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CSE 246 Data Structures and Algorithms

CSE 246 Data Structures and Algorithms. Lecture# 7. Stack Application. Run time Stack procedures Parenthesis Validation Postfix Calculator Interpret infix with precedence. Stack Application.

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CSE 246 Data Structures and Algorithms

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  1. CSE 246Data Structures and Algorithms Lecture# 7

  2. Stack Application • Run time Stack procedures • Parenthesis Validation • Postfix Calculator • Interpret infix with precedence Quratulain

  3. Stack Application • Almost invariably, programs compiled from modern high level languages (even C!) make use of a stack frame for the working memory of each procedure or function invocation. • When any procedure or function is called, a number of words - the stack frame - is pushed onto a program stack. Quratulain

  4. Visualizing Recursion • To understand how recursion works, it helps to visualize what’s going on. • To help visualize, we will use a common concept called the Stack. • A stack basically operates like a container of trays in a cafeteria. It has only two operations: • Push: you can push something onto the stack. • Pop: you can pop something off the top of the stack.

  5. Stacks The diagram below shows a stack over time. We perform two pushes (of 2 and then of 8), and one pop. 8 2 2 2 Time 3: Pop: Gets 8 Time: 0 Empty Stack Time 1: Push “2” Time 2: Push “8” Time 4: Pop: Gets 2

  6. Arithmetic Expression • Infix, Postfix and Prefix notations are three different but equivalent ways of writing expressions. Quratulain

  7. Arithmetic Expressions Infix Expressions An expression in which every binary operation appears between its operands Example: (i) a+b “+” is a binary operation and a and b are its operands (ii) (a+b)*c Prefix Expressions An expression in which operator comes before its operands Example: (i) a+b = +ab (ii) (a+b)*c = *+abc (iii) a+(b*c) =+a*bc Postfix Expressions An expression in which operator comes after its operands Example: (i) a+b = ab+ (ii) (a+b)*c = ab+c* (iii) a+(b*c) = abc*+ Quratulain

  8. Arithmetic Expression validate • Pushing an item on to stack correspond to opening a scope, and popping an item from the stack corresponds to closing a scope. • When the stack is empty and scope ender encountered, so the parenthesis pattern is invalid. Quratulain

  9. Infix notation • Infix notation: A * ( B + C ) / D • Infix notation needs extra information to make the order of evaluation of the operators clear: rules built into the language about operator precedence and associativity, and brackets ( ) to allow users to override these rules. Quratulain

  10. Postfix notation • The order of evaluation of operators is always left-to-right, and brackets cannot be used to change this order. • Operators act on values immediately to the left of them. • RPN has the advantage of being extremely easy, and therefore fast, for a computer to analyze. Quratulain

  11. Postfix • In the 1920's, Jan Lukasiewicz developed a formal logic system which allowed mathematical expressions to be specified without parentheses by placing the operators before (prefix notation) or after (postfix notation) the operands. • postfix notation for a calculator keyboard. • computer scientists realized that RPN or postfix notation was very efficient for computer math. • As a postfix expression is scanned from left to right, operands are simply placed into a last-in, first-out (LIFO) stack and operators may be immediately applied to the operands at the bottom of the stack. • Another advantage is consistency between machines. Quratulain

  12. Practical implications • Calculations proceed from left to right • There are no brackets or parentheses, as they are unnecessary. • Operands precede operator. They are removed as the operation is evaluated. • When an operation is made, the result becomes an operand itself (for later operators) • There is no hidden state. No need to wonder if you hit an operator or not. Quratulain

  13. Precedence of Operators • The five binary operators are: addition, subtraction, multiplication, division and exponentiation. • The order of precedence is (highest to lowest) • Exponentiation  • Multiplication/division *, / • Addition/subtraction +, -

  14. Example • The calculation: ((1 + 2) * 4) + 3 can be written down like this in RPN: • The expression is evaluated in the following way (the Stack is displayed after Operation has taken place): • Input Stack Operation 1 1 Push operand 2 1, 2 Push operand + 3 Addition 4 3, 4 Push operand * 12 Multiplication 3 12,3 Push operand + 15 Addition 1 2 + 4 * 3 + Quratulain

  15. Infix to Postfix conversion Algorithm Opstk = the empty stack while(not end of input) { symb=next input char; if (symbol is operand) add symb in postfix string else { while(!empty(opstk)&& prcd(stacktop(opstk), symb)) { topsymb=pop(opstk); add topsymb to postfix string; } } while(!empty(opstk)) { topsymb=pop(opstk); add topsymb to postfix string; } Quratulain

  16. Algorithm for postfix evalution Opndstk = the emty stack; While (not end of input) { Symb = next input character; If (symb is an operand) Push(opndstk,symb); Else { Opnd2=pop(opndstk); Opnd1=pop(opndstk); Value = result of applying symb to opnd1 and opnd2; Push(opndstk, value); } } Return (pop(opndstk)); Quratulain

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