Implementing Stacks & Queues

IKI 10100: Data Structures & Algorithms

Ruli Manurung(acknowledgments to Denny & Ade Azurat)

Fasilkom UI

Ruli Manurung (Fasilkom UI) IKI10100: Lecture 6th Mar 2007

2Ruli Manurung (Fasilkom UI) IKI10100: Lecture 6th Mar 2007

ADT Stacks Basic operations Examples of use Implementations

• Array-based and linked list-basedADT Queues Basic operations Examples of use Implementations

• Array-based and linked list-basedStack Applications Balanced Symbol Checker Postfix Machines

Summary

Outline

Linear Data Structures

A collection of components that are arranged along one dimension, i.e in a straight line, or linearly.

Stack: a linear data structure where access is restricted to the most recently inserted item.

Queue: a linear data structure where access is restricted to the least recently inserted item.

Both of these abstract data types can be implemented at a lower level using a list: either an array or a linked list.

The last item added is pushed (added) to the stack.

The last item added can be popped (removed) from the stack.

The last item added can be topped (accessed) from the stack.

These operations all take constant time: O(1).

A typical stack interface:

void push(Thing newThing);

void pop();

Thing top();

Stack Implementation: Array

A stack can be implemented as an array A and an integer top that records the index of the top of the stack.

For an empty stack, set top to -1.

When push(X) is called, increment top, and write X to A[top].

When pop() is called, decrement top.

When top() is called, return A[top].

Example

top(-1)

A top(0) A

B top(1)

Push A Push B

MyStack myStack = new MyStack();

myStack.push(A);

myStack.push(B);

Array Doubling

When array-based stack is constructed, instantiate an array with a “default” size.

When the array underlying the stack is full (not the stack itself!), we can increase the array through array doubling.

Allocate a new array twice the size, and copy the old array to the first half of the new array:

Thing[] newA = new Thing[oldA.length*2];

for(int ii=0; ii<oldA.length; ii++) newA[ii] = oldA[ii];

oldA = newA;

Running Time

Without array doubling, all stack operations take constant time – O(1).

With array doubling, push() may be O(N), but this happens quite rarely: array doubling due to data size N must be preceded by N/2 push() non-doubling calls. Effectively, still constant time Amortization.

Stack Implementation: Array

public class MyArrayStack<T>{ private T[] array; private int topOfStack; private static final int DEFAULT_CAPACITY = 10;

public MyArrayStack() … public boolean isEmpty() … public void makeEmpty() … public T top() … public void pop() … public T topAndPop() … public void push(T x) …

private void doubleArray() …}

Stack Implementation: Linked List

First item in list = top of stack (if empty: null)push(Thing x): Create a new node containing x Insert it as the first elementpop(): Delete first item (i.e. move “top” to the second

item)top(): Return the data of the first element

d c b a

topOfStack

Stack Implementation: Linked List

public class MyLinkedListStack<T>{ private ListNode<T> topOfStack;

public MyLinkedListStack() … public boolean isEmpty() … public void makeEmpty() … public T top() … public void pop() … public T topAndPop() … public void push(T x) …}

Last item added is enqueued (added) to the back.First item added is dequeued (removed) from the front.First item added can be accessed: getFront.These operations all take constant time – O(1).

A typical queue interface:void enqueue(Thing newThing);void dequeue();Thing getFront();

Queue Implementation: Simple Idea

Store items in an array A

Maintain index: back

Front of queue = A[0]

Back of queue = A[back]

Enqueue is easy & fast: store at A[back], back++

Dequeue is inefficient: A[1] to A[back] needs to be shifted (and back--) O(N)

enqueue(X) enqueue(Y) enqueue(Z) dequeue()

Queue Implementation: Better Idea

Add another index: front, which records the front of the queue

Dequeue is now done by incrementing front

Both enqueue and dequeue are now O(1).

