Vlsi physical design automation on partitioning

VLSI Physical Design Automation
Introduction , partitioning

Sushil kundu
Roll No:20084056
Registration No: 1954
Dept. of Applied Electronics
& Instrumentation Engg.,
University Institute of
Technology burdwan

Intended Audience

• VLSI CAD (also known as EDA – electronic design
automation) students, in particular for chip implementation
(physical design)
• Circuit designers to understand how tools work behind
the scene
• Process engineers to tune process that is more
circuit/physical design friendly
• Mathematical/Computer Science majors who want to find
tough problems to solve
– Lots of VLSI physical design problems can be
formulated into combinatorial optimization or
mathematical programming problems.
– Actually, most CAD problems are NP-complete ->
heuristics

Objective of this Lecture
To review the materials used in fabrication of VLSI devices.
To review the structure of devices and process involved in
fabricating different types of VLSI circuits.
To review the basic algorithm concepts.
 Understand the process of VLSI layout design.
 Study the basic algorithms used in layout design of VLSI
circuits.
 Learn about the physical design automation techniques used
in the best-known academic and commercial layout systems.

Physical Design
• Converts a circuit description into a geometric
description.
– This description is used for fabrication of the chip.
• Basic steps in the physical design cycle:
1. Partitioning
2. Floorplanning
3. placement
4. Routing
5. Compaction

So, what is Partitioning?

System Level Partitioning System

PCBs

Board Level Partitioning

Chips

Chip Level Partitioning
Subcircuits
/ Blocks

6

Why partition ?
• Ask Lord Curzon 
– The most effective way to solve problems of high
complexity : Parallel CAD Development
• System-level partitioning for multi-chip designs
– Inter-chip interconnection delay dominates system
performance
• IO Pin Limitation
• In deep-submicron designs, partitioning defines local
and global interconnect, and has significant impact on
circuit performance

7

Importance of Circuit Partitioning
 Divide-and-conquer methodology
 The most effective way to solve problems of high complexity
E.g.: min-cut based placement, partitioning-based test
generation,…
 System-level partitioning for multi-chip designs
inter-chip interconnection delay dominates system
performance.
 Circuit emulation/parallel simulation
partition large circuit into multiple FPGAs (e.g. Quickturn), or
multiple special-purpose processors (e.g. Zycad).
 Parallel CAD development
Task decomposition and load balancing
 In deep-submicron designs, partitioning defines local and
global interconnect, and has significant impact on circuit
performance
…… ……

Objectives
• Since each partition can correspond to a chip, interesting
objectives are:
– Minimum number of partitions
• Subject to maximum size (area) of each partition
– Minimum number of interconnections between partitions
• Since they correspond to off-chip wiring with more
delay and less reliability
• Less pin count on ICs (larger IO pins, much higher
packaging cost)
– Balanced partitioning given bound for area of each
partition

9

Partitioning:
 Partitioning is the task of dividing a circuit into smaller parts .
The objective is to partition the circuit into parts, so that the size
of each component is within prescribed ranges and the number of
connections between the components is minimized .
 Different ways to partition correspond to different circuit
implementations . Therefore, a good partitioning can significantly
improve circuit performance and reduce layout costs .
• Decomposition of a complex system into smaller subsystems
– Done hierarchically
– Partitioning done until each subsystem has manageable size
– Each subsystem can be designed independently
• Interconnections between partitions minimized
– Less hassle interfacing the subsystems
– Communication between subsystems usually costly

Partitioning of a Circuit

Input size: 48

Cut 1=4 Cut 2=4
Size 1=15 Size 2=16 Size 3=17

Hierarcahical Partitioning
• Levels of partitioning:
– System-level partitioning:
Each sub-system can be designed as a single PCB
– Board-level partitioning:
Circuit assigned to a PCB is partitioned into sub-circuits
each fabricated as a VLSI chip
– Chip-level partitioning:
Circuit assigned to the chip is divided into manageable sub-
circuits
NOTE: physically not necessary

Delay at Different Levels of Partitions

x
A
B
D

C
10x

PCB1 PCB2
20x

13

Partitioning: Formal Definition
• Input:
– Graph or hypergraph
– Usually with vertex weights
– Usually weighted edges
• Constraints
– Number of partitions (K-way partitioning)
– Maximum capacity of each partition
OR
maximum allowable difference between partitions
• Objective
– Assign nodes to partitions subject to constraints
s.t. the cutsize is minimized
• Tractability
- Is NP-complete 

14

Circuit Representation

• Netlist: B
– Gates: A, B, C, D
A
– Nets: {A,B,C}, {B,D}, {C,D}

C D
• Hypergraph:
– Vertices: A, B, C, D
– Hyperedges: {A,B,C}, {B,D}, {C,D}
B
– Vertex label: Gate size/area
A
– Hyperedge label:
Importance of net (weight)
C D

Circuit Partitioning: Formulation

Bi-partitioning formulation:
Minimize interconnections between partitions

c(X,X’)

