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GSEIS - LME
Logic Synthesis in IC
Design and Associated Tools
Review-II
Wang Jiang Chau
Grupo de Projeto de Sistemas
Eletrônicos e Software Aplicado
Laboratório de Microeletrônica – LME
Depto. Sistemas Eletrônicos
Universidade de São Paulo
Escola Politécnica da Universidade de São Paulo
GSEIS - LME
Data Structures
 Data Type: integer, boolean, etc.
Set of values that objects can assume, their common
data representation and set of operations on them.
Abstract Data Type (ADT): graphs, trees, lists
Mathematical model with defined operations
Data Structures: arrays, pointers
Data types and their relationships, used to implement
ADTs
Obs. Very common The term Data structure is used
to mean ADT
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Abstract Data Types (ADTs)- 1
The concept of abstraction means:
 We know what a data type can do
 How it is done is hidden for the user
With an ADT, users are not concerned with how the task
is done but rather with what it can do.
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Abstract Data Types (ADTs)- 2
 Mathematical model with associated operations.
 The implementation is not defined
 Two ADTs with the same model, but with different
set of operations are considered distinct ones (the
implementation may be different depending on the
set of operations
 Lists, stacks, queues, graphs, trees, heaps
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List ADT
 A sequence of zero or more elements
A1, A2, A3, … AN
 N: length of the list
 A1: first element
 AN: last element
 Ai: position i
 If N=0, then empty list
 Linearly ordered


Ai precedes Ai+1
Ai follows Ai-1
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Operations
 printList: print the list
 makeEmpty: create an empty list
 find: locate the position of an object in a list


list: 34,12, 52, 16, 12
find(52)  3
 insert: insert an object to a list

insert(x,3)  34, 12, 52, x, 16, 12
 remove: delete an element from the list

remove(52)  34, 12, x, 16, 12
 findKth: retrieve the element at a certain position
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Implementation of a List
 Choose a data structure to represent the list ADT

E.g. arrays, records, etc.
 Each operation associated with the list is
implemented by one or more subroutines
 Two standard implementations for the list ADT


Array-based
Linked list
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Lists with Arrays
 Elements are stored in contiguous array
positions
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Array Implementation
 Requires an estimate of the maximum size of the
list

waste space
 printList and find: O(n)
 findKth: O(1)
 insert and delete: O(n)

e.g. insert at position 0 (making a new element)


e.g. delete at position 0


requires first pushing the entire array down one spot to make
room
requires shifting all the elements in the list up one
On average, half of the lists needs to be moved for
either operation
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Lists with Pointers (Linked Lists)
 Ensure that the list is not stored contiguously


use a linked list
a series of structures that are not necessarily adjacent in
memory
 Each node contains the element and a pointer to a
structure containing its successor
the last cell’s next link points to NULL
 Compared to the array implementation,
the pointer implementation uses only as much space as is needed
for the elements currently on the list
but requires space for the pointers in each cell
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Linked Lists
A
B

C
Head
 A linked list is a series of connected nodes
 Each node contains at least


A piece of data (any type)
Pointer to the next node in the list
 Head: pointer to the first node
 The last node points to NULL
node
A
data
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pointer
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Pointer Implementation
 Requires no estimate of the maximum size of the
list

No wasted space
 printList and find: O(n)
 findKth: O(n)
 insert and delete: O(1)

e.g. insert at position 0 (making a new element)


e.g. delete at position 0


Insert does not require moving the other elements
requires no shifting of elements
Insertion and deletion becomes easier, but finding the
Kth element moves from O(1) to O(n)
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The Stack ADT
 The Stack ADT stores
 Auxiliary stack
arbitrary objects
operations:
 Insertions and deletions
 object top(): returns the last
follow the last-in first-out
inserted element without
(LIFO) scheme
removing it
 integer size(): returns the
 Think of a spring-loaded coin
number of elements stored
dispenser
 boolean isEmpty():
 Main stack operations:


