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Learning Objectives:To introduce the stack data structure.To be able to implement the most common operations

on stacks.Apply stacks to solve common problems

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Outlines1. INTRODUCTION2. STL3. ERROR PROCESSING4. SPECIFICATION FOR METHODS5. CLASS STACK6. REVERSING A LIST7. REVERSING A WORD.8. RPN CALCULATOR9. BRACKET MATCHING10. EVALUATING ARITHMETIC EXPRESSIONS

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1. INTRODUCTIONA stack is a data structure in which all

insertions and deletions of entries are made at one end, called the top of the stack. The last entry which is inserted is the first one that will be removed.

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1. INTRODUCTION

Push box Q onto empty stackPush box A onto stackPop a box from stackPop a box from stack

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2. STL The standard template library (STL) in C++ contains much

useful information, many functions, and many classes. The STL provides convenient implementations for many

common data structures, including almost all the data structures we shall study.

We can include the STL stack implementation into our programs with the directive:

#include <stack>

and then we can utilize initially empty stack objects and apply methods called push, pop, top, and empty.

In C++, a template construction allows us to create data structures whose entries have different types. Example:

stack<double> numbers

stack<int> numbers

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2. STLThe STL provides several

implementations of various data structures such as stacks.

The code in a client program should not depend on a particular choice of implementation, but the performance of the final program may very much depend on the choice of implementation.

A second, optional template parameter selects the implementation.

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3. ERROR PROCESSING We must recognize that a method might be applied

illegally by a client, e.g. a client might try to pop an empty stack.

We shall use a single enumerated type called Error_ code to report errors from all of our programs and functions.

The enumerated type Error code is part of the utility package

Stacks use three values of an Error code:successoverflowunderflow

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4. SPECIFICATION FOR METHODS Error_code Stack ::pop( );

Pre: None.Post: If the Stack is not empty, the top of

the Stack is removed. If the Stack is empty, an Error code of underflow is returned and the Stack is left unchanged.

Error_code Stack ::push(const Stack_entry &item);Pre: None.Post: If the Stack is not full, item is added

to the top of the Stack. If the Stack is full, an Error code of overflow is returned and the Stack is left unchanged.

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4. SPECIFICATION FOR METHODS Error_code Stack ::top(Stack_entry &item) const;

Pre: None.Post: The top of a nonempty Stack is

copied to item. A code of fail is returned if the Stack is empty.

bool Stack ::empty( ) const;Pre: None.Post: A result of true is returned if the

Stack is empty, otherwise false is returned.

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5. Class Stack We set up an array to hold the entries in the stack and a counter to indicate how

many entries there are.typedef int Stack_entry;const int maxstack = 10; //small value for testingclass Stack {public:

Stack( );bool empty( ) const;Error code pop( );Error code top(Stack_entry &item) const;Error code push(const Stack_entry &item);

private:int count;Stack_entry entry[maxstack];

}; The declaration of private visibility for the data makes it is impossible for a client to

access the data stored in a Stack except by using the methods push( ), pop( ), and top( ).

Important result: A Stack can never contain illegal or corrupted data. In general, data is said to be encapsulated if it can only be accessed by a controlled set of functions.

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5. CLASS STACK Empty stack

Push the first entry (item)

N items on the stack

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5. CLASS STACK Error_code Stack ::push(const Stack_entry &item)/* Pre: None.

Post: If the Stack is not full, item is added to the top of the Stack . If the Stack is full, an Error_code of overflow is returned and the Stack is left unchanged.*/

{Error_code outcome = success;if (count >= maxstack)

outcome = overflow;else

entry[count++] = item;return outcome;}

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5. CLASS STACKError_code Stack ::pop( )

/* Pre: None.

Post: If the Stack is not empty, the top of the Stack is removed. If the Stack is empty, an Error_code of underflow is returned. */

{

Error_code outcome = success;

if (count == 0)

outcome = underflow;

else

--count;

return outcome;

}

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5. CLASS STACKError_code Stack ::top(Stack_entry &item) const

/* Pre: None.

Post: If the Stack is not empty, the top of the Stack is returned in item . If the Stack is empty an Error_code of underflow is returned. */

{

Error_code outcome = success;

if (count == 0)

outcome = underflow;

else

item = entry[count . 1];

return outcome;

}

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5. CLASS STACKbool Stack ::empty( ) const

/* Pre: None.

Post: If the Stack is empty, true is returned. Otherwise false is returned. */

{

bool outcome = true;

if (count > 0) outcome = false;

return outcome;

}

Stack ::Stack( )

/* Pre: None.

