searching a floorplan which has best area utilization or say minimum dead

area from all possible placement of 2, 3 or 4 blocks. For generation of all possible

placements of 2, 3 or 4 blocks first we generate the set PO of all possible ordering of

blocks. Let say for Set of 2 blocks {A, B}, PO = {AB, BA} represent all possible ordering

of two blocks. For three blocks {A, B, C}, PO = {ABC, ACB, BAC, BCA, CAB, CBA}

represent all possible ordering of three blocks. And similarly set PO for 4 blocks is also

generated. Since A block can be rotated by + 90os, we have two orientation vertical and

horizontal orientation for a block. So count of all possible pattern of placement (PPP) for

two blocks is equal to 2! * 22 = 8, Let say PPP = {AB, BA, AB, BA, AB, BA, AB,

BA} represents possible pattern of placement. Here Aand B represent rotation by +

90os for BlockA andB respectively and similarly count of all possible pattern of placement

(PPP) for three blocks equal to 3! * 23 = 48 and its equal to 4! * 24 = 384 for four

blocks.

BC

AB

C

A

BC

A

B

C

A

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In the following section we identify unique structure that can hold 2, 3 or 4 blocks in LB

compact floorplan. We have removed other redundant structures that always produce a

floorplan with same floorplan area, FA when placed with possible pattern of placement

(PPP) for two, three or four blocks. When blocks according to possible pattern of

placement are placed in unique structures we received set of all possible placement sayPP

= {F1, F2, , FK}. Here we have one unique structure for two blocks, two unique

structures for three blocks and six unique structures for four blocks. Next three sections

describe how we have identified the unique structures. Now for setPPwe have, | PP | =

k = 2! * 22 * 1 = 8, for two blocks, | PP | = k = 3! * 23* 2 = 96, for three blocks and |

PP | = k =4! * 24 * 6 = 2304.

From setPPwe search for floorplanFi which has smallest area, where, 0 i k. if two

or more floorplans have equal and minimum area then a floorplan with aspect ratio near to

1.0 is selected. Thus | PP | represent number placement to be considered before selecting

one.

In case of soft blocks, blocks A, B, C, D can take any one of the shape form its given set

of aspect ratios, which increases number of possible placement. Let sayAAR is set of aspect

ratios for blockA, BAR is set of aspect ratios for block B, CAR is set of aspect ratios for

block C,DAR is set of aspect ratios for block D then size of possible placement set PP, get

scaled proportional to value of | AAR |, | BAR |, | CAR |, | DAR |. Thus size of possible

placement set PP, | PP | = k = 2! * 22 * 1 * | AAR | * | BAR |, for two blocks, | PP | = k

= 3! * 23 * 2 * | AAR | * | BAR | * | CAR |, for three blocks and | PP | = k = 4! * 24 * 6 *

| AAR | * | BAR | * | CAR | * | DAR |. Once a soft block get placed in floorplan of 2, 3 or 4

blocks, its aspect ratio get fixed and its no longer a soft blocks now. And floorplanFthat

we received after placing 2,3 or 4 blocks together has also fixed aspect ratio because we

are selecting floorplanFfrom setPPaccording to it smallest area value and if two or more

floorplans have equal and minimum area then a floorplan with aspect ratio nearer to 1.0 is

selected.

3.4.1 Two Block Placements

In this section we have identifies LB-compact unique structure for placing 2 blocks

together in a floorplan. While placing two blocks together we can only have slicing

structures. Non-slicing structure can not possible for placing two blocks together. For

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producing all possible slicing structures for blocks, we have used a binary tree with a root

node and two children. A root node is operator and its children are two blocks. This

suggests placement of two children in a way that placement of right child is with respect to

left child and according to operator in LB-compact manner. In slicing structure we have

two-operator horizontal placement operator sayHand vertical placement operator say V.

Let O is set of operator for slicing structure then set O is define as O = {H, V}. Here

horizontal placement means two blokes are placed in side-by-side or adjacent in LB-

compact manner. And vertical placements mean two blocks are placed one above other in

LB-compact manner.

In fig. 3.4.1 (a) shows a binary tree of two blocks, while in fig 3.4.1 (b) and fig. 3.4.1 (c)

show horizontal and vertical placement respectively derived from binary by placing value

of operator as O1 = {H, V}.

