Download Stacks and Linked Lists

Stacks and Linked Lists
Abstract Data Types (ADTs)
• An ADT is an abstraction of a data structure that
specifies
– Data stored
– Operations on the data
– Error conditions associated with operations
Abstract Data Types (ADTs)
• An ADT is an abstraction of a data structure that
specifies
– Data stored
– Operations on the data
– Error conditions associated with operations
• Example: Registering for classes
– The data stored are the courses in your schedule
– The operations supported are
• Register(course)
• Unregister(course)
• ForceRequest(course)
– Error conditions:
• Registering for multiple classes meeting at the same time
Stacks
Stacks
• Stacks store arbitrary objects
(Pez in this case)
Stacks
• Stacks store arbitrary objects
(Pez in this case)
• Operations
– push(e): inserts an element to the
top of the stack
Stacks
• Stacks store arbitrary objects
(Pez in this case)
• Operations
– push(e): inserts an element to the
top of the stack
Stacks
• Stacks store arbitrary objects
(Pez in this case)
• Operations
– push(e): inserts an element to the
top of the stack
– pop(): removes and returns the top
element of the stack
Stacks
• Stacks store arbitrary objects
(Pez in this case)
• Operations
– push(e): inserts an element to the
top of the stack
– pop(): removes and returns the top
element of the stack
Stacks
• Stacks store arbitrary objects
(Pez in this case)
• Operations
– push(e): inserts an element to the
top of the stack
– pop(): removes and returns the top
element of the stack
Stacks
• Stacks store arbitrary objects
(Pez in this case)
• Operations
– push(e): inserts an element to the
top of the stack
– pop(): removes and returns the top
element of the stack
– top(): returns a reference to the
top element of the stack, but
doesn’t remove it
Stacks
• Stacks store arbitrary objects
(Pez in this case)
• Operations
– push(e): inserts an element to the
top of the stack
– pop(): removes and returns the top
element of the stack
– top(): returns a reference to the
top element of the stack, but
doesn’t remove it
• Optional operations
– size(): returns the number of
elements in the stack
– empty(): returns a bool indicating if
the stack contains any objects
Stack Exceptions
• Attempting to execute an operation of ADT may
cause an error condition called an exception
• Exceptions are said to be “thrown” by an
operation that cannot be executed
• In the Stack ADT, pop and top cannot be
performed if the stack is empty
• Attempting to execute pop or top on an empty
stack throws an EmptyStackException
Exercise: Stacks
• Describe the output and final structure of the stack after
the following operations:
– Push(8)
– Push(3)
– Pop()
– Push(2)
– Push(5)
– Pop()
– Pop()
– Push(9)
– Push(1)
Applications of Stacks
• Direct applications
– Page-visited history in a Web browser
– Undo sequence in a text editor
– Saving local variables when one function calls another, and
this one calls another, and so on.
• Indirect applications
– Auxiliary data structure for algorithms
– Component of other data structures
C++ Run-time Stack
• The C++ run-time system keeps
track of the chain of active
functions with a stack
• When a function is called, the
run-time system pushes on the
stack a frame containing
– Local variables and return value
– Program counter, keeping track of
the statement being executed
• When a function returns, its
frame is popped from the stack
and control is passed to the
method on top of the stack