push(object): inserts an element
object pop(): removes and
returns the last inserted element
indicates whether no
elements are stored
 Suited to arrays (the top element is the kth one) !!
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The Queue ADT
 The Queue ADT stores arbitrary  Auxiliary queue operations:
objects
 object front(): returns the
element at the front without
 Insertions and deletions follow the
removing it
first-in first-out (FIFO) scheme
 integer size(): returns the
 Insertions are at the rear of the
number of elements stored
queue and removals are at the front
 boolean isEmpty(): indicates
of the queue (think of a line in a
whether no elements are stored
cashier)
 Exceptions
 Main queue operations:
 Attempting the execution of

enqueue(object): inserts an element
dequeue or front on an empty
at the end of the queue
queue throws an

object dequeue(): removes and
EmptyQueueException
returns the element at the front of
the queue
 Suited to pointers (both ends need to be controlled) !!
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Graph ADT
 A graph is a pair (V, E), where
 V is a set of nodes, called vertices
 E is a collection of pairs of vertices, called edges
 Example:
 A vertex represents an airport and stores the three-letter
airport code
 An edge represents a flight route between two airports and
stores the mileage of the route
PVD
ORD
SFO
LGA
HNL
LAX
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DFW
MIA
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Edge Types
 Directed edge (arc)




ordered pair of vertices (u,v)
first vertex u is the origin
second vertex v is the destination
e.g., a flight
ORD
flight
AA 1206
PVD
ORD
849
miles
PVD
 Undirected edge


unordered pair of vertices (u,v)
e.g., a flight route
 Directed graph


all the edges are directed
e.g., route network
 Undirected graph


all the edges are undirected
e.g., flight network
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Operations
 Vertices and edges

are positions

store elements
 Accessor methods

endVertices(e): the two
endvertices of e

opposite(v, e): the vertex
opposite of v on e

areAdjacent(v, w): true iff
v and w are adjacent

replace(v, x): replace
element at vertex v with x

replace(e, x): replace
element at edge e with x
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 Update methods

insertVertex(o): insert a
vertex storing element o

insertEdge(v, w, o): insert
an edge (v,w) storing
element o

removeVertex(v): remove
vertex v (and its incident
edges)

removeEdge(e): remove
edge e
 Iterator methods

incidentEdges(v): edges
incident to v

vertices(): all vertices in
the graph

edges(): all edges in the
graph
GSEIS - LME
Graphs with Arrays
Edge
connectivity
SFO
1 1 1 1
1
1 1 1
1 1 1
1
1 1
1 1
1
1
SFO
ORD
LAX
Vertices
DFW
MIA
PVD
PVD
ORD
LGA
HNL
LAX
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DFW
MIA
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Array Implementation
 Requires an estimate of the maximum size of the
vertices

waste space (besides memory size is quadratic to the
number of vertices)
 areAdjacent: O(1)
 IncidentEdges: O(n)
 insert and removeVertex: O(n)- similar to lists, but
both arrays must be re-arranged
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Graphs with Arrays of Lists
 type lisgraph= array [1, 2, …, nnodes] of record
value: information
neighbors: linked_list
 Node[4]= {value=DFW;
neighbors= {LAXORDLGAMIA}
Obs. record  structure
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Array of Lists Implementation
 Requires also an estimate of the maximum size of the
vertices

waste space
 A graph with few edges favors this scheme
 areAdjacent: O(n)
 IncidentEdges: O(k) (depends on the size of the lists)
 insert and removeVertex: O(n), but only one array
must be re-arranged
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Rooted Tree ADT
 A tree is a collection of nodes
 The collection can be empty
 (recursive definition) If not empty, a tree
consists of a distinguished node r (the root), and
zero or more nonempty subtrees T1, T2, ...., Tk,
each of whose roots are connected by a directed
edge from r
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Terminology
 Root: unique node without a
parent
 Internal node: node with at least
one child (A, B, C, F)
 External node (a.k.a. leaf): node
without children (E, I, J, K, G, H,
D)
 Ancestors of a node: parent,
grandparent, great-grandparent, …
 Descendant of a node: child,
grandchild, great-grandchild, etc.
 Depth of a node: number of
ancestors
 Height of a tree: maximum depth
of any node (3)
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 Subtree: tree consisting of a
node and its descendants
A
B
E
C
F
I
J
G
K
D
H
subtree
GSEIS - LME
Operations
 We use positions to
abstract nodes
 Generic methods:
 integer size()
 boolean isEmpty()
 Iterator elements()
 Iterator positions()
 Accessor methods:
 position root()
 position parent(p)
 positionIterator
children(p)
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 Query methods:
 boolean isInternal(p)
 boolean isExternal(p)
 boolean isRoot(p)
 Update method:
 object replace (p, o)
 Additional methods may
be defined by data
structures implementing
the Tree ADT
GSEIS - LME
Binary Tree ADT
 A binary tree is a set T of nodes such that
either


