Chapter 04 Stacks

7/30/2019 Chapter 04 Stacks

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Chapter 4: Stacks

Introduction:

Consider the problems on pp. 170-1

(1) Model the discard pile in a card

game

(2) Model a railroad switching

yard

(3) Parentheses checker(4) Calculate and display base-

two representation

of base-ten numbers; e.g.,

2610 = ???????2

Siding

Main Track


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In the last problem:Remainders are generated in right-to-left order. We need to

"stack" them up, then print them out from top to bottom.

In these problems we need a

"last-discarded-first-removed,"

"last-pushed-onto-first-removed,"

"last-stored-first-removed, ""last-generated-first-displayed"

structured data type.

In summary ... a LIFO (Last-In-First-Out) structure.

LIFO


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Stack as an ADT

A stack is anorderedcollection of data items in whichaccess is

possible only at one end(called thetop of the stack).

Basic operations:

1. Construct a stack (usually empty)

2. Check if stackis empty

3. Push: Add an element at the top of the stack

4. Top: Retrievethetop element of the stack

5. Pop: Removethetop element of the stack

Terminology is from spring-loaded

stack of plates in a cafeteria:Adding a plate pushes those below it

down in the stackRemoving a plate pops those below it

up one position.


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Example use of StackBASE-CONVERSION ALGORITHM

(See p. 171-2)

/* This algorithm displays the base-2 representation

of a base-10 number.

Receive: a positive integer number

Output: the base-two representation ofnumber.

---------------------------------------------------------------*/

1. Create an empty stack to hold the remainders.

2. While number 0:

a. Calculate the remainderthat results when

numberis divided by 2.

b. Push remainderonto the stack of remainders.

c. Replace numberby the integer quotient of

numberdivided by 2.


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3. While the stack of remainders is notempty:

a. Retrieve and remove the remainder

from the top of the stack of

remainders.

b. Display remainder.

Example (contd.)


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/* Program that uses a stack to convert the base-ten

* representation of a positive integer to base two.** Input: A positive integer* Output: Base-two representation of the number***************************************************/

#include "Stack.h" // our own -- for STL version#include using namespace std;

int main()

{unsigned number, // the number to be converted

remainder; // remainder when number is// divided by 2

Stack stackOfRemainders; // stack of remainders

char response; // user response

Base10 to Base2 Conversion


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do{cout > number;

while (number != 0){remainder = number % 2;stackOfRemainders.push(remainder);number /= 2;

}cout


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Building a Stack ClassTwo steps:1. Design the class; and

2. Implement the class.

Identify the operations needed to manipulate "real-world" objects

being modeled by class.

Operations are described:

Independent of class implementation.

In a manner independent of any specific representation of object

(because we have no idea what data members will be available).

1. Designing a Stack Class


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//***** Constructors *****

Construction: Initializes an empty stack

//***** Mutators *****

Push operation: Modifies a stack by adding a value to top of

stackPop operation: Modifies a stack by removing the top value

of the stack

//***** Accessors *****Top operation: Retrieves the value at the top of the stack

Empty operation: Determines if stack contains any values

To help with debugging, add early on:

Output: Displays all the elements stored in the stack.

Stack Operations:


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2. Implementing a Stack ClassDefine data members: consider storage structure(s)

Attempt #1: Use an array with the top of the stack at position 0.e.g., Push 75, Push 89, Push 64, Pop

+ features: This models the operation of the stack of plates.

features: Not efficient to shift the array elements up and down in the array.

0123

4

0123

4

75

Push 75

0123

4

89

Push 89

750123

4

89

Push 64

75

64 0123

4

89

Pop

75

I l ti Cl


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Implementing Stack Class

Refined

Keep thebottom of stack at position 0. Maintain a "pointer"myTop to the top

of the stack.

Instead of modeling a stack of plates, model a stack of books (or a discard pile in a

card game.)

7589

43210

Push 75 Push 89

7589

Push 64

75

64

Pop43210

43210

43210

43210

8975

64

myTopmyTop

myTopmyTop

myTop

myTop = -1 myTop = 0 myTop = 1 myTop = 2 myTop = 1

Note: No moving of array elements.

Note: Wedon't clear

this.


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Provide:

An array data member to hold the stack elements. An integer data member to indicate the top of the stack.