But what happens if enqueue and dequeue array.length-1 items?

enqueue(X)enqueue(Y)enqueue(Z) dequeue()

front front

dequeue()

Queue Implementation: “Circular” Array

After the array.length-1-th item is enqueued, the underlying array is full, even though the queue is not logically, it should be (almost?) empty.

Solution: wraparound

Re-use cells at beginning of array that are ‘empty’ due to dequeue.

When either front or back is incremented and points “outside array” (≥array.length), reset to 0.

Circular Example

Both front and back indexes “wraparound” the array. Think of the array as a circle…

front back

P Q R S

frontback

P Q R S

front back

Java Implementation

Fairly straightforward. Basically, maintain Front Back Number of items in queue

When is the underlying array really full?

How do we do array doubling?

Queue Implementation: Array

public class MyArrayQueue<T>{ private T[] array; private int front,back,currentSize; private static final int DEFAULT_CAPACITY = 10;

public MyArrayQueue() … public boolean isEmpty() … public void makeEmpty() … public T getFront() … public void dequeue() … public T getFrontAndDequeue() … public void enqueue(T x) …

private void doubleQueue() … private int increment(int x) …}

Queue Implementation: Linked List

Maintain 2 node references: front & back

An empty queue: front = back = null.

enqueue(Thing X): Create a new node N containing X If queue empty: front = back = N Else append N and update back

dequeue(): Delete first item (referenced by front)

getFront(): Return data of first element

a b c d

front back

Queue Implementation: Linked List

public class MyLinkedListQueue<T>{ private ListNode<T> front; private ListNode<T> back;

public MyLinkedListQueue() … public boolean isEmpty() … public void makeEmpty() … public T getFront() … public void dequeue() … public T getFrontAndDequeue() … public void enqueue(T x) …}

Stack Application: Balanced Symbols

Java uses pairs of symbols to denote blocks: ( and ) { and } [ and ]

An opening symbol must be matched with a closing symbol: { [ ( ) ] } is OK { ( [ ) ] } is NOT OK

A closing symbol, e.g. ‘)’, must match the most recently seen opening symbol, e.g. ‘(‘.

This can be done using a stack.

Balanced Symbol Checker Algorithm

Make an empty stack

Read symbols: If symbol is an opening symbol, push onto stack. If symbol is a closing symbol:

• If stack is empty, report error!

• Else, (top and) pop stack. If popped symbol is not a matching opening symbol, report error!

At the end, if the stack is not empty, report error!

Say we are given an arithmetic expression, and we are asked to evaluate it.Typically, we use infix expressions, of the form operand1 OPERATOR operand2Example: 1 + 2 4 * 3

To process infix expressions we need precedence and associativity: 1 + 2 * 3 10 - 4 - 3 2 ^ 3 ^ 3 1 - 2 - 4 ^ 5 * 3 * 6 / 7 ^ 2 ^ 2

We use brackets: (1-2)-((((4^5)*3)*6)/(7^(2^2)))

Stack Application: Simple Calculator

Postfix Notation

Typically, we use postfix expressions, of the form operand1 operand2 OPERATOR

Example: 1 2 + 4 3 *

With postfix expressions, precedence is unambiguous: 1 - 2 - 4 ^ 5 * 3 * 6 / 7 ^ 2 ^ 2, or (1-2)-((((4^5)*3)*6)/(7^(2^2))), becomes 1 2 - 4 5 ^ 3 * 6 * 7 2 2 ^ ^ / -

Postfix Machines

A postfix machine evaluates a postfix expression: If an operand is seen, push it onto stack If an operator is seen, pop appropriate number of

operands, evaluate operator, push result onto stack

When the complete postfix expression is evaluated, the stack should contain exactly one item: the result.

Both versions, array and linked-list, run in O(1)

Linked-list implementation requires extra overhead due to next reference at each node

(Circular) array implementation of queues can be quite tricky

Array space doubling needs memory at least 3x size of actual data.

Summary

Implementing Stacks & Queues

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