Under condition of exhaustive search with all possible ordering of blocks A and B with for

each block + 90os rotation allowed both of structures from fig 3.4.1 (b) and fig. 3.4.1 (c)

produce same minimum floorplan areaFA = Fh * Fw.But difference is that one is horizontal

Fh = max (Ah,Bh)Fw = Aw + Bw

Fh = Ah + BhFw = max (Aw, Bw)

Fig. 3.4.1 (a)

Fig. 3.4.1 (b) Fig. 3.4.1 (c)

O1

A

B

A

B

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composition and other is vertical composition which achieved by floorplan rotation by +

90os and rearranging each block in floorplan once again with 90os. Thus we identified only

one unique placement structure and its two compositions for two blocks placement. Table

3.4.1 presents a unique placement structure and its two compositions for two blocks

placement.

Vertical Composition Horizontal composition

Table 3.4.1 unique placement structure and its two compositions for two blocks placement

3.4.2 Three Block Placements


together in a floorplan. While placing three blocks together we can only have slicing

structures. Non-slicing structure cant be possible for placing three blocks together. For

producing all possible slicing structures for three blocks, we have used two binary trees

with leaf nodes represent block and all other internal nodes are operator. These binary

trees have two operators to arrange three blocks. Fig. 3.4.2 (a) and fig. 3.4.2 (b) show a

binary tree of three blocks, we have two operators O1 = {H, V} and O2 = {H, V}

Fig 3.4.2 (c), (d), (e) and (f) show placement derived from a binary tree (in Fig. 3.4.2 (a))

by placing value of operator as O1 O2 = {HH, HV, VH, VV} and similarly Fig 3.4.2 (g), (h),

(i) and (j) show placement derived from a binary tree available in Fig. 3.4.2 (b).

Fh

= Ah

+ Bh

F = max (A , B )

A

B

Fh

= max (Ah,B

h)

Fw

= Aw

+ Bw

A B

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O2

O1

O1

O2

Fig. 3.4.2 (a) Fig. 3.4.2 (b)

A B C

Fh

= max (Ah,B

h, C

h)

F = A+ B + C

A B

C

Fh

= max (Ah,B

h) + C

h

Fw

= max (Aw

+ Bw, C

w)

Fig. 3.4.2 (d)Fig. 3.4.2 (c)

C

A

B

Fh

= max (Ah

+ Bh, C

h)

Fw

= max (Aw

+ Bw) + C

w

A

B

C

Fh

= Ah

+ Bh

+ Ch

Fw

= max (Aw, B

w, C

wFig. 3.4.2 (e)

Fig. 3.4.2 (f)

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Under condition of exhaustive search with all possible ordering of blocks A, B and C with

each block + 90os rotation allowed there are few redundant structures from fig 3.4.2 (c) to

fig. 3.4.2 (i) always produce same floorplan area FA = Fh * Fw. Thus we identified two

unique placement structures and its two compositions for three blocks placement. Table

3.4.2 presents two compositions of two unique placement structures for three blocks

placement.


A B C

Fh

= max (Ah,B

h, C

h)

F = A+ B + C

B

C

A

Fh

= max (Ah, B

h+ C

h)

Fw

= Aw

+ max (Bw, C

w)

Fig. 3.4.2 (g) Fig. 3.4.2 (h)

A

B

C

Fh

= Ah

+ Bh

+ Ch

=

B C

A

Fh

= Ah

+ max (Bh, C

h)

F = max (A , B + C )

Fig. 3.4.2 (i) Fig. 3.4.2 (j)

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Table 3.4.1 two compositions of two unique placement structures for three blocks

3.4.3 Four Block placements


together in a floorplan. While placing for blocks together we have slicing structures as well

as Non-slicing structure. For producing all possible slicing structures for four blocks, we

have used five different binary trees with leaf nodes represent block and all other internal

nodes are operator. These binary trees have three operators to arrange four blocks. Fig.

3.4.3 (a), Fig. 3.4.3 (b), Fig. 3.4.3 (c), Fig. 3.4.3 (d) and fig. 3.4.3 (e) show a binary tree

of four blocks, we have three operators O1 = {H, V}, O2 = {H, V} and O3 = {H, V}.

