DATA STRUCTURE & ALGORITHMS
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Transcript of DATA STRUCTURE & ALGORITHMS
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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
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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
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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
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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
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Pseudocode for Array of StacksProcedure 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.
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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.
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Stack ExerciseExercise 1:Consider the following stack of characters, where STACK is allocated N = 8memory 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)
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Stack ExerciseExercise 2:Consider the following stack, where STACK is allocated N = 6 memorycells :
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
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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
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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 operator14 + 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 IncludingData Structures, Fourth Edition