CSE 246 Data Structures and Algorithms

CSE 246Data Structures and Algorithms Spring2011 Lecture#11

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

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

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.

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

Stacks and Methods • When you run a program, the computer creates a stack for you. • Each time you invoke a method, the method is placed on top of the stack. • When the method returns or exits, the method is popped off the stack. • The diagram on the next page shows a sample stack for a simple Java program.

Stacks and Methods square() main() main() main() Time 4: Pop: main() returns a value. method exits. Time 3: Pop: square() returns a value. method exits. Time: 0 Empty Stack Time 1: Push: main() Time 2: Push: square() This is called an activation record or stack frame. Usually, this actually grows downward.

Stacks and Recursion • Each time a method is called, you push the method on the stack. • Each time the method returns or exits, you pop the method off the stack. • If a method calls itself recursively, you just push another copy of the method onto the stack. • We therefore have a simple way to visualize how recursion really works.

Back to the Simple Recursion Program • Here’s the code again. Now, that we understand stacks, we can visualize the recursion. public class Recursion { public static void main (String args[]) { count(0); System.out.println(); } public static void count (int index) { System.out.print(index); if (index < 2) count(index+1); } }

Stacks and Recursion in Action count(2) count(1) count(1) count(0) count(0) count(0) … main() main() main() main() Time 3: Push: count(1) Time 4: Push: count(2) Time: 0 Empty Stack Time 1: Push: main() Time 2: Push: count(0) Times 5-8: Pop everything one by one Inside count(0): print (index);  0 if (index < 2) count(index+1); Inside count(1): print (index);  1 if (index < 2) count(index+1); Inside count(2): print (index);  2 if (index < 2) count(index+1); This condition now fails! Hence, recursion stops, and we proceed to pop all methods off the stack,one by one.

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

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

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

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

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

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

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

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 +, -

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

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

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

A Stack Interface in Java • The stack data structure is included as a "built-in" class in the java.util package of Java. • it is instructive to learn how to design and implement a stack "from scratch.“ • Implementing an abstract data type in Java involves two steps • Define interface • Define exceptions for any error conditions that can arise. • Provide a concrete class that implements the methods of the interface associated with that ADT. Quratulain

Stack interface public interface Stack <E> { public int size(); public booleanisEmpty(); public E top() throws EmptyStackException; public void push( E element); public E pop() throws EmptyStackException; } Quratulain

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Implementing a Stack with a Generic Linked List Left as homework Quratulain

CSE 246 Data Structures and Algorithms