A

B

C

Fh

= Ah

+ Bh

+ Ch

F = max A , B , C

A B C

Fh

= max (Ah,B

h, C

h)

F = A+ B + C

B C

A

Fh

= Ah

+ max (Bh, C

h)

F = max A , B + C

B

CA

Fh

= max (Ah, B

h+ C

h)

Fw

= Aw

+ max (Bw, C

w)

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There exist no such procedure for producing all possible for non-slicing structures. But we

have one LB-compact unique non-slicing structure possible for placing four blocks.

Fig 3.4.3 (a1) to Fig 3.4.3 (a8) show placement derived from a binary tree in Fig. 3.4.3 (a)

and similarly Fig 3.4.3 (b1) to Fig 3.4.3 (b8) show placement derived from a binary tree in

Fig. 3.4.3 (b) and then so on up to a binary tree in Fig. 3.4.3 (e). These placements are

derived after placing value of operator as O1 O2 O3= {HHH, HHV, HVH, HHVV, VHH,

VHV, VVH, VVV}.

Fig. 3.4.3 (a)

O2

O1

O3

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A B C D A B C

D

Fh

= max (Ah,B

h, C

h, D

h)

Fw

= Aw

+ Bw

+ Cw

+ Dw

Fh

= max (Ah,B

h, C

h+ D

h)

Fw

= Aw

+ Bw

+ max (Cw, D

w)

Fig. 3.4.3 (a2)Fig. 3.4.3 (a1)

A B

C D

A B

C

D

Fh

= max (Ah,B

h) + max (C

h, D

h)

Fw

= max (Aw

+ Bw, C

w+ D

w)

Fh

= max (Ah,B

h) + C

h+ D

h

F = max (A+ B , C , D

)

Fig. 3.4.3 (a4)Fig. 3.4.3 (a3)

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C D

A

B

A

B

C

D

Fh

= Ah

+ Bh

+ max (Ch

+ Dh)

F = max (A , B , C + D )

Fh

= Ah

+ Bh

+ Ch

+ Dh

F = max A , B , C , DFig. 3.4.3 (a8)Fig. 3.4.3 (a7)

C DA

B

A

B

C

D

Fh

= max (Ah

+ Bh, C

h, D

h)

Fw

= max (Aw

+ Bw) + C

w+ D

w

Fh

= max (Ah

+ Bh, C

h+ D

h)

F = max A+ B + max C D

Fig. 3.4.3 (a6)Fig. 3.4.3 (a5)

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Fig. 3.4.3 (b)

O2

O1

O3

Fig. 3.4.3 (b2)Fig. 3.4.3 (b1)

A B C DA B C

D

Fh

= max (Ah,B

h, C

h, D

h)

F = A + B + C + D

Fh

= max (Ah

,

Bh

, Ch

) + Dh

F = max A+ B + C D

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Fig. 3.4.3 (b4)Fig. 3.4.3 (b3)

A B D

C

A

C

D

Fh

= max (max (Ah,B

h) + C

h, D

h)

F = max (A+ B , C ) + D

Fh

= max (Ah,B

h) + C

h+ D

h

F = max (A+ B , C , D )

Fig. 3.4.3 (b6)Fig. 3.4.3 (b5)

C DA

BC

D

A

B

Fh = max (Ah +Bh, Ch, Dh)

Fw

= max (Aw

+ Bw) + C

w+ D

w

Fh

= max (Ah

+B

h, C

h) + D

h

F = max (A+ B , C , D )

Fig. 3.4.3 (b8)Fig. 3.4.3 (b7)

A

B

C

D

A

B

CD

Fh = Ah + Bh + Ch + DhFw = max (Aw, Bw, Cw, Dw)

Fh = max (Ah +Bh + Ch, Dh)Fw = max (Aw + Bw, Cw) + Dw

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O1

O3

O2

Fig. 3.4.3 (c)

Fig. 3.4.3 (c2)Fig. 3.4.3 (c1)

A B C D A B C

D

Fh

= max (Ah,B

h, C

h, D

h)

F = A+ B + C

+ D

Fh

= max (Ah, B

h, C

h+ D

h)

F = A+ B + max (C , D )

Fig. 3.4.3 (c4)Fig. 3.4.3 (c3)

C D

AB B

CD

Fh

= max (Ah, B

h+ max (C

h, D

h))

F = A+ max B C + D

Fh

= max (Ah, B

h+ C

h+ D

h)

= +

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Fig. 3.4.3 (c5)