main() {
int i;
i = 5;
foo(i);
bar
PC = 1
m=6
foo(int j)
{
int k;
k = j+1;
bar(k);
}
foo
PC = 3
j=5
k=6
bar(int m)
{
…
}
main
PC = 2
i=5
}
Array-based Stack
• A simple way of
implementing the
Stack ADT uses an
array
• We add elements
from left to right
• A variable keeps track
of the index of the
top element
Algorithm size()
return t + 1
Algorithm empty()
return size () == 0
Algorithm pop()
if empty() then
throw EmptyStackException
else
t  t  1
return S[t + 1]
…
S
0
1
2
t
Array-based Stack (cont.)
• The array storing the
stack elements may
become full
• A push operation will
then throw a
FullStackException
– Limitation of the
array-based
implementation
– Not intrinsic to the
Stack ADT
Algorithm push(e)
if t = S.length  1 then
throw FullStackException
else
t  t + 1
S[t]  e
…
S
0
1
2
t
Performance and Limitations
(array-based implementation of stack ADT)
• Performance
– Let n be the number of elements in the stack
– The space used is O(n)
– Each operation runs in time O(1)
• Limitations
– The maximum size of the stack must be defined a
priori , and cannot be changed
– Trying to push a new element into a full stack causes
an implementation-specific exception
Growable Array-based Stack
• In a push operation, when
the array is full, instead of
throwing an exception, we
can replace the array with a
larger one
• How large should the new
array be?
– incremental strategy:
increase the size by a
constant c
– doubling strategy: double the
size
Algorithm push(o)
if t = S.length  1
then
A  new array of
size …
for i  0 to t do
A[i]  S[i]
S  A
t  t + 1
S[t]  o
Comparison
• We compare the incremental strategy and the
doubling strategy by analyzing the total time
T(n) needed to perform a series of n push
operations
• Assume that we start with an empty stack
represented by an array of size 1
• We call amortized time of a push operation
the average time taken by a push over the
series of operations, i.e., T(n)/n
Incremental Strategy Analysis
• We replace the array k = n/c times
• The total time T(n) of a series of n push
operations is proportional to
• n + c + 2c + 3c + 4c + … + kc =
• n + c(1 + 2 + 3 + … + k) =
• n + ck(k + 1)/2
• Since c is a constant, T(n) is O(n + k2) = O(n2)
• The amortized time of a push operation is O(n)
Doubling Strategy Analysis
• We replace the array k = log2 n times
• The total time T(n) of a series of n push
operations is proportional to
• n + 1 + 2 + 4 + 8 + …+ 2k =
• n + 2k + 1 1 = 3n 1
• T(n) is O(n)
• The amortized time of a push operation is O(1)
Stack Interface in C++
• Requires the
definition of class
EmptyStackException
• Most similar STL
construct is vector
template <class Type>
class Stack {
public:
int size();
bool isEmpty();
Type& top()
throw(EmptyStackException);
void push(Type e);
Type pop()
throw(EmptyStackException);
};
Array-based Stack in C++
template <class Type>
class ArrayStack
{
private:
int capacity; // stack capacity
Type *S;
// stack array
int t;
// top of stack
public:
ArrayStack(int c) : capacity(c) {
S = new Type [ capacity ];
t = -1;
}
bool isEmpty() { return t < 0; }
Type pop() throw(EmptyStackException) {
if ( isEmpty ( ) )
throw EmptyStackException(“Popping from empty stack”);
return S [ t-- ];
}
//… (other functions omitted)
Singly Linked List
• A singly linked list is a
structure consisting of a
sequence of nodes
• A singly linked list stores
a pointer to the first node
(head) and last (tail)
• Each node stores
head
next
node
elem
– element
– link to the next node
tail