Post: The stack is initialized to be empty. */

{

count = 0;

}

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6. REVERSING A LIST#include <stack>int main(){

int n; double item; stack<double> numbers;cout<<“Type an integer n: ”cin>>n;cout<<“Enter “<< n <<“ numbers: \n”for(int i=0; i<n; i++){

cin>>item;numbers.push(item);

}cout<<endl<<endl;while(!numbers.empty()){

cout<<numbers.top() << “ “;numbers.pop();

}cout<<endl<<endl;

}

Type an integer n: 4Enter 4 numbers: 10121416

16 14 12 10

101012

101214

10121416

10121416

101214

1012

10

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7. REVERSING A WORDRead in a word and then write it out backward.NATTANESIOTROTTORTOISE.Pseudocode

//reversing a wordWhile(there are more characters of the word to read)

Read a character, and push the char onto the stackWhile(the stack is not empty)

Write the top char to the screen, and pop it off the stack

Write Top(N) then pop

NAN

TAN

Push N

Push A Push T

AN

Write Top(T) then pop

Output:TAN

Write Top(A) then pop

N

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8. RPN CALCULATOR In a Reverse Polish Notation (RPN) calculator,

the operands (usually numbers) are entered before an operation (like addition, subtraction, multiplication, or division) is specified

(a b +), (4 5 6 * +), …etc // a, b, 4, 5, 6 are operands, + and * are operators.

The operands are pushed onto a stack. When an operation is performed, it pops its operands from the stack

and pushes its result back onto the stack. 4 5 6 * +

4

45

456

5*6=30

430

4+30=34

4 3434

12

3

4 5 67

8

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8. RPN CALCULATORtypedef double Stack_entry;

#include "stack.h"

int main( )

/* Post: The program has executed simple arithmetic commands entered by the user.

Uses: The class Stack and functions introduction, instructions ,do command ,and get command. */

{

Stack stored_numbers;

introduction( );

instructions( );

while (do_command(get_command( ), stored_numbers));

}

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8. RPN CALCULATORchar get_command( )

{

char command; bool waiting = true;

cout << " Select command and press < Enter > :";

while (waiting) {

cin >> command;

command = tolower(command); //include header file : cctype

if (command == ‘?’ || command == ‘=‘ || command == ‘+’ ||

command == ‘-’ || command == ‘*’ || command == ‘/’ ||

command == ‘q’)

waiting = false;

else {

cout << " Please enter a valid command:" <<endl

<< " [?]push to stack [=]print top" <<endl

<< " [+] [ . ] [*] [/] are arithmetic operations" <<endl

<< " [Q]uit." <<endl;

}//else

}//while

return command;

}//get_command

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8. RPN CALCULATORbool do command(char command, Stack &numbers)

/* Pre: The first parameter species a valid calculator command.

Post: The command specified by the first parameter has been applied to the Stack

of numbers given by the second parameter. A result of true is returned unless

command == ‘q’ . */

{ double p, q;

switch (command) {

case ‘?’: cout << " Enter a real number: " <<flush; cin >> p;

if (numbers.push(p) == overflow) cout << " Warning: Stack full, lost number" <<endl; break;

case ‘=‘: if (numbers.top(p) == underflow) cout << " Stack empty" <<endl;

else cout << p << endl; break;

case ‘+’: if (numbers.top(p) == underflow) cout << " Stack empty" <<endl;

else {

numbers.pop( );

if (numbers.top(q) == underflow) {

cout << " Stack has just one entry" <<endl;

numbers.push(p); }

else { numbers.pop( );

if (numbers.push(q C p) == overflow)

cout << " Warning: Stack full, lost result" <<endl; }

} break;

// Add options for further user commands.

case ‘q’: cout << " Calculation finished.\n"; return false;

} return true; }

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9. BRACKET MATCHING We develop a program to check that brackets are

correctly matched in an input text file. We limit our attention to the brackets {, }, ( , ), [, and ]. We read a single line of characters and ignore all input

other than bracket characters. Algorithm: Read the le character by character. Each

opening bracket ( , [, or{ that is encountered is considered as unmatched and is stored until a matching bracket can be found. Any closing bracket ), ], or} must correspond, in bracket style, to the last unmatched opening bracket, which should now be retrieved and removed from storage. Finally, at the end of the program, we must check that no unmatched opening brackets are left over.

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9. BRACKET MATCHING A program to test the matching of brackets needs to

process its input le character by character, and, as it works its way through the input, it needs some way to remember any currently unmatched brackets.

These brackets must be retrieved in the exact reverse of their input order, and therefore a Stack provides an attractive option for their storage.

Our program need only loop over the input characters, until either a bracketing error is detected or the input le ends.Whenever a bracket is found, an appropriate Stack operation is applied.

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9. BRACKET MATCHINGint main( )/* Post: The program has notied the user of any bracket mismatch in the standard input file. Uses: The class Stack . */{ Stack openings; char symbol; bool is matched = true;

while (is matched && (symbol = cin.get( )) != 0\n0) {if (symbol == ‘{‘ || symbol == ‘(‘ || symbol == ‘[‘)

openings.push(symbol);if (symbol == ‘}’ || symbol == ‘)’ || symbol == ‘]’) {

if (openings.empty( )) {cout << " Unmatched closing bracket " <<symbol

<< " detected." <<endl;Is_matched = false;}else {char match;openings.top(match);openings.pop( );is matched = (symbol == ‘}’ && match == ‘{‘) || (symbol == ‘)’ && match == ‘(‘) || (symbol == ‘]’ && match == ‘[‘);if (!is matched)

cout << " Bad match " <<match << symbol << endl;}}}

if (!openings.empty( ))cout << " Unmatched opening bracket(s) detected." <<endl;

