Question

(Infix-to-Postfix Converter) Stacks are used by compilers to help in the process of evaluating expressions and generating machine-language code. In this and the next exercise, we investigate how compilers evaluate arithmetic expressions consisting only of constants, operators and parentheses. Humans generally write expressions like $3+4$ and $7/9$ in which the operator $(+\text{ or }/\text{ here })$ is written between its operands - this is called infix notation. Computers "prefer" postfix notation, in which the operator is written to the right of its two operands. The preceding infix expressions would appear in postfix notation as $34+$ and $79/,$ respectively. To evaluate a complex infix expression, a compiler would first convert the expression to postfix notation and evaluate the postfix version. Each of these algorithms requires only a single left-toright pass of the expression. Each algorithm uses a stack object in support of its operation, but each uses the stack for a different purpose. In this exercise, you'll write a Java version of the infix-to-postfix conversion algorithm. In the next exercise, you'll write a Java version of the postfix expression evaluation algorithm. In a later exercise, you'll discover that code you write in this exercise can help you implement a complete working compiler. Write class InfixToPostfixconverter to convert an ordinary infix arithmetic expression (assume a valid expression is entered) with single-digit integers such as \[ (6+2) * 5-8 / 4 \] to a postfix expression. The postfix version (no parentheses are needed) of the this infix expression is \[ 62+5 * 84 /- \] The program should read the expression into StringBuffer infix and use one of the stack classes implemented in this chapter to help create the postfix expression in StringBuffer postfix. The algorithm for creating a postfix expression is as follows: a) Push a left parenthesis ' (' onto the stack. b) Append a right parenthesis ' )' to the end of infix. c) While the stack is not empty, read infix from left to right and do the following: If the current character in infix is a digit, append it to postfix. If the current character in infix is a left parenthesis, push it onto the stack. If the current character in infix is an operator: Pop operators (if there are any) at the top of the stack while they have equal or higher precedence than the current operator, and append the popped operators to postfix. Push the current character in infix onto the stack. If the current character in infix is a right parenthesis: Pop operators from the top of the stack and append them to postfix until a left parenthesis is at the top of the stack. Pop (and discard) the left parenthesis from the stack. The following arithmetic operations are allowed in an expression: $+$ addition $-$ subtraction multiplication / division $\wedge$ exponentiation $\mathrm{\%}$ remainder The stack should be maintained with stack nodes that each contain an instance variable and a reference to the next stack node. Some methods you may want to provide are as follows: a) Method convertToPostfix, which converts the infix expression to postfix notation. b) Method isoperator, which determines whether $c$ is an operator. c) Method precedence, which determines whether the precedence of operator1 (from the infix expression $)$ is less than, equal to or greater than that of operator 2 (from the stack). The method returns true if operator1 has lower precedence than operator2. Otherwise, false is returned. d) Method peek (this should be added to the stack class), which returns the top value of the stack without popping the stack.

Short answer

Create an InfixToPostfixConverter class with appropriate methods and a stack to manage operators. Process the infix expression character by character, applying stack operations based on the character's type, and ultimately produce a postfix expression.

Step-by-step solution

Initialization

Create a Java class named InfixToPostfixConverter. Initialize a Stack object to store operators and determine their precedence. Create two StringBuffer objects, one for the infix expression called infix and one for the postfix expression called postfix.

Append Parentheses

Modify the infix expression by pushing a left parenthesis '(' onto the stack and appending a right parenthesis ')' to the end of the infix expression to ensure proper handling of precedence.

Processing the Infix Expression

Iterate through each character of the infix expression. For each character, use a switch or if-else structure to determine its type (digit, operator, or parenthesis) and apply the appropriate logic based on its type.

Handling Digits

If the current character is a digit, append it directly to postfix.

Handling Left Parenthesis

If the current character is a left parenthesis '(', push it onto the stack.

Handling Operators

If the current character is an operator, use the stack to handle the operator precedence. Pop and append to postfix all operators from the stack that have greater than or equal precedence to the current operator. Then, push the current operator onto the stack.

Handling Right Parenthesis

If the current character is a right parenthesis ')', pop operators from the stack and append them to postfix until a left parenthesis is encountered. Then, pop and discard the left parenthesis from the stack.

Implement Helper Methods

Implement the helper methods isOperator to check if a character is an operator, precedence to compare the precedence of two operators, and peek to look at the top value of the stack without popping it.

Convert to Postfix

After processing the entire infix expression, the stack should be empty, and the postfix expression should be complete in the postfix StringBuffer. The method convertToPostfix will return the final postfix expression.

Key concepts

Stack Data Structure

The stack data structure is an essential concept in computer science and a crucial tool in various computing tasks, including expression evaluation and syntax parsing. Essentially, a stack functions much like a stack of books; you can only take the top item off or add a new item on top, which is often summarized as a 'Last In, First Out' (LIFO) principle.

A stack is used in infix-to-postfix conversion for handling operators and ensuring that their precedence is honored correctly in the resulting expression. In Java, a stack can be implemented using the Stack class or with custom stack classes, typically utilizing nodes to maintain each element and a pointer to the next element, ensuring a dynamic data structure that can grow and shrink as needed.

For the infix-to-postfix conversion algorithm, the stack momentarily stores operators encountered in the infix expression. When an operator with lower precedence than the one on the stack top is encountered, or a right parenthesis is read, operators are popped from the stack and added to the postfix expression until the correct condition is met — either the stack is empty, a left parenthesis surfaces, or an operator with lower precedence is at the stack's top.

Operator Precedence

Operator precedence determines the order in which operators are evaluated in expressions. Just as in standard arithmetic, where multiplication is performed before addition, in infix expressions, certain operators must be evaluated before others to maintain the correct result.

In the conversion process from infix to postfix, operator precedence is vital. To handle this, each operator is assigned a precedence level. For instance, multiplication and division have higher precedence than addition and subtraction. This hierarchy ensures that when converting to postfix, the expression evaluates correctly. If the current operator in the infix expression has lower precedence than the operator on the stack top, the operator on the stack is popped and appended to the postfix expression.

The method precedence in the InfixToPostfixConverter class plays a crucial role, as it facilitates the correct placement of operators in the postfix expression. When a new operator is encountered, the method determines whether it should be pushed onto the stack or if operators should be popped from the stack and appended to the postfix expression.

Java Programming

The Java programming language provides a rich set of features that make tasks like infix-to-postfix conversion implementable in a clear and concise way. Java's object-oriented nature enables the creation of classes and methods that can represent and contain the functionality needed for such algorithms.

In the context of the InfixToPostfixConverter class, Java allows the encapsulation of the entire conversion logic within a single class, which can then be instantiated and utilized as needed. The creation of helper methods such as isOperator, which checks if a character is an operator, or precedence, which compares operator precedence, adds to the readability and maintainability of the code.

Moreover, StringBuffer objects in Java are used for creating mutable strings, which are perfect for building the postfix expression incrementally during the conversion process. Java provides the StringBuilder or StringBuffer for such purposes where strings are modified frequently, emphasizing Java's utility in creating efficient and effective solutions for complex problems like expression conversion.