Leonard
Sheldon
Howard
Raj
Singly Linked List Node in C++
template <class Type>
class SLinkedListNode {
public:
Type elem;
next
SLinkedListNode<Type> *next;
node
elem
};

Leonard
Sheldon
Howard
Raj
Singly Linked List
• A singly linked list is a structure consisting of a
sequence of nodes
• Operations
– insertFront(e): inserts an element on the front of
the list
– removeFront(): returns and removes the element
at the front of the list
– insertBack(e): inserts an element on the back of
the list
– removeBack(): returns and removes the element
at the end of the list
Inserting at the Front
1. Allocate a new node
2. Have new node point to old head
3. Update head to point to new node
head
tail

Leonard
Sheldon
Howard
Raj
Inserting at the Front
1. Allocate a new node
2. Have new node point to old head
3. Update head to point to new node
head
tail

Penny

Leonard
Sheldon
Howard
Raj
Inserting at the Front
1. Allocate a new node
2. Have new node point to old head
3. Update head to point to new node
head
tail

Penny
Leonard
Sheldon
Howard
Raj
Inserting at the Front
1. Allocate a new node
2. Have new node point to old head
3. Update head to point to new node
tail
head

Penny
Leonard
Sheldon
Howard
Raj
Inserting at the Front
1. Allocate a new node
2. Have new node point to old head
3. Update head to point to new node
head

tail

Inserting at the Front
1. Allocate a new node
2. Have new node point to old head
3. Update head to point to new node
head

tail


Raj
Inserting at the Front
1.
2.
3.
4.
Allocate a new node
Have new node point to old head
Update head to point to new node
If tail is NULL, update tail to point to the head
head
tail
node

