Stack and Queue

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Implementing and Using Stacks

many slides taken from Mike Scott, UT Austin

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Stacks Stacks are a straight forward and simple data

structure Access is allowed only at one point of the

structure, normally termed the top of the stack, so the operations are limited:– push (add item to stack)– pop (remove top item from stack)– top (get top item without removing it)– clear– isEmpty– size

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Stack OperationsAssume a simple stack for integers.

Stack s = new Stack();

s.push(12);

s.push(4);

s.push( s.top() + 2 );

s.pop()

s.push( s.top() );

//what are contents of stack?

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Stack OperationsWrite a method to print out contents of stack in reverse order.

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Common Stack ErrorStack s = new Stack();// put stuff in stackfor(int i = 0; i < 7; i++)

s.push( i );// print out contents of stack // while emptying itfor(int i = 0; i < s.size(); i++)

System.out.println( s.pop() );// Output? Why?// Crossover error

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Corrected VersionStack s = new Stack();// put stuff in stackfor(int i = 0; i < 7; i++)

s.push( i );// print out contents of stack // while emptying itint limit = s.size();for(int i = 0; i < limit; i++)

System.out.println( s.pop() );

//or

// while( !s.isEmpty() )

// System.out.println( s.pop() );

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Balanced Symbol Checking In processing programs and working with

computer languages there are many instances when symbols must be balanced{ } , [ ] , ( )

A stack is useful for checking symbol balance. When a closing symbol is found it must match the most recent opening symbol of the same type.

Algorithm?

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Algorithm for Balanced Symbol Checking

Make an empty stack read symbols until end of file

– if the symbol is an opening symbol push it onto the stack

– if it is a closing symbol do the following• if the stack is empty report an error• otherwise pop the stack. If the symbol popped does

not match the closing symbol report an error

At the end of the file if the stack is not empty report an error

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Algorithm in practice list[i] = 3 * ( 44 - method( foo( list[ 2 * (i + 1) +

foo( list[i - 1] ) ) / 2 *) - list[ method(list[0])];

Processing a file– Tokenization: the process of scanning an input

stream. Each independent chunk is a token.

Tokens may be made up of 1 or more characters

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Mathematical CalculationsWhat is 3 + 2 * 4? 2 * 4 + 3? 3 * 2 + 4?

The precedence of operators affects the order of operations. A mathematical expression cannot simply be evaluated left to right. A challenge when evaluating a program.Lexical analysis is the process of interpreting a program. Involves Tokenization

What about 1 - 2 - 4 ^ 5 * 3 * 6 / 7 ^ 2 ^ 2

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Infix and Postfix Expressions The way we are used to writing expressions

is known as infix notation Postfix expression does not require any

precedence rules 3 2 * 1 + is postfix of 3 * 2 + 1 evaluate the following postfix expressions

and write out a corresponding infix expression:2 3 2 4 * + * 1 2 3 4 ^ * +

1 2 - 3 2 ^ 3 * 6 / + 2 5 ^ 1 -

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Evaluation of Postfix Expressions Easy to do with a stack given a proper postfix expression:

– get the next token– if it is an operand push it onto the stack– else if it is an operator

• pop the stack for the right hand operand• pop the stack for the left hand operand• apply the operator to the two operands• push the result onto the stack

– when the expression has been exhausted the result is the top (and only element) of the stack

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Infix to Postfix Convert the following equations from infix to

postfix:2 ^ 3 ^ 3 + 5 * 1

2 3 3 ^ ^ 5 1 * +

11 + 2 - 1 * 3 / 3 + 2 ^ 2 / 3

11 2 + 1 3 * 3 / - 2 2 ^ 3 / +

Problems:

parentheses in expression

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Infix to Postfix Conversion Requires operator precedence parsing algorithm

– parse v. To determine the syntactic structure of a sentence or other utterance

Operands: add to expression

Close parenthesis: pop stack symbols until an open parenthesis appears

Operators:

Pop all stack symbols until a symbol of lower precedence appears. Then push the operator

End of input: Pop all remaining stack symbols and add to the expression

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Simple ExampleInfix Expression: 3 + 2 * 4

PostFix Expression:

Operator Stack:

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Simple ExampleInfix Expression: + 2 * 4

PostFix Expression: 3

Operator Stack:

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Simple ExampleInfix Expression: 2 * 4

PostFix Expression: 3

Operator Stack: +

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Simple ExampleInfix Expression: * 4

PostFix Expression: 3 2

Operator Stack: +

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Simple ExampleInfix Expression: 4

PostFix Expression: 3 2

Operator Stack: + *

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Simple ExampleInfix Expression:

PostFix Expression: 3 2 4

Operator Stack: + *

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Simple ExampleInfix Expression:

PostFix Expression: 3 2 4 *

Operator Stack: +

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Simple ExampleInfix Expression:

PostFix Expression: 3 2 4 * +

Operator Stack:

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Example1 - 2 ^ 3 ^ 3 - ( 4 + 5 * 6 ) * 7

Show algorithm in action on above equation

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Applications of Stacks Direct applications

– Page-visited history in a Web browser– Undo sequence in a text editor– Chain of method calls in the Java Virtual

Machine– Validate XML

Indirect applications– Auxiliary data structure for algorithms– Component of other data structures

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Method Stack in the JVM

The Java Virtual Machine (JVM) keeps track of the chain of active methods with a stack

When a method is called, the JVM pushes on the stack a frame containing– Local variables and return value– Program counter, keeping track of

the statement being executed When a method ends, its frame

is popped from the stack and control is passed to the method on top of the stack

Allows for recursion

main() {int i = 5;foo(i);}

foo(int j) {int k;k = j+1;bar(k);}

bar(int m) {…}

bar PC = 1 m = 6

foo PC = 3 j = 5 k = 6

main PC = 2 i = 5

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Implementing a stack need an underlying collection to hold the elements

of the stack 2 basic choices

– array (native or ArrayList)– linked list

array implementation

linked list implementation

Some of the uses for a stack are much more interesting than the implementation of a stack

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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

S0 1 2 t

…

Algorithm size()return t + 1

Algorithm pop()if isEmpty() then

throw EmptyStackException else

t t 1return S[t + 1]

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Queues

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Queues Closely related to Stacks Like a line

– In Britain people don’t “get in line” they “queue up”.

Queues are a first in first out data structure– FIFO (or LILO, but that sounds a bit silly)

Add items to the end of the queue Access and remove from the front Used extensively in operating systems

– Queues of processes, I/O requests, and much more

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Queue operations add(Object item)

– a.k.a. enqueue(Object item)

Object get()– a.k.a. Object front()

Object remove()– a.k.a. Object dequeue()

boolean isEmpty() Specify in an interface, allow varied

implementations

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Queue Example

Operation Output Q enqueue(5) – (5)enqueue(3) – (5, 3)dequeue() 5 (3)enqueue(7) – (3, 7)dequeue() 3 (7)front() 7 (7)dequeue() 7 ()dequeue() “error” ()isEmpty() true ()enqueue(9) – (9)enqueue(7) – (9, 7)size() 2 (9, 7)enqueue(3) – (9, 7, 3)enqueue(5) – (9, 7, 3, 5)dequeue() 9 (7, 3, 5)

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Applications of Queues Direct applications

– Waiting lists, bureaucracy– Access to shared resources (e.g., printer)– Multiprogramming

Indirect applications– Auxiliary data structure for algorithms– Component of other data structures

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Implementation 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 elementr index immediately past the rear element

Array location r is kept empty

Q0 1 2 rf

normal configuration

Q0 1 2 fr

wrapped-around configuration

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Queue Operations

We use the modulo operator (remainder of division)

Algorithm size()return (N f + r) mod N

Algorithm isEmpty()return (f r)

Q0 1 2 rf

Q0 1 2 fr

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Queue Operations (cont.)

Algorithm enqueue(o)if size() = N 1 then

throw FullQueueException else

Q[r] or (r + 1) mod N

Operation enqueue throws an exception if the array is full

This exception is implementation-dependent

Q0 1 2 rf

Q0 1 2 fr

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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 Nreturn o

Q0 1 2 rf

Q0 1 2 fr

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Queue Interface in Java

Java interface corresponding to our Queue ADT

Requires the definition of class EmptyQueueException

No corresponding built-in Java class, however there is PriorityQueue

public interface Queue {

public int size();

public boolean isEmpty();

public Object front()throws EmptyQueueException;

public void enqueue(Object o);

public Object dequeue() throws EmptyQueueException;

}

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Application: Round Robin Schedulers

We can implement a round robin scheduler using a queue, Q, by repeatedly performing the following steps:

1. e = Q.dequeue()

2. Service element e

3. Q.enqueue(e)

The Queue

Shared Service

1. Deque the next element

3. Enqueue the serviced element

2 . Service the next element

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Implementing a Queue Given the internal storage container and

choice for front and back of queue what are the Big O of the queue operations?

ArrayList LinkedList LinkedList (Singly Linked) (Doubly

Linked)

enqueue

front

dequeue

isEmpty

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Priority Queue ImplementationSequence-based Priority Queue

Implementation with an unsorted list

Performance:– insert takes O(1) time

since we can insert the item at the beginning or end of the sequence

– removeMin and min take O(n) time since we have to traverse the entire sequence to find the smallest key

Implementation with a sorted list

Performance:– insert takes O(n) time

since we have to find the place where to insert the item

– removeMin and min take O(1) time, since the smallest key is at the beginning

4 5 2 3 1 1 2 3 4 5

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Priority Queue Applications Priority-based OS process scheduler