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 (+ or / here) is written between its operandsthis 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 3 4 + and 7 9 /, respectively.
To evaluate a complex infix expression, a compiler would first convert the expression to postfix notation and evaluate the postfix version of the expression. Each of these algorithms requires only a single left-to-right pass of the expression. Each algorithm uses a stack object in support of its operation, and in each algorithm the stack is used for a different purpose.
In this exercise, you will write a C++ version of the infix-to-postfix conversion algorithm. In the next exercise, you will write a C++ version of the postfix expression evaluation algorithm. Later in the chapter, you will discover that code you write in this exercise can help you implement a complete working compiler.
Write a program that converts 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 of the preceding infix expression is
6 2 + 5 * 8 4 / -
The program should read the expression into character array infix and use modified versions of the stack functions implemented in this chapter to help create the postfix expression in character array postfix. The algorithm for creating a postfix expression is as follows:
- Push a left parenthesis '(' onto the stack.
- Append a right parenthesis ')' to the end of infix.
- 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, copy it to the next element of 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 insert the popped operators in 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 insert them in 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:
[Note: We assume left to right associativity for all operators for the purpose of this exercise.] The stack should be maintained with stack nodes, each containing a data member and a pointer to the next stack node.
Some of the functional capabilities you may want to provide are:
- function convertToPostfix that converts the infix expression to postfix notation
- function isOperator that determines whether c is an operator
- function precedence that determines whether the precedence of operator1 is less than, equal to or greater than the precedence of operator2 (the function returns1, 0 and 1, respectively)
- function push that pushes a value onto the stack
- function pop that pops a value off the stack
- function stackTop that returns the top value of the stack without popping the stack
- function isEmpty that determines if the stack is empty
- function printStack that prints the stack