B C D

A A

C

D

B

Fh

= Ah

+ max (Bh, C

h, D

h)

F = max (A , B + C + D )

Fh

= Ah

+ max (Bh, C

h+ D

h)

F = max (A , B + max (C , D ))

Fig. 3.4.3 (c5) Fig. 3.4.3 (c6)

DC

B

A

Fig. 3.4.3 (c8)Fig. 3.4.3 (c7)

A

B

C

D

Fh = Ah + Bh + Ch + DhFw = max (Aw, Bw, Cw, Dw)

Fh = Ah + Bh + max (Ch, Dh)Fw = max (Aw, Bw, Cw + Dw)

Fig. 3.4.3 (c7) Fig. 3.4.3 (c8)

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Fig. 3.4.3 (d)

O3

O1

O2

Fig. 3.4.3 (d2)Fig. 3.4.3 (d1)

A B C D A B C

D

Fh

= max (Ah,B

h, C

h, D

h)

F = A + B + C + D

Fh

= max (Ah, B

h, C

h) + D

h

F = max (A + B + C , D )

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B

CD

D

B

C

Fh

= max (Ah, B

h+ C

h, D

h)

=

Fh

= max (Ah, B

h+ C

h) + D

h

= +

Fig. 3.4.3 (d3) Fig. 3.4.3 (d4)

Fig. 3.4.3 (d6)Fig. 3.4.3 (d5)

B CD

A

B C

D

A

Fh

= max (Ah

+ max (Bh, C

h), D

h)

F = max (A + B + C ) + D

Fh

= Ah

+ max (Bh, C

h) + D

h

F = max A B + C D

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Fig. 3.4.3 (d8)Fig. 3.4.3 (d7)

A

B

C

D

A

B

C

D

Fh

= Ah

+ Bh

+ Ch

+ Dh

F = max (A , B , C , D )

Fh

= max (Ah

+ Bh

+ Ch, D

h)

F = max (A , B , C ) + D

Fig. 3.4.3 (e)

O1

O2

O3

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Fig. 3.4.3 (e2)Fig. 3.4.3 (e1)

A B C D A B C

D

Fh

= max (Ah,B

h, C

h, D

h)

F = A+ B + C

+ D

Fh

= max (Ah, max (B

h, C

h) + D

h)

F = A + max B + C D

Fig. 3.4.3 (e4)Fig. 3.4.3 (e3)

B

CD

B

C

D

Fh = max (Ah, Bh + Ch, Dh)= + +

Fh

= max (Ah, B

h+ C

h+ D

h)

= +

Fig. 3.4.3 (e6)Fig. 3.4.3 (e5)

B C D

A

B C

D

A

Fh

= Ah

+ max (Bh, C

h, Dh)

F = max (A , B + C + D )F

h= A

h+ max (B

h, C

h) + D

h

F = max A B + C D

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Under condition of exhaustive search with all possible ordering of blocks A, B, C and D

with for each block + 90os rotation allowed there are few redundant structures from fig

3.4.3 (a, a1-a8) to fig. 3.4.3 (e, e1-a8) always produce same floorplan area FA = Fh * Fw.

Thus we identified five unique LB-compact placement structures and its two

compositions. In addition to this we have one more unique non-slicing LB-compact

placement structure its two compositions. Table 3.4.3 presents two compositions of six

unique placement structures four blocks placement.


Fig. 3.4.3 (e8)Fig. 3.4.3 (e7)

A

B

C

D

A

B

C

D

Fh

= Ah

+ max (Bh

+ Ch, D

h)

F = max A max B C + D

Fh

= Ah

+ Bh

+ Ch

+ Dh

F = max (A , B , C , D )

Fh

= Ah

+ Bh

+ Ch

+ Dh

F = max (A , B , C , D )

A

B

C

D

A B C D

Fh

= max (Ah,B

h, C

h, D

h)

F = A+ B + C

+ D

C D

A

B

Fh

= Ah

+ Bh

+ max (Ch, D

h)

F = max A B C + D

A BC

D

Fh

= max (Ah,B

h, C

h+ D

h)

F = A+ B + max C D

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Non-slicing

B C D

A

Fh

= Ah

+ max (Bh, C

h, D

h)

F = max A B + C + D

B

C

D

A

Fh

= max (Ah, B

h+ C

h+ D

h)

F = A+ max (B , C , D )