X X’

• Minimum cut: min c(x, x’)
• minimum bisection: min c(x, x’) with |x|= |x’|
• minimum ratio-cut: min c(x, x’) / |x||x’|

16

A Bi-Partitioning Example

a c 100 e
100 100
100 100
9
min-cut
4

b 10 d 100 f

mini-ratio-cut min-bisection

Min-cut size=13
Min-Bisection size = 300
Min-ratio-cut size= 19

Ratio-cut helps to identify natural clusters
17

Iterative Partitioning Algorithms

• Greedy iterative improvement method (Deterministic)
– [Kernighan-Lin 1970]

• Simulated Annealing (Non-Deterministic)

18

Restricted Partition Problem

• Restrictions:
– For Bisectioning of circuit
– Assume all gates are of the same size
– Works only for 2-terminal nets

• If all nets are 2-terminal, hypergraph  graph
b b
a a

c d c d
Hypergraph Graph
Representation Representation
19

Problem Formulation
• Input: A graph with
– Set vertices V (|V| = 2n)
– Set of edges E (|E| = m)
– Cost cAB for each edge {A, B} in E
• Output: 2 partitions X & Y such that
– Total cost of edge cuts is minimized
– Each partition has n vertices
• This problem is NP-Complete!!!!!

20

A Trivial Approach
• Try all possible bisections and find the best one
• If there are 2n vertices,
# of possibilities = (2n)! / n!2 = nO(n)

• For 4 vertices (a,b,c,d), 3 possibilities
1. X={a,b} & Y={c,d}
2. X={a,c} & Y={b,d}
3. X={a,d} & Y={b,c}
• For 100 vertices, 5x1028 possibilities
• Need 1.59x1013 years if one can try 100M
possbilities per second

21

Definitions
• Definition 1: Consider any node a in block X. The
contribution of node a to the cutset is called the external cost
of a and is denoted as Ea, where
Ea =Σcav (for all v in Y)
• Definition 2: The internal cost Ia of node a in X is defined
as follows:
Ia =Σcav (for all v in X)

Example

• External cost (connection) Ea = 2
• Internal cost Ia = 1

X b
Y
c
a d

Idea of KL Algorithm

• Da = Decrease in cut value if moving a = Ea-Ia
– Moving node a from block X to block Y would decrease the
value of the cutset by Ea and increase it by Ia

X b
Y X b
Y
c
c
a d a d

Da = 2-1 = 1
Db = 1-1 = 0

• Note that we want to balance two partitions
• If switch A & B, gain(A,B) = DA+DB-2cAB
– cAB : edge cost for AB

X B
Y X B Y
C C
D
A A D
gain(A,B) = 1+0-2 = -1


• Start with any initial legal partitions X and Y
• A pass (exchanging each vertex exactly once) is described
below:
1. For i := 1 to n do
From the unlocked (unexchanged) vertices,
choose a pair (A,B) s.t. gain(A,B) is largest
Exchange A and B. Lock A and B.
Let gi = gain(A,B)
2. Find the k s.t. G=g1+...+gk is maximized
3. Switch the first k pairs
• Repeat the pass until there is no
improvement (G=0)

Example

X Y X Y
1 4 4 1
2 5
2 5

3 6 3 6

Original Cut Value = 9 Optimal Cut Value = 5

A good step-by-step example in SY book

Time Complexity of KL

• For each pass,
– O(n2) time to find the best pair to exchange.
– n pairs exchanged.
– Total time is O(n3) per pass.
• Better implementation can get O(n2log n) time per pass.

• Number of passes is usually small.

Recap of Kernighan-Lin’s Algorithm
 Pair-wise exchange of nodes to reduce cut size
 Allow cut size to increase temporarily within a
pass
Compute the gain of a swap
Repeat Perform a feasible swap of max gain
Mark swapped nodes “locked”; u v a
Update swap gains;
Until no feasible swap; v u
Find max prefix partial sum in gain sequence g1,
locked
g2, …, gm
Make corresponding swaps permanent.

 Start another pass if current pass reduces the
cut size
(usually converge after a few passes)

Other Partitioning Methods
• KL and FM have each held up very well
• Min-cut / max-flow algorithms
– Ford-Fulkerson – for unconstrained
partitions
• Ratio cut
• Genetic algorithm
• Simulated annealing

References and Copyright
 Textbooks referred (none required)
 [Mic94] G. De Micheli
“Synthesis and Optimization of Digital Circuits”
McGraw-Hill, 1994.
 [CLR90] T. H. Cormen, C. E. Leiserson, R. L. Rivest
“Introduction to Algorithms”
MIT Press, 1990.
 [Sar96] M. Sarrafzadeh, C. K. Wong
“An Introduction to VLSI Physical Design”
McGraw-Hill, 1996.
 [She99] N. Sherwani
“Algorithms For VLSI Physical Design Automation”
Kluwer Academic Publishers, 3rd edition, 1999.

Vlsi physical design automation on partitioning

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