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Applications of Stack (Data Structure).pdf
- 1. Applications of Stack The following Applications of stack are: 1. Stack is used by compilers to check for balancing of parentheses, brackets and braces. 2. Stack is used to evaluate a postfix expression. 3. Stack is used to convert an infix expression into postfix/prefix form. 4. In recursion, all intermediate arguments and return values are stored on the processors stack. 5. During a function call the return address and arguments are pushed onto a stack and on return they are popped off.
- 2. Expression evaluation and conversion The most frequent application of stacks is in the evaluation of arithmetic expressions. An arithmetic expression is made of operands, operators. Operand is the quantity on which a mathematical operation is performed. Operand may be a variable like x, y, z or a constant like 5, 4, 6 etc. Operator is a symbol which signifies a mathematical or logical operation between the operands. Examples of familiar operators include +, -, *, /, ^ etc. When high-level programming languages came into existence, one of the major difficulties faced by computer scientists was to generate machine language instructions that could properly evaluate any arithmetic expression. A complex assignment statement such as
- 3. Might have several meanings, and even if the meanings were uniquely defied, it is still difficult to generate a correct and reasonable instruction sequence. The fist problem in understanding the meaning of an expression is to decide the order in which the operations are to be carried out. This demands that every language must uniquely define such an order.
- 4. Another example: To calculate this expression for values 4, 3, 7 for A, B, C respectively we must follow certain in order to have the right result : The answer is not correct; multiplication is to be done before the addition, because multiplication has higher precedence over addition. This means that an expression is calculated according to the operators precedence not the order as they look like. Thus expression A + B * C can be interpreted as A + (B * C). Using this alternative method we can convey to the computer that multiplication has higher precedence over addition.
- 5. To fix the order of evaluation, assign each operator a priority, and evaluated from left to right.
- 6. Let us consider the expression By using priorities rules, the expression X is rewritten as We manually evaluate the innermost expression first, followed by the next parenthesized inner expression, which produces the result. Another application of stack is calculation of postfix expression. There are basically three types of notation for an expression (mathematical expression; An expression is defined as the number of operands or data items combined with several operators.) 1. Infix notation 2. Prefix notation 3. Postfix notation
- 7. Infix The infix notation is what we come across in our general mathematics, where the operator is written in-between the operands. For example: The expression to add two numbers A and B is written in infix notation as: Prefix The prefix notation is a notation in which the operator(s) is written before the operands. The same expression when written in prefix notation looks like: Postfix In the postfix notation the operator(s) are written after the operands, so it is called the postfix notation (post means after), it is also known as suffix notation. The above expression if written in postfix expression looks like:
- 8. A + B * C Infix Form A + (B * C) Parenthesized expression A + (B C *) Convert the multiplication A (B C *) + Convert the addition A B C * + Postfix form The rules to be remembered during infix to postfix or prefix conversion are: 1. Parenthesize the expression starting from left to right. 2. During parenthesizing the expression, the operands associated with operator having higher precedence are first parenthesized. For example in the above expression B * C is parenthesized first before A + B. 3. The sub-expression (part of expression), which has been converted into postfix, is to be treated as single operand. 4. Once the expression is converted to postfix form, remove the parenthesis.
- 9. Exercise A * B + C / D (A * B) + (C / D) Parenthesized expression (A B*) + (C / D) Convert the multiplication (A B*) + (C D/) Convert the division (A B*) (C D/) + Convert the addition A B*C D/ + Postfix form
- 10. Converting infix to prefix expression Example: A * B + C / D Infix Form (A * B) + (C/D) Parenthesized expression (*A B) + (C/D) Convert Multiplication (*A B) + (/CD ) Convert Division +(*A B) (/CD ) Convert addition +*A B /CD Prefix form Note: the precedence rules for converting an expression from infix to prefix is identical to postfix. The only change from postfix conversion is that the operator is placed before the operands rather than after them.
- 11. Conversion from infix to postfix (another method): Procedure to convert from infix expression to postfix expression is as follows: 1. Scan the infix expression from left to right. 2. a) If the scanned symbol is left parenthesis, push it onto the stack. b) If the scanned symbol is an operand, then place directly in the postfix expression (output). c) If the symbol scanned is a right parenthesis, then go on popping all the items from the stack and place them in the postfix expression till we get the matching left parenthesis. d) If the scanned symbol is an operator, then go on removing all the operators from the stack and place them in the postfix expression, if and only if the precedence of the operator which is on the top of the stack is greater than (or greater than or equal) to the precedence of the scanned operator and push the scanned operator onto the stack otherwise, push the scanned operator onto the stack. The three important features of postfix expression are: 1. The operands maintain the same order as in the equivalent infix expression. 2. The parentheses are not needed to designate the expression unambiguously. 3. While evaluating the postfix expression the priority of the operators is no longer relevant.
- 12. Example: Convert the following infix expression to the equivalent postfix notation. (30+23)*(43-21)/(84+7)
- 13. Need for prefix and postfix expression The evaluation of an infix expression using a computer needs proper code generation by the compiler without any ambiguity and is difficult because of various aspects such as the operators priority . This problem can be overcome by writing or converting the infix expression to an alternate notation such as the prefix or the postfix. ((A - (B + C)) * (C+ D / B )) ^ B + C Infix expression A B C + - C D B / + * B ^ C + Postfix expression Lets: A = 6 B = 2 C = 3 D = 8 6 2 3 + - 3 8 2 / + * 2 ^ 3 + Postfix
- 14. Evaluate the following expression in postfix 6 2 3 + - 3 8 2 / + * 2 ^ 3 +