A B

C D

Fh

= max (Ah,B

h) + max (C

h, D

h)

F = max (A+ B , C

+ D

)

A

B

C

D

Fh

= max (Ah

+ Bh, C

h+ D

h)

F = max (A+ B ) + max (C , D )

A

C

DB

Fh

= Ah

+ max (Bh, C

h+ D

h)

Fw

= max (Aw, B

w+ max (C

w, D

w))

C DA

B

Fh

= max (Ah, B

h+ max (C

h, D

h))

Fw

= Aw

+ max (Bw, C

w+ D

w)

A B

D

C

AC

BD

Fh =max (Bh + Dh, max (Ah, Bh) + Ch )Fw = max (max (Aw, Cw) + Bw, Cw + Dw )

Fh =max (Ah + Bh, max (Ah, Ch) + Dh )Fw = max (max (Aw, Bw) + Cw, Bw + Dw )

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Table 3.4.1 two compositions of six unique placement structures for four blocks

3.5 Algorithm

We propose two algorithms for floorplanning. Both having complexity O(n) but first

algorithm requires recursive call in order of log4 n while second algorithm requiredrecursive call in order of n. We have taken bottom-up, recursive approach in these

algorithms. These algorithms use efficient exhaustive search procedure as explain in last

section for placing two, three or four rectangular blocks in a floorplan. Both algorithms

fall in class of constructive algorithm rather than class of iterative algorithm and work in

bottom-up constructive manner and they are recursive by nature. These algorithms

designed with concept that In case of placing few blocks together in non overlapping

manner, we can achieve better area utilization if blocks are having their area value in

neighbourhood if area values are arrange in order.

3.5.1 Algorithm-I

This algorithm starts with given blocks b1, b2, , bn, before initiating recursive call, first

blocks are arranged in ascending order according to their area. Let say ordered list of

blocks as ab1, ab2, , abn. Then list of composite blocks is generated from ordered list of

blocks as ab1, ab2, , abn.. Here a composite block is a block that which generate after

placing 2, 3 or 4 blocks together using exhaustive search procedure. A composite block

also generated from placing 2, 3 or 4 composite blocks together. Let say list of composite

blocks as cb1, cb2, , cbk. Here k = n / 4 if n mod 4 = 0 otherwise k = n / 4+1 . In list of

composite blocks cb1 generated from first four blocks of order list ab1, ab2, , abn, cb1

generated from next four blocks and so on up to cbk, generated from last four blocks from

our order list ab1, ab2, , abn,if n mod 4 = 0 otherwise cbk generated from {abn-2, abn-1,

abn} if n mod 4 = 3, { abn-1, abn} if n mod 4 = 2 or {abn} if n mod 4 = 1. Here for

generating composite blocks list, blocks are selected in-group of four from order list of

blocks starting from smallest area and then up to end of list. So last the composite block

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may have 4,3, 2 or 1 blocks or block according to number of blocks in list. Then the

selected group of four blocks are place using exhaustive search procedure, which

generates a composite block. A composite block has same property as a hard block define

previously. Thus this approach once again applies to new list of composite blocks. Before

initiating same recursive procedure, composite blocks are ordered according to area in

aviable list. This recursive procedure is stooped when only one composite block remains in

the list. And this composite block is our floorplan rectangle, which envelops n blocks in

non-overlapping manner.

This bottom up constructive approach provides us floorplan rectangle but exact co-

ordinates of each blocks has been not assigned. So with each returning from recursive call

in top-down way each composite block assign co-ordinated to its constitute blocks or

composite blocks according to rotation, ordering of blocks and LB-compact unique

structure used to generate that composite block.

At the end of algorithm we have rectangle R which contains blocks b1, b2, , bn in non

overlapping manner and floorplan F = {(x1i, y1i, x2i, y2i) | 1 I n}, means each block

has bottom-left corners being assigned to (x1i, y1i) and top-right corners being assigned to

(x2i, y2i).

Algorithm: algorithm-I (listOfBlocks)

Input: listOfBlocks blocks with height, width and aspect ratio range in case

of soft blocks.

Output: listOfBlocks with each block having fix co-ordinates and aspectratio.

FloorplanH Floorplan Height.FloorplanW Floorplan Width.