Raj
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
head
tail

Leonard
Sheldon
Howard
Raj
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
head
tail

Leonard
Sheldon
Howard
Raj
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
head
tail

Leonard
Sheldon
Howard
Raj
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
head
tail

Leonard
Sheldon
Howard
Raj
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
head
tail

Sheldon
Howard
Raj
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
head
tail

Sheldon
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
head

tail

Sheldon
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
head

tail

Sheldon
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
head

tail

Sheldon
Removing at the Front
1. Update head to point to next node in the list
2. Return elem of previous head and delete the
node
3. If head is NULL, update tail to NULL
head

tail

Inserting at the Back
1. Allocate a new node
2. If tail is NULL, update head and tail to point to
the new node; otherwise
1. Have the old tail point to the new node
2. Update tail to point to new node
head
tail

Leonard
Sheldon
Howard
Inserting at the Back
1. Allocate a new node
2. If tail is NULL, update head and tail to point to
the new node; otherwise
1. Have the old tail point to the new node
2. Update tail to point to new node
head
tail


Leonard
Sheldon
Howard
Raj
Inserting at the Back
1. Allocate a new node
2. If tail is NULL, update head and tail to point to
the new node; otherwise
1. Have the old tail point to the new node
2. Update tail to point to new node
head
tail

Leonard
Sheldon
Howard
Raj
Inserting at the Back
1. Allocate a new node
2. If tail is NULL, update head and tail to point to
the new node; otherwise
1. Have the old tail point to the new node
2. Update tail to point to new node
head
tail

Leonard
Sheldon
Howard
Raj
Removing at the Back
•
•
No efficient way of doing so (O(n))
Typically would not use a singly linked-list if this
operation is commonly used
head
tail

Leonard
Sheldon
Howard
Raj
Stack with a Singly Linked List
• We can implement a stack with a singly linked list
• The top element of the stack is the first node of the list
• The space used is O(n) and each operation of the Stack
ADT takes O(1) time
nodes

top t
elements
Stack Summary
• Stack Operation Complexity for Different
Array
Array
Singly
Implementations
Fixed-Size Expandable (doubling
strategy)
Linked
List
Pop()
O(1)
O(1)
O(1)
Push(o)
O(1)
O(n) Worst Case
O(1) Best Case
O(1) Average Case
O(1)
Top()
O(1)
O(1)
O(1)
Size(), isEmpty()
O(1)
O(1)
O(1)
Queues
Queues
• Queues store arbitrary objects
• Insertions are at the end of the
queue and removals are at the
front of the queue
• Main queue operations:
• Auxiliary queue operations:
– enqueue(e): inserts an element
at the end of the queue
– dequeue(): removes and returns
the element at the front of the
•
queue
– front(): returns the element at
the front without removing it
– size(): returns the number of
elements stored
– isEmpty(): returns a boolean
value indicating if there are no
elements in the queue
Exceptions
– Attempting to execute dequeue
or front on an empty queue
throws an EmptyQueueException
Exercise: Queues
• Describe the output and final structure of the queue
after the following operations:
– enqueue(8)
– enqueue(3)
– dequeue()
– enqueue(2)
– enqueue(5)
– dequeue()
– dequeue()
– enqueue(9)
– enqueue(1)
Applications of Queues
• Direct applications
– Waiting lines
– Access to shared resources (e.g., printer)
– User input in a game
• Indirect applications
– Auxiliary data structure for algorithms
– Component of other data structures
Array-based Queue
• Use an array of size N in a circular fashion
• Two variables keep track of the front and rear
– f index of the front element
– r index immediately past the rear element
• Array location r is kept empty
normal configuration
Q
0 1 2
f
r
wrapped-around configuration
Q
0 1 2
r
f
Queue Operations
• We use the modulo
operator
(remainder of
division)
Algorithm size()
return (N - f + r) mod N
Algorithm isEmpty()
return (f = r)
Q
0 1 2
f
0 1 2
r
r
Q
f
Queue Operations (cont.)
• Operation enqueue throws an
exception if the array is full
• This exception is
implementation-dependent
Algorithm enqueue(o)
if size() = N  1 then
throw FullQueueException
else
Q[r]  o
r  (r + 1) mod N
Q
0 1 2
f
0 1 2
r
r
Q
f
Queue Operations (cont.)
• Operation dequeue
throws an exception
if the queue is
empty
• This exception is
specified in the
queue ADT
Algorithm dequeue()
if isEmpty() then
throw EmptyQueueException
else
o  Q[f]
f  (f + 1) mod N
return o
Q
0 1 2
f
0 1 2
r
r
Q
f
Performance and Limitations
- array-based implementation of queue ADT
• Performance
– Let n be the number of elements in the queue
– The space used is O(n)
– Each operation runs in time O(1)
• Limitations
– The maximum size of the queue must be defined a
priori , and cannot be changed
– Trying to enqueue a new element into a full queue
causes an implementation-specific exception
Growable Array-based Queue
• In an enqueue operation, when the array is full,
instead of throwing an exception, we can
replace the array with a larger one
• Similar to what we did for an array-based stack
• The enqueue operation has amortized running
time
– O(n) with the incremental strategy
– O(1) with the doubling strategy
Exercise
• Describe how to implement a queue using a
singly-linked list
– Queue operations: enqueue(x), dequeue(),
size(), isEmpty()
– For each operation, give the running time
Queue with a Singly Linked List
•
We can implement a queue with a singly linked list
– The front element is stored at the head of the list
– The rear element is stored at the tail of the list
•
•
The space used is O(n) and each operation of the Queue ADT takes O(1)
time
NOTE: we do not have the limitation of the array based implementation
on the size of the stack b/c the size of the linked list is not fixed, I.e., the
queue is NEVER full.
head
tail