Problems: We need an array declaration of the formArrayElementTypemyArray[ARRAYCAPACITY];

What type should be used?Solution (for now): Use the typedef mechanism:

typedef int StackElement;// put this before the class declaration

What about the capacity?const int STACK_CAPACITY = 128;// put this before the class declaration

Now we can declare the array data member in the private section:StackElement myArray[STACK_CAPACITY];

Stack's Data Members


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1. typedef makes StackElement a synonym for int

2. Want a stack of reals, or characters, or . . . ? Change thetypedeftypedef double StackElementType;

or

typedef char StackElementType;

or . . .When class library is recompiled, array elements will be double or char or . . .

3. An alternative to using and changing a typedef: Atemplateto build a Stack class is written whose element type is left unspecified.

The element type is passed as a specialkind ofparameter at

compile time.

This is the approach used in the Standard Template Library (STL)and in

most of the other C++ libraries.

Notes


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4. The typedef and declaration ofSTACK_CAPACITY outside the class

declaration makes them:

easy to find when they need changing

accessible throughout the class and in any file that #includes Stack.h

without qualification

If we put typedef ofStackElement and declaration of the constant

STACK_CAPACITY in public section of the class declaration, they can be

accessed outside the class, but require qualification:

Stack::StackElementStack::STACK_CAPACITY (for class)s.STACK_CAPACITY (for object)

5. If we were to make STACK_CAPACITY a data member, we would probablymake it a static data member:

static const int STACK_CAPACITY = 128;

This makes it a property of the class usable by all class objects, but they

do not have their own copies ofSTACK_CAPACITY.


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/* Stack.h provides a Stack class.* Basic operations:* Constructor: Constructs an empty stack* empty: Checks if a stack is empty* push: Modifies a stack by adding a value at the top* top: Accesses the top stack value; leaves stack* unchanged

* pop: Modifies a stack by removing the value at the* top* display: Displays all the stack elements* Class Invariant:* 1. The stack elements (if any) are stored in positions* 0, 1, . . ., myTop of myArray.

* 2. -1


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#ifndef __STACK_H

#define __STACK_H

const int STACK_CAPACITY = 128;typedef int StackElement;

class Stack

{/***** Function Members *****/

public:. . .

/***** Data Members *****/

private:StackElement myArray[STACK_CAPACITY];int myTop;

}; // end of class declaration. . .

#endif

Stack.h (contd.)


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Stack's Function MembersConstructor:

class Stack{

public:

/*--- Constructor ---Precondition: A stack has been declared.Postcondition: Stack has been constructed

as an empty stack.*/Stack();

...} // end of class declaration

Simple

enough to

inline?

inline Stack::Stack() { myTop = -1; }

Yes

NOTE THE

DOCUMENTATION


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A declaration

Stack s;

will construct s as follows:

s 0 1 2 3 4 . . . 127

myArray ? ? ? ? ? . . . ?

myTop -1

A sample declaration


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Receives Stack containing it as a function member (perhaps implicitly)

Returns: True if stack is empty, false otherwise.

Member function? Yes

Yes

YesConst function? (Shouldn't alter data members)?

Simple enough to inline?

class Stack{

public:. . .

/* --- Is the Stack empty? ---* Returns: true if the Stack containing this* function is empty and false otherwise

***************************************************/bool empty() const;

. . .};// end of class declaration

inline bool Stack::empty() const

{ return (myTop == -1); }

Empty Member Function


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#include #include

#include "Stack.h"using namespace std;int main(){

Stack s;cout


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Receives Stack containing it as a function member (perhaps implicitly)

Value to be added to stack

Returns: Modified stack (perhaps implicitly)

Member function? YesProbably notNoConst function? Simple enough to inline?

Add to class declaration:

/* --- Add a value to the stack ---*

* Receive: A value to be added to a Stack* Postcondition: Value is added at top of stack* provided there's space* Output: "Stack full" message if no space

*************************************************/

void push(const StackElement & value);

Push member function


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void Stack::push(const StackElement & value){

if (myTop < STACK_CAPACITY - 1){ //Preserve stack invariant++myTop;

myArray[myTop] = value;} // or simply,myArray[++myTop] = value;

elsecerr


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Add at bottom of driver:

for (int i = 1; i


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Receives: Stack containing it as a function member (perhaps implicitly)

an ostream

Output: Contents of Stack, from the top down.

Member function? Yes YesYesConst function? Simple enough to inline?