01 ArrangeBlocksInAscOrderOfArea (listOfBlocks);

02 If NumberOfBlocks (listOfBlocks) = 1 then03 SetCordinateOfSubBlocks (listOfBlocks);

04 FloorplanH = firstBlock (listOfBlocks).Height;05 FloorplanW = firstBlock (listOfBlocks).Width;

06 Return;

07 End If

08 newListOfCompositeBlocks = CreateCompositeBlocks (listOfBlocks);

09 Call algorithm-I (newListOfCompositeBlocks);

10 SetCordinateOfSubBlocks (newListOfCompositeBlocks);

3.5.2 Algorithm-II

This algorithm starts with given blocks b1, b2, , bn, before initiating recursive call, first

blocks are arranged in ascending order according to their area. Let say ordered list of

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blocks as ab1, ab2, , abn. Then from list of blocks first four blocks are selected and a new

composite block is generated from it. Let say cb1234as it is generated fromb1, b2, b3, and b4

after exhaustive search procedure. The composite block added to list of order blocks after

replacing its constituted in order list. The composite block is inserted in order list

according to its area so that order is maintained in the list. Since a composite block has

property same as a hard block. Thus this approach once again applies to new list available

after adding new composite block. Thus with each recursive call 4 blocks are replaced with

1 composite block, hence size of list reduce by 3 at each recursive call. This recursive

procedure is stooped when only one composite block remains in the list. And this

composite block is our floorplan rectangle, which envelops n blocks in non-overlapping

manner.

This bottom up constructive approach provides us floorplan rectangle but exact co-

ordinates of each blocks has been not assigned. So with each returning from recursive call

in top-down way each composite block assign co-ordinated to its constitute blocks or

composite blocks according to rotation, ordering of blocks and LB-compact unique

structure used to generate that composite block.

At the end of algorithm we have rectangle R which contains blocks b1, b2, , bn in non

overlapping manner and floorplan F = {(x1i, y1i, x2i, y2i) | 1 I n}, means each block

has bottom-left corners being assigned to (x1i, y1i) and top-right corners being assigned to

(x2i, y2i).

Algorithm: algorithm-II (listOfBlocks)Input: listOfBlocks blocks with height, width and aspect ratio range in case

of soft blocks.

Output: listOfBlocks with each block having fix co-ordinates and aspect

ratio.FloorplanH Floorplan Height.

FloorplanW Floorplan Width.

01 ArrangeBlocksInAscOrderOfArea (listOfBlocks);

02 If NumberOfBlocks (listOfBlocks) = 1 then

03 SetCordinateOfSubBlocks (listOfBlocks);

04 FloorplanH = firstBlock (listOfBlocks).Height;

05 FloorplanW = firstBlock (listOfBlocks).Width;06 Return;

07 End If

08 newCompositeBlock = CreateOneCompositeBlock

(getFirstFourOrLessBlocks (listOfBlocks));

09 InsertNewBlockInList (newCompositeBlock, listOfBlocks);10 Call algorithm-II (listOfBlocks);

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11 SetCordinateOfSubBlocks (listOfBlocks);

3.5.3 Comparisons between Algorithm-I and Algorithm-II

In algorithm-I, numbers of recursive call are in order of log4 n while in algorithm-II,

numbers of recursive calls are equal to (n-2)/3 hence in order ofn. If both algorithms are

evaluated with respect to numbers of exhaustive search procedures required, both are

same in this respect. Numbers of exhaustive search procedures required is equal to (n-2)/3

thus it is in order ofn (O (n)).

In case of algorithm-II, composite block is inserted in order list of blocks according to its

area, while in Algorithm-II sorting is used to arrange the list of composite blocks.

Algorithm-I can easily implemented on distributed environment for better performance

because in this algorithm. We can divide our problem size n in two problems of each size

n/2.

3.6 Experimental Results

We have implemented the algorithm-I and algorithm-II in the C++ programming language

on a PC with Intel PIV 1.8 GHz CPU and 256 MB memory. We have compared algorithm-

I and algorithm-II with SP [6], O-tree [1], B*-tree [4], Enhanced O-tree [9], CBL [3],

TCG [2], TCG-S, FAST-SP [13] and GPE [11] based on the five MSNC benchmark

circuits. All of these algorithms are iterative algorithm. So they are taking much more time

then our algorithm and also producing better results in area utilization, while our algorithm

producing satisfactory results in area utilization and taking very less time. Here we have

compared algorithms only for hard blocks placement.