T is empty, or
T is partitioned into three disjoint subsets:


A single node r, the root
Two possibly empty sets that are binary trees, called left
and right subtrees of r
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Binary Trees - Example
 Operations are similar to the general trees’ ones !!
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Binary Search Trees
 A binary search tree

A binary tree that has the following properties for
each node n



n’s value is greater than all values in its left subtree TL
n’s value is less than all values in its right subtree TR
Both TL and TR are binary search trees
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Binary Search Trees - Example
 Alphabetical ordering !
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Array Implementation - 1
 An array-based representation of a complete tree
 A binary tree is represented by using an array
of tree nodes
 If the binary tree is complete and remains
complete, then, a memory-efficient array-based
implementation can be used
 Requires the creation of a free list which keeps
track of available nodes
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Array Implementation - 2
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Pointer Implementation - 1
 A very simple way to implement a tree is to have
each node store a reference to its left and right
children
 An alternative form is to have each node store a
reference to its parent
 Could also be used to store information about
cities on roads, circuits on a board, etc.
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Pointer Implementation - 2
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Greedy Algorithms
Algorithm is greedy if :
 it builds up a solution in small steps
 it chooses a decision at each step myopically to
optimize some underlying criterion
Analyzing optimal greedy algorithms by showing that:
 in every step it is not worse than any other algorithm,
or
 every algorithm can be gradually transformed to the
greedy one without hurting its quality
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Minimum Spanning Tree Problem
 A minimum spanning tree is a least-cost subset of the
edges of a graph that connects all the nodes
6
4
2
4
1
3
3
2
3
5
2
3
4
2
3
4
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Edge
6-5 (2)
5-3 (2)
4-3 (2)
5-4 (3)
3-2 (3)
3-1 (3)
Connected Components
{1},{2},{3},{4},{5},{6}
{1},{2},{3},{4},{5,6}
{1},{2},{4},{3,5,6}
{1},{2},{3,4,5,6}
rejected
{1},{2,3,4,5,6}
{1,2,3,4,5,6}
GSEIS - LME
Greedy Algorithm – General Model
 Set C of candidates (not used)
 Solution Set S= 
While S  final_solution and C   {
x is a “maximized” element of C; select (x)
C  C-{x}
If (S {x}) is acceptable then SS {x}
}
If S= final_solution then return (S) }
else return (there is no
solution)
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Divide and Conquer
 A divide and conquer algorithm consists of two
parts:


Divide the problem into smaller subproblems of the
same type, and solve these subproblems recursively
Combine the solutions to the subproblems into a
solution to the original problem
 Traditionally, an algorithm is only called
“divide and conquer” if it contains at least two
recursive calls
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D & C- Example and Counter-example
 Quick Sort:
Partition the array into two parts (smaller numbers in one part, larger
numbers in the other part)


Quicksort each of the parts

No additional work is required to combine the two sorted parts
 Binary tree Look-up:

