data structure & algorithms chapter 3: stacks. 2 objectives in this chapter, you will: learn...

DATA STRUCTURE & ALGORITHMS

CHAPTER 3: STACKS

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ObjectivesIn this chapter, you will:• Learn about stacks• Examine various stack operations• Discover stack applications

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Stacks• Stack: list of homogenous elements

– Addition and deletion occur only at one end, called the top of the stack• Example: in a cafeteria, the second tray can be

removed only if first tray has been removed

– Last in first out (LIFO) data structure• Operations:

– Push: to add an element onto the stack– Pop: to remove an element from the stack

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Stacks (continued)

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Stack Operations• In the abstract class stackADT:

– initializeStack– isEmptyStack– isFullStack– push– top– pop

Stack ApplicationStack applications classified into 4 broad categories: • Reversing data – e.g. reverse a list & convert decimal

to binary.– Eg. 26 = 110102

• Parsing data – e.g. translate a source program to machine language.

• Postponing data usage – e.g. evaluation, transformation.

• Backtracking – e.g. computer gaming, decision analysis, expert systems.

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Implementation of Stacks as Arrays

• First element can go in first array position, the second in the second position, etc.

• The top of the stack is the index of the last element added to the stack

• Stack elements are stored in an array• Stack element is accessed only through top• To keep track of the top position, use a

variable called stackTop

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Implementation of Stacks as Arrays

• Because stack is homogeneous– You can use an array to implement a stack

• Can dynamically allocate array– Enables user to specify size of the array

• The class stackType implements the functions of the abstract class stackADT

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Implementation of Stacks as Arrays

• C++ arrays begin with the index 0– Must distinguish between:

• The value of stackTop• The array position indicated by stackTop

• If stackTop is 0, the stack is empty• If stackTop is nonzero, the stack is not

empty– The top element is given by stackTop - 1

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Implementation of Stacks as Arrays

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Initialize Stack

Empty Stack• If stackTop is 0, the stack is empty

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Full Stack• The stack is full if stackTop is equal to maxStackSize

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Push• Store the newItem in the array component

indicated by stackTop• Increment stackTop• Must avoid an overflow

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Push

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Return the Top Element

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Pop• Simply decrement stackTop by 1• Must check for underflow condition

Stacks Example• Example Suppose the following 6 elements are pushed, in order, onto

an empty stacks.

A, B, C, D, E, F We write the stack: STACK: A, B, C, D, E, F

Pseudocode for Array of Stacks

Procedure 3.1 PUSH(STACK, TOP, MAXSTK, ITEM)

This procedure pushes an ITEM onto a stack.

1. [Stack already filled ?]

If TOP = MAXSTK, then: Print:OVERFLOW, and

Return.

2. Set TOP := TOP + 1. [Increses TOP by 1]

3. Set STACK[TOP] :=ITEM. [Inserts ITEM in new TOP

position.

4. Return.

Pseudocode for Array of Stacks

• Procedure 3.2 POP(STACK, TOP, ITEM)

This procedures deletes the top element of STACK

and assigns it to the variable ITEM.

1. [Stacks has an item to be removed?]

If TOP = 0, the Print: UNDERFLOW, and RETURN.

2. Set ITEM := STACK[TOP].[Assigns TOP elements

to ITEM.]

3. Set TOP := TOP – 1.[ Decreases TOP by 1]

4. Return.

Stack ExerciseExercise 1:Consider the following stack of characters, where STACK is allocated N = 8

memory cells :

STACK : A,C,D, F, K, _, _ , _

( For notational convenience, we use “_” to denote an empty memory cell).

Describe the stack as the following operations take place :

(a)POP (STACK, ITEM ) (e) POP (STACK, ITEM)

(b)POP (STACK,ITEM) (f) PUSH(STACK, R)

(c) PUSH (STACK, L) (g) PUSH(STACK, S)

(d) PUSH (STACK, P) (h) POP(STACK, ITEM)

Stack ExerciseExercise 2:Consider the following stack, where STACK is allocated N = 6 memory

cells :

STACK : A, D, E, F, G, _______.