}

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9. BRACKET MATCHINGInput: ( ( ) ())

(

(

( (

(

( (

Initial empty stack

Read and

push 1st (Read and

push 2nd (

Read 1st )

and pop (

Read and

push 3rd (

Read 2nd )

and pop (Read 3rd ) and

pop last (

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10. Evaluating Arithmetic (Infix) Expressions

Input: ( ( ( 6 + 9 ) / 3 ) * ( 6 – 4 ) )

Numbers: push onto the number stack

Operators: push onto the operator’s stack

Right parenthesis, ‘)’: take the two top numbers from the number stack and the top operator form the operator’s stack, the two numbers are evaluated and the result is pushed back to the numbers stack.

Left parenthesis, ‘(‘: throw away.

96 +

6+9=15

15

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Input: ( ( ( 6 + 9 ) / 3 ) * ( 6 – 4 ) )

315 /

15/3=5

5

Input: ( ( ( 6 + 9 ) / 3 ) * ( 6 – 4 ) )

465

-*

6-4=225 *

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25 *

2*5=10

10

At this point, there is no more input, and there is only one number in the numbers stack, so:

10 is the output

Input: ( ( ( 6 + 9 ) / 3 ) * ( 6 – 4 ) )

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11. Evaluating Postfix Expressions

5

The result of the computation is: 12

Input: : 5 3 2 * + 4 - 5 +

35

235

65 11

411 7

57 12

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11. Evaluating Postfix ExpressionsPseudocode to evaluate postfix:

1. Initialize a stack of numbers

2. Do

If (the input is a number)

Push onto the stack

Else

Use top and pop to get the two numbers off the stack

Evaluate the two numbers using the operator

(use the second number popped as the left operand)

Push the result back onto the stack

While (there is more input to read)

3. Pop the only number off the stack, which is the value of the postfix expression

Input: : 5 3 2 * + 4 - 5 +

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12. Converting Infix into Postfix Expressions

If the Infix is fully parenthesized:

move each operator to the location of the right parenthesis corresponding to that operator and then remove all parenthesis.

Example: Infix: ( ( ( A + 7 ) * ( B / C ) ) – ( 2 * D ) ): ( ( ( A + 7 ) * ( B / C ) ) – ( 2 * D ) )

Postfix: A 7 : A 7 ++ B C / * 2 D * – B C / * 2 D * –

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If the Infix is fully parenthesized (pseudocode):1. Initialize a stack of characters

2. Do

If (input is ‘(‘ )

Push onto the stack

Else if (input is a number or operand (A,B,C,…) )

Write to the output

Else if (input is an operator (*, /, + ,-) )

Push onto the stack

Else if (input is ‘)’ )

Discard ‘)’

Write top of stack to the output

Pop the stack (an operator)

Pop the stack (‘(‘)

While (there is more input)

3. At this point the stack should be empty otherwise print error message.

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A7+BC/*

A7+BC/


If the Infix is fully parenthesized (pseudocode):

Infix: ( ( ( A + 7 ) * ( B / C ) ) – ( 2 * : ( ( ( A + 7 ) * ( B / C ) ) – ( 2 * D ) )D ) )

(((

(((

Output: A

+(((

A 7

A 7 +

+(((

*((

(*((

A7+B

/(*((

A7+BC

/(*((

*((

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A7+BC/*2D*


If the Infix is fully parenthesized (pseudocode):

Infix: ( ( ( A + 7 ) * ( B / C ) ) – ( 2 * : ( ( ( A + 7 ) * ( B / C ) ) – ( 2 * D ) )D ) )

––(

(––( A7+BC/*2

*(––(

A7+BC/*2D

*(––(

A7+BC/*2D*-

–(

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+3X*Y 12-+

3X*Y 12-3X*

12. Converting Infix into Postfix Expressions If the infix expression is NOT fully parenthesized

Infix: 3 * X + ( Y – 12 ) – Z: 3 * X + ( Y – 12 ) – Z

3

*

3X

* +(+

3X*Y

––(+

3X*Y 12

–(+

––

3X*Y 12-+Z ––––

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12. Converting Infix into Postfix Expressions If the infix expression is NOT fully parenthesized (pseudocode)

1. Initialize a stack of characters2. Do

If (input is ‘(‘ )Push onto the stack

Else if (input is a number or operand (A,B,C,…) )Write to the output

Else if (input is an operator (*, /, + ,-) )if the p(top) > p(input)do

push input onto the stackelse

print top of stack and poppush input

Else if (input is ‘)’ )Discard ‘)’Write top of stack to the output, (keep printing and poping until ‘(‘ is

encountered) While (there is more input)

3. At this point the stack should be empty otherwise print error message.

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NOTES Common uses of stack in computer

science: To keep track of function calls To keep track of local variables To analyze syntax of a program

Education

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