/* --- Display values stored in the stack ---** Receive: The ostream out* Output: Stack's contents, from top down, to out***************************************************/void display(ostream & out) const;

Add toStack.hinline voidStack::display(ostream & out) const{for (int i = myTop; i >= 0; i--)out


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Modify driver:/*for (int i = 1; i


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/* --- Return value at top of the stack ---* Return: value at the top of nonempty Stack* Output: "Stack empty" message if stack empty***************************************************/StackElement top() const;

Receives: Stack containing it as a function member (perhaps implicitly)

Returns: Value at the top of the stack

Output: Error message if stack is empty

Member function? Yes

No?YesConst function?

Simple enough to inline?

Top Member Function


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StackElement Stack::top() const

{ if (myTop >= 0) // or if (!empty())return (myArray[myTop]);

else{

cerr


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Receives Stack containing it as a function member (perhaps implicitly)Returns: Stack with top element removed (perhaps implicitly)

Output: Error message if stack is empty.

Member function? Yes YesNoConst function? Simple enough to inline?

class Stack

. . ./* --- Remove value at top of the stack ---* Postcondition: Top value of stack (if any) has* been removed.* Output: "Stack-empty" message if stack is empty.**************************************************/void pop();

. . .}; // end of class declaration

Pop Member Function

Pop (contd )


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inline void Stack::pop(){

if (myTop >= 0) // Preserve stack invariant

myTop--;elsecerr


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Application of Stacks: Run-timeStack

Whenever a function begins execution (i.e., is activated), an

activation record(or stack frame) is created to store the

current environmentfor that function. Its contents include:

parameterscaller's state information (saved)

(e.g., contents of registers, return address)

local variables

temporary storage

What kind of data structure should be used to store these so

that they can be recovered and the system reset when the

function resumes execution?


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Problem: FunctionA can call FunctionB

FunctionB can call FunctionC

. . .When a function calls another function, it interrupts its own

execution and needs to be able to resume its execution in the

same state it was in when it was interrupted.

When FunctionCfinishes, control should return to FunctionB.When FunctionB finishes, control should return to FunctionA.

So, the order of returns from a function is the reverse of

function invocations; that is, LIFO behavior.

Use a stack to store the activation records. Since it is manipulatedat run-time, it is called the run-time stack.

Exampleon slide 33


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What happens when a function is called?1. Push a copy of its activation record onto the run-time stack

2. Copy its arguments into the parameter spaces

3. Transfer control to the address of the function's body

The top activation record in the run-time stack isalways that of

the function currently executing.

What happens when a function terminates?

1. Pop activation record of terminated function from therun-time stack

2. Use new top activation record to restore the

environment of the interrupted function and resume

execution of the interrupted function.

Run-time Stack Process

E l


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Trace run-time stack for the

following:. . .int main(){ . . .f2(...);f3(...);

}

void f1(...) {. . .}void f2(...){... f1(...); ...}void f3(...){... f2(...); ...}

Examplesint factorial(int n){if (n < 2)return 1;

elsereturn n * factorial(n-1);

}

What happens to the run-time stack when the

following statement executes?

int answer = factorial(4);

This pushing and popping of the run-time stack is the real

overhead associated with function calls that inlining avoids

by replacing the function call with the body of the function.


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Application of Stacks: RPN

1. What is RPN (Reverse Polish Notation)?A notation for arithmetic expressions in which operators are written

after the operands. Expressions can be written without using

parentheses.

Developed by Polish logician, Jan Lukasiewics, in 1950's

Infix notation: operators written between the operandsPostfix " (RPN): operators written after the operandsPrefix " : operators written before the operands

Examples:

INFIX RPN (POSTFIX) PREFIX

A + BA * B + CA * (B + C)A - (B - (C - D))A - B - C - D

A B + + A BA B * C +A B C + *

A B C D - - -

A B - C - D -

+ * A B C* A + B C

- A - B - C D

- - - A B C D


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Evaluating RPN Expressions

"By hand" (Underlining technique):1. Scan the expression from left to right to find an operator.

2. Locate ("underline") the last two preceding operands

and combine them using this operator.

3. Repeat until the end of the expression is reached.

Example:2 3 4 + 5 6 - - *

2 8 *

2 8 * 16

2 7 5 6 - - *2 7 5 6 - - *2 7 -1 - *

2 7 -1 - *

2 3 4 + 5 6 - - *


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Stack Algorithm: (p.195)

Receive: An RPN expression.