Leonard
Sheldon
Howard
Raj
Informal C++ Queue Interface
• Informal C++
interface for our
Queue ADT
• Requires the
definition of class
EmptyQueueException
• No corresponding
built-in STL class
template <class Type>
class Queue {
public:
int size();
bool isEmpty();
Type& front()
throw(EmptyQueueException);
void enqueue(Type e);
Type dequeue()
throw(EmptyQueueException);
};
Queue Summary
• Queue Operation Complexity for Different
Implementations
Array
Array
Fixed-Size Expandable (doubling
strategy)
List
SinglyLinked
dequeue()
O(1)
O(1)
O(1)
enqueue(o)
O(1)
O(n) Worst Case
O(1) Best Case
O(1) Average Case
O(1)
front()
O(1)
O(1)
O(1)
Size(), isEmpty()
O(1)
O(1)
O(1)
Double-Ended Queues
•
•
•
The Double-Ended Queue, or
•
Deque, ADT stores arbitrary
objects. (Pronounced ‘deck’)
Richer than stack or queue ADTs.
Supports insertions and deletions
at both the front and the end.
Main deque operations:
– insertFirst(object o): inserts
element o at the beginning of the
deque
– insertLast(object o): inserts
element o at the end of the deque
•
– removeFirst(): removes and
returns the element at the front
of the deque
– removeLast(): removes and
returns the element at the end of
the deque
Auxiliary deque operations:
– first(): returns the element at the
front without removing it
– last(): returns the element at the
front without removing it
– size(): returns the number of
elements stored
– isEmpty(): returns a Boolean value
indicating whether no elements are
stored
Exceptions
– Attempting to execute
removeFirst,removeLast, front, or
last on an empty deque throws an
EmptyDequeException
Doubly Linked List
• A doubly linked list is a structure
consisting of a sequence of
nodes
• A doubly linked list stores a
pointer to a special head/tail
node
• Each node stores
next
prev
elem
node
– element
– link to the prev, next node
tail
head
Leonard
Sheldon
Howard
Raj
Doubly Linked List
• A doubly linked list is a structure
consisting of a sequence of
nodes
• A doubly linked list stores a
pointer to a special head/tail
node
• Each node stores
– element
– link to the prev, next node
head
tail
next
prev
elem
node
Doubly Linked List Node in C++
template <class Type>
class DLinkedListNode {
public:
Type elem;
next
prev
DLinkedListNode<Type> *prev, *next;
elem
};
node
tail
head
Leonard
Sheldon
Howard
Raj
Doubly Linked List
• A doubly linked list is a structure consisting of a
sequence of nodes
• Operations
– insertFront(e): inserts an element on the front of the list
– removeFront(): returns and removes the element at the
front of the list
– insertBack(e): inserts an element on the back of the list
– removeBack(): returns and removes the element at the end
of the list
• Private operations
– add(n, e): inserts the element after the node n
– remove(n): returns and removes the element stored in the
node n
Adding a Node
1. Allocate a new node
2. Have new node point to the previous and next
nodes
3. Update the previous and next nodes to point to
the new node
tail
head
Leonard
Sheldon
Howard
Raj
Adding a Node
1. Allocate a new node
2. Have new node point to the previous and next
nodes
3. Update the previous and next nodes to point to
the new node
Bernadette
tail
head
Leonard
Sheldon
Howard
Raj
Adding a Node
1. Allocate a new node
2. Have new node point to the previous and next
nodes
3. Update the previous and next nodes to point to
the new node
Bernadette
tail
head
Leonard
Sheldon
Howard
Raj
Adding a Node
1. Allocate a new node
2. Have new node point to the previous and next
nodes
3. Update the previous and next nodes to point to
the new node
Bernadette
tail
head
Leonard
Sheldon
Howard
Raj
Adding a Node
1. Allocate a new node
2. Have new node point to the previous and next
nodes
3. Update the previous and next nodes to point to
the new node
tail
head
Leonard
Sheldon
Howard
Bernadette
Raj
Adding a Node
1. Allocate a new node
2. Have new node point to the previous and next
nodes
3. Update the previous and next nodes to point to
the new node
Sheldon
head
tail
Adding a Node
1. Allocate a new node
2. Have new node point to the previous and next
nodes
3. Update the previous and next nodes to point to
the new node
Sheldon
head
tail
Adding a Node
1. Allocate a new node
2. Have new node point to the previous and next
nodes
3. Update the previous and next nodes to point to
the new node
Sheldon
head
tail
Adding a Node
1. Allocate a new node
2. Have new node point to the previous and next
nodes
3. Update the previous and next nodes to point to
the new node
head
tail
Sheldon
Removing a Node
1. Have the prev node’s next point to the next of
the current node
2. Have the next node’s prev point to the prev of
the current node
3. Delete the current node
tail
head
Leonard
Sheldon
Howard
Raj
Removing a Node
1. Have the prev node’s next point to the next of
the current node
2. Have the next node’s prev point to the prev of
the current node
3. Delete the current node
tail
head
Leonard
Sheldon
Howard
Raj
Removing a Node
1. Have the prev node’s next point to the next of
the current node
2. Have the next node’s prev point to the prev of
the current node
3. Delete the current node
tail
head
Leonard
Sheldon
Howard
Raj
Removing a Node
1. Have the prev node’s next point to the next of
the current node
2. Have the next node’s prev point to the prev of
the current node
3. Delete the current node
tail
head
Leonard
Sheldon
Howard
Raj
Removing a Node
1. Have the prev node’s next point to the next of
the current node
2. Have the next node’s prev point to the prev of
the current node
3. Delete the current node
tail
head
Leonard
Sheldon
Raj
Deque with a Doubly Linked List
• We can implement a deque with a doubly linked list
– The front element is pointed to by head
– The rear element is pointed to by tail
• The space used is O(n) and each operation of the
Deque ADT takes O(1) time
tail
head
Leonard
Sheldon
Howard
Raj
Performance and Limitations
- doubly linked list implementation of deque ADT
• Performance
– Let n be the number of elements in the deque
– The space used is O(n)
– Each operation runs in time O(1)
• Limitations
– NOTE: we do not have the limitation of the array
based implementation on the size of the deque b/c
the size of the linked list is not fixed, I.e., the deque is
NEVER full.
Deque Summary
• Deque Operation Complexity for Different
Implementations
Array
FixedSize
Array
Expandable
(doubling strategy)
List
SinglyLinked
List
DoublyLinked
removeFirst(),
removeLast()
O(1)
O(1)
O(1) –
removeFirst,
O(n) –
removeLast
O(1)
insertFirst(o),
InsertLast(o)
O(1)
O(n) Worst Case
O(1) Best Case
O(1) Average Case
O(1)
O(1)
first(), last
O(1)
O(1)
O(1)
O(1)
size(),
isEmpty()
O(1)
O(1)
O(1)
O(1)