The area and runtime comparisons among SP [6] (on SUN Sparc Ultra-I), O-tree [1] (on a

200 MHz SUN Sparc Ultra-I workstation with 521 MB memory), B*-tree [4] (on a 200

MHz SUN Sparc Ultra-I workstation with 256 MB memory), Enhanced O-tree [9] (on a

SUN Sparc Ultra-60), CBL [3] (on a SUN Sparc Ultra-20), TCG [2] (on a 433 MHz SUN

Sparc Ultra-60 workstation with 1GB memory), TCG-S [16] (on a 433 MHz SUN Sparc

Ultra-60 workstation with 1GB memory), FAST-SP [13](on ultra1)and GPE [11] (on a

PC with Intel PIII 800 MHz CPU and 128 MB memory) is provided from Table 3.6.3a to

Table 3.6.3e.

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Area utilization and runtimes for algorithm-I and algorithm-II are shown in Table 3.6.1 and

Table 3.6.2 respectively. Their comparisons are available in Table 3.6.3. Appendix A.1 and

Appendix A.2 contain circuit layout generated by algorithm-I and algorithm-II

respectively.

MCNCCircuit

ModuleCount

Width(mm)

Height(mm)

FloorplanArea(mm2)

TotalModulesAreas(mm2)

DeadArea(%)

Time(s)

ami49 49 10.4720 3.8220 40.0240 35.4454 11.4395


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MCNCCircuit

ModuleCount

MinimumArea(mm2

)

SP O-tree

FloorplanArea(mm2)

DeadArea(%)

Time(s)

FloorplanArea(mm2)

DeadArea(%)

Time(s)

ami49 49 35.4454 38.842 8.7446 1580 37.6 5.7303 7428

ami33 33 1.1564 1.22 5.2131 676 1.25 7.488 1430

hp 11 8.8306 9.93 11.071 5 9.21 4.1194 57

xerox 10 19.3503 20.69 6.4751 15 20.1 3.7298 118

apte 9 46.5616 48.12 3.2385 13 47.1 1.1430 38

Table 3.6.3a Area utilization and runtime for SP and O-tree

MCNC

Circuit

Module

Count

Minimum

Area(mm

2

)

B*-tree Enhanced O-tree

FloorplanArea(mm2)

DeadArea(%)

Time(s)

FloorplanArea(mm2)

DeadArea(%)

Time(s)

ami49 49 35.4454 36.80 3.6809 4752 37.73 6.0551 406

ami33 33 1.1564 1.27 8.9448 3417 1.24 6.7419 118

hp 11 8.8306 8.947 1.3009 55 9.16 3.5960 19

xerox 10 19.3503 19.83 2.4190 25 20.16 4.0163 38

apte 9 46.5616 46.92 0.7638 7 46.92 0.7638 11

Table 3.6.3b Area utilization and runtime for B*-tree and Enhanced O-tree

MCNCCircuit

ModuleCount

MinimumArea(mm2

)

CBL TCG

FloorplanArea(mm2)

DeadArea(%)

Time(s)

FloorplanArea(mm2)

DeadArea(%)

Time(s)

ami49 49 35.4454 38.58 8.1249 65 36.77 3.6023 434ami33 33 1.1564 1.20 3.63333 36 1.20 3.6333 306

hp 11 8.8306 NA NA NA 8.947 1.3009 20

xerox 10 19.3503 20.96 7.6798 30 19.83 2.4190 18

apte 9 46.5616 NA NA NA 46.92 0.7638 1

Table 3.6.3c Area utilization and runtime for CBL and TCG

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MCNCCircuit

ModuleCount

MinimumArea(mm2

)

TCG-S FAST-SP

FloorplanArea(mm2)

DeadArea(%)

Time(s)

FloorplanArea(mm2)

DeadArea(%)

Time(s)

ami49 49 35.4454 36.40 2.6225 369 36.50 2.8893 31

ami33 33 1.1564 1.185 2.4135 84 1.205 4.0331 20

hp 11 8.8306 8.947 1.3009 7 8.947 1.3009 6

xerox 10 19.3503 19.796 2.2514 5 19.80 2.2712 14

apte 9 46.5616 46.92 0.7638 1 46.92 0.7638 1

Table 3.6.3d Area utilization and runtime for TCG-S and FAST-SP

Table 3.6.3e Area utilization and runtime for GPE

We have also generated three test cases for checking our algorithm for placement of soft

blocks. In case-I, half numbers of blocks are soft and they are selected randomly. Range of

their aspect ratio is from 1.0 to 2.0 (with + 90os rotation allowed) with 0.1 as increment.