Compare the key to the value in the root
 If the two values are equal, report success
 If the key is less, search the left subtree
 If the key is greater, search the right subtree
 This is not a divide and conquer algorithm because, although there are
two recursive calls, only one is used at each level of the recursion
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Traversing Graphs and Trees- DFS
A
B
D
C
E
H
L
M N O
F
I
G
J
P
K
Q
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 A depth-first search (DFS)
explores a path all the way to a
leaf before backtracking and
exploring another path
 For example, after searching
A, then B, then D, the search
backtracks and tries another
path from B
 Node are explored in the order
ABDEHLMNIOPC
FGJKQ
 N will be found before either I
or J
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How to Depth-First Search
 Start at root_node (any first node if graph)
 If (node  marked) then dfs(node)
// mark is important in graphs
dfs (v)
mark v;
for each node w adjacent to v
if (w  marked) then dfs(w);
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Traversing Graphs and Trees- BFS
A
B
D
C
E
H
L
M N O
F
I
G
J
P
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K
Q
 A breadth-first search (BFS)
explores nodes nearest the root
before exploring nodes further
away
 For example, after searching
A, then B, then C, the search
proceeds with D, E, F, G
 Node are explored in the order
ABCDEFGHIJKL
MNOPQ
 J will be found before N
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How to Breadth-First Search
 Start at root_node (any first node if graph)
 If (node  marked) then bfs(node)
// mark is important in graphs
bfs (v)
ENQUEUE (v, Q) ; // Q is a queue
while (Q  )
u FIRST(Q);
DEQUEUE (u, Q);
for each w adjacent to u
if (w  marked) then
mark w; ENQUEUE (w,Q)
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Exploring Graphs-1
Pre-order traversal (with dfs)
A
B
D
H
C
E
I
F
G
J
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 Node processing first
 Left node (and descendents)
processing
 Other children node (and
descendents)
processing
Sequence of processing:
A, B, D, E H, I, J, C, F, G
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Exploring Graphs-2
Post-order traversal (with dfs)
A
B
D
H
C
E
I
F
G
J
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 Left node (and descendents)
processing first
 Other children node (and
descendents) processing
 Node processing
Sequence of processing:
D, H, I, J, E B, F, G, C, A
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Exploring Graphs-3
In-order traversal (with dfs)
A
B
D
H
C
E
I
F
G
J
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 Left node (and descendents)
processing first
 Node processing
 Other children node (and
descendents) processing
Sequence of processing:
D, B, H, E I, J, A, F, C,G
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Branch and Bound Algorithms
 Branch and bound algorithms are generally used for optimization problems
 As the algorithm progresses, a tree of subproblems is formed
 The original problem is considered the “root problem”
 A method is used to construct an upper and lower bound for a given
problem
 At each node, apply the bounding methods
 If the bounds match, it is deemed a feasible solution to that particular
subproblem
 If bounds do not match, partition the problem represented by that node,
and make the two subproblems into children nodes
 Continue, using the best known feasible solution to trim sections of the
tree, until all nodes have been solved or trimmed
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Branch and Bound Algorithms- Example
 Traveling salesman problem: A salesman has to visit each of n
cities once each, and wants to minimize total cost traveled
1
2
3
4
5
0 14 4 10 20
14 0
7
8
4
0
7 16
5
11 7
9
7
0
2
18 7 17 4
0
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Cost of a direct travel from
one city to another:
Departing from 1 and arriving
in 5 : 20
Departing from 5 and arriving
in 1 : 18
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Example- Computing Partial Costs-1




To specify partial paths
To explore the most promising conditions first
Define the probable minimal cost for arriving
Define the probable minimal cost for departing
4
2
1
7
2
2
3
4
5
4
2
 OBS.
From A to B, half of the value
depends on the arrival at B and
half on the departure froma A
4
2
2
2
2
2
4
2
4
2
2
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5
4
2
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Example- Computing Partial Costs-2
 Suppose the journey starts at city 1
 The initial cost (actually eual for any starting city is 40/2 = 20
 Next step- computing any one of the possible paths:


12: (in this case, we know that (13,4,5), (3,4,52) and (21) are not
possible anymore and we re- compute the nodes costs
Cost= 14 (2) +17 (others) = 31
1
2
4
2
1
N.A.
X
N.A.
2
7
7
7
3
4
5
0 14 X
X
X
0
7
8
4
X
0
7 16
11 X
9
0
2
18 X 17 4
0
2
4
2
2
2
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2
3
4
4
2
5
4
2
2
2
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Example- Computing Partial Costs-3
1
Bound 20
1,2
Bound 31
1,3,2
Bound 24
1,3,2,4
Bound 37
1,3
Bound 24
1,3,4
Bound 30,5
1,3,5
Bound 40,5
1,3,2,5
Bound 31
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1,4
Bound 29
1,4,2
Bound 40
1,5
Bound 41
1,4,3
Bound 41,5
1,4,5,2
Bound 30
1,4,5
Bound 29
1,4,5,3
Bound 48