Return: A stack whose top element is the value of RPN expression

(unless an error occurred).

1. Initialize an empty stack.

2. Repeat the following until the end of the expression is

encountered:

a. Get next token (constant, variable, arithmetic operator) in

the RPN expression.

b. If token is an operand, push it onto the stack.

If it is an operator, then:

(i) Pop top two values from the stack. If stack

does not contain two items, signal error due to a

malformed RPN and terminate evaluation.(ii)Apply the operatorto these two values.

(iii) Push the resulting value back onto the stack.

3. When the end of expression encountered, its value is on top

of the stack (and, in fact, must be the only value in the stack).

Push TEMP#

Generate code:

LOAD operand1op operand2

STORE TEMP#:

A B C + D E - - *

Sample RPN Evaluation (p 196)


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Sample RPN Evaluation (p. 196)Example: 2 3 4 + 5 6 - - *

Push 2Push 3Push 4Read+

Pop4, Pop3,Push 7Push 5Push 6Read -

Pop6,Pop5,Push -1

Read-Pop-1, Pop7,

Push 8Read*

Pop8, Pop2,

Push 16

234

27

56

27

-1

28

16

3 + 4 = 7

5 - 6 = -1

7 - -1 = 8

2 * 8 = 16


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Which interpretation for the -?

Example: 5 3 - -

5 3 - -5 -3 -8

5 3 - -2 --2

Use a different symbol:

5 3 ~ -

OR

5 3 - ~

Unary minus causes problems

Have ~ stand for negative

C ti I fi t RPN


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Converting Infix to RPN1. By hand: Represent infix expression as an expression tree:

A * B + C

+

C*

A B

A * (B + C) ((A + B) * C) / (D - E)

*

A +

B CC

A B

D E

*

+

/

-

D

x y

xD y

Root Node

Edge Children Nodes

(Binary Tree2 kids)

T th t i L f Ri h P/


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C

A B

D E

*

+

/

-

Traverse the tree inLeft-Right-Parent

order (postorder) to get RPN:

C

A B

D E

*

+

/

-

Traverse tree in Parent-Left-Right

order (preorder) to get prefix:

Traverse tree inLeft-Parent-Right

order (inorder) to get infixmust insert ()'s

A B + C * D E - /

- D E

(A + B)( * C)/( )(D - E) C

A B

D E

*

+

/

-

/ * + A B C


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2. By hand: "Fully parenthesize-move-erase" method:

1. Fully parenthesize the expression.

2. Replace each right parenthesis by the corresponding operator.

3. Erase all left parentheses.

Examples:

A * B + C

((A B * C +

A B * C +

A * (B + C)

(A (B C + *

A B C + *

Exercise:

((A + B) * C) / (D - E)

((A * B) + C) (A * (B + C) )


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3. Use a stack! Stack Algorithm: (p.199)

1. Initialize an empty stack of operators.

2. While no error has occurred and end of infix expression not reached

a. Get next token in the infix expression.b. Iftoken is

(i) a left parenthesis: Push it onto the stack.

(ii) a right parenthesis: Pop and display stack elements until a left

parenthesis is encountered, but do not display

it. (Error if stack empty with no left

parenthesis found.)

(iii) an operator: If stack empty or token has higher priority

than top stack element, push token onto stack.

(Left parenthesis in stack has lower priority

than operators)

Otherwise, pop and display the top stackelement; then repeat the comparison oftoken

with new top stack item.

(iv) an operand: Display it.

3. When end of infix expression reached, pop and display stack items until

stack is empty.

( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))( )/( ( ))


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(-

/

Example:Push ( OutputDisplayAPush +

Display BPush*Display CRead )

Pop *, Display *,Pop +, Display +, Pop (

Push /Push (Display DPush -Push (Display E

Push -Display FRead )

Pop -, Display -, Pop (

Read )Pop -, Display -, Pop (

(

+*

(+

(-(-

(

(A+B*C)/(D-(E-F))

A

ABCAB

ABC*ABC*+

ABC*+D

ABC*+DE

ABC*+DEF

ABC*+DEF-

ABC*+DEF--

(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))

/

/

(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))(A+B*C)/(D-(E-F))

Documents

Chapter 04 Stacks