Table 3.6.4 shows pattern of hard and soft Blocks in test CASE I. In Case-II, half

numbers of blocks are soft and they are complement of blocks in Case-I. It means thoseblocks, which are soft in case-I, are hard in case-II and visa versa. Range of their aspect

ratio is from 1.0 to 2.0 (with + 90os rotation allowed) with 0.1 as increment. Table 3.6.5

shows pattern of hard and soft Blocks in test CASE II. In Case-III, all blocks are soft.

Range of their aspect ratio is from 1.0 to 2.0 (with + 90 os rotation allowed) with 0.1 as

increment.

MCNCCircuit

Hard, SoftBlocks

Pattern of Hard and Soft Blocks.Arranged in ascending order of area, 1 and 0 represents hard

MCNCCircuit

ModuleCount

MinimumArea(mm2

)

GPE

Floorplan

Area(mm2)

Dead

Area(%)

Time(s)

ami49 49 35.4454 36.45 2.7561 247

ami33 33 1.1564 1.18 2 81

hp 11 8.8306 9.12 3.1732 2

xerox 10 19.3503 20.14 3.9210 2

apte 9 46.5616 46.90 0.7215 1

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Count and soft blocks respectively.

ami49 (25,24) {1000001011010110000110010101101110111010111001000}

ami33 (17,16) {001101100101010000101111100101001}

hp (6,5) {000001111011}

xerox (5,5) {0001111100}

apte (5,4) {1000001111}Table 3.6.4 Pattern of Hard and Soft Blocks in test CASE -I

MCNCCircuit

Hard, SoftBlocksCount

Pattern of Hard and Soft Blocks.Arranged in ascending order of area, 1 and 0 represents hard

and soft blocks respectively.

ami49 (24,25) {0111110100101001111001101010010001000101000110111}

ami33 (16,17) {110010011010101111010000011010110}

hp (5,6) {111110000100}

xerox (5,5) {1110000011}apte (4,5) {0111110000}

Table 3.6.5 Pattern of Hard and Soft Blocks in test CASE -II

Area utilization and runtime of algorithm-I for case-I, case-II and case-III are shown in

Table 3.6.6, Table 3.6.7 and Table 3.6.8 respectively. And for algorithm-II its available in

Table 3.6.9, Table 3.6.10 and Table 3.6.11. The comparisons between algorithm-I and

algorithm II with respect to case-I, case-II and case-III are available in Table 3.6.12 and

Table 3.6.13. Appendix B.1 and Appendix B.2 contain circuit layout generated byalgorithm-I and algorithm-II respectively for case-I, case-II and case-III.

Circuit ModuleCount

Width(mm) Height(mm) FloorplanArea(mm)


DeadArea (%)

Time(s)

ami49 49 9.7980 4.1190 40.3580 35.4454 12.1724 9

ami33 33 1.5030 0.9960 1.4970 1.1564 22.7483 31

hp 11 5.8330 1.6090 9.3853 8.8306 5.9104 2

xerox 10 9.0030 2.4380 21.9493 19.3503 11.8410 31

apte 9 5.5230 9.7810 54.0205 46.5616 13.8074 0

Table 3.6.6 Area utilization and runtime for algorithm-I applied on case-I

Circuit Module

Count


TotalModules

Areas(mm2)

DeadArea(%

)

Time(s)

ami49 49 10.6370 3.7150 39.5165 35.4454 10.3021 9

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ami33 33 1.8160 0.7320 1.3293 1.1564 13.0039 6

hp 11 10.4830 1.0080 10.5669 8.8306 16.4314 30

xerox 10 3.2070 7.6440 24.5143 19.3503 21.0653 3

apte 9 5.3780 10.4530 56.2162 46.5616 17.1741 2

Table 3.6.7 Area utilization and runtime for algorithm-I applied on case-II

Circuit ModuleCount



DeadArea(%)

Time(s)

ami49 49 9.9360 3.9850 39.5950 35.4454 10.4800 367

ami33 33 1.8590 0.73

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Thesis on Floorplanning VLSI by Ramesh

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