(a) PUSH(STACK, K)

(b) POP(STACK, ITEM )

(c) PUSH(STACK, L)

(d) PUSH(STACK,S)

(e) POP(STACK, ITEM)

(f) PUSH(STACK, M)

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Copy Stack

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Stack Header File• Place definitions of class and functions

(stack operations) together in a file

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Linked Implementation of Stacks• Array only allows fixed number of elements• If number of elements to be pushed exceeds

array size– Program may terminate

• Linked lists can dynamically organize data• In a linked representation, stackTop is

pointer to top element in stack

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Default Constructor• Initializes the stack to an empty state when a

stack object is declared– Sets stackTop to NULL

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Empty Stack and Full Stack• In the linked implementation of stacks, the

function isFullStack does not apply– Logically, the stack is never full

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Initialize Stack

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Push• The newElement is added at the beginning

of the linked list pointed to by stackTop

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Push (continued)

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Push (continued)

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Push (continued)

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Push (continued)• We do not need to check whether the stack

is full before we push an element onto the stack

C++ Programming: Program Design Including Data Structures, Fourth Edition 39

Return the Top Element

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.Pop

• Node pointed to by stackTop is removed

C++ Programming: Program Design Including Data Structures, Fourth Edition 42

Pop (continued)

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Copy Stack

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Copy Stack

• Notice that this function is similar to the definition of copyList for linked lists

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Application of Stacks: Postfix Expressions Calculator

• Infix notation: usual notation for writing arithmetic expressions– The operator is written between the operands– Example: a + b– The operators have precedence

• Parentheses can be used to override precedence

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Application of Stacks: Postfix Expressions Calculator

• Prefix (Polish) notation: the operators are written before the operands– Introduced by the Polish mathematician Jan

Lukasiewicz• Early 1920s

– The parentheses can be omitted– Example: + a b

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Application of Stacks: Postfix Expressions Calculator

• Reverse Polish notation: the operators follow the operands (postfix operators)– Proposed by the Australian philosopher and

early computer scientist Charles L. Hamblin• Late 1950's

– Advantage: the operators appear in the order required for computation

– Example: a + b * c • In a postfix expression: a b c * +

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Application of Stacks: Postfix Expressions Calculator

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Application of Stacks: Postfix Expressions Calculator

• Postfix notation has important applications in computer science– Many compilers first translate arithmetic

expressions into postfix notation and then translate this expression into machine code

• Evaluation algorithm:– Scan expression from left to right– When an operator is found, back up to get the

operands, perform the operation, and continue

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Application of Stacks: Postfix Expressions Calculator

• Example: 6 3 + 2 * =– Read first symbol

• 6 is a number push it onto the stack

– Read next symbol• 3 is a number push it onto the stack

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Application of Stacks: Postfix Expressions Calculator

• Example: 6 3 + 2 * =– Read next symbol

• + is an operator (two operands) pop stack twice, perform operation, put result back onto stack

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• Example: 6 3 + 2 * =– Read next symbol

• 2 is a number push it onto the stack

Application of Stacks: Postfix Expressions Calculator

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• Example: 6 3 + 2 * =– Read next symbol

• * is an operator pop stack twice, perform operation, put result back onto stack

Application of Stacks: Postfix Expressions Calculator

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• Example: 6 3 + 2 * =– Read next symbol

• = is an operator indicates end of expression• Print the result (pop stack first)

Application of Stacks: Postfix Expressions Calculator

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Application of Stacks: Postfix Expressions Calculator

• Symbols can be numbers or anything else:– +, -, *, and / are operators

• Pop stack twice and evaluate expression• If stack has less than two elements error

– If symbol is =, the expression ends• Pop and print answer from stack• If stack has more than one element error

– If symbol is anything else• Expression contains an illegal operator

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Application of Stacks: Postfix Expressions Calculator

• Examples:7 6 + 3 ; 6 - =

• ; is an illegal operator

14 + 2 3 * =• Does not have enough operands for +

14 2 3 + =• Error: stack will have two elements when we

encounter equal (=) sign

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Application of Stacks: Postfix Expressions Calculator

• We assume that the postfix expressions are in the following form:

#6 #3 + #2 * =– If symbol scanned is #, next input is a number– If the symbol scanned is not #, then it is:

• An operator (may be illegal) or• An equal sign (end of expression)

• We assume expressions contain only +, -, *, and / operators

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Conclusions• Stack: items are added/deleted from one end

– Last In First Out (LIFO) data structure– Operations: push, pop, initialize, destroy, check for

empty/full stack– Can be implemented as array or linked list– Middle elements should not be accessed

• Postfix notation: operators are written after the operands (no parentheses needed)

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ReferencesC++ Programming: Program Design Including

Data Structures, Fourth Edition