The shunting yard algorithm is used to convert the infix notation to reverse Polish notation. The postfix notation is also known as the Reverse Polish Notation (RPN). The algorithm was named a “Shunting yard” because its activity is similar to a railroad shunting yard. It is a method for representing expressions in which the operator symbol is placed after the arguments being operated on. Polish notation, in which the operator comes before the operands. Australian philosopher and computer scientist suggested placing the operator after the operands and hence created reverse Polish notation.  Dijkstra developed this algorithm

Representation and Interpretation:

Brackets are not required to represent the order of evaluation or grouping of the terms. RPN expressions are simply evaluated from left to right and this greatly simplifies the computation of the expression within computer programs. As an example, the arithmetic expression.

Interpreting from left to right the following two executions can be performed

1. If the value appears next in the expression push the current value in the stack.
2. Now, if the operator appears next, pop the topmost two elements from the stack, execute the operation, and push back the result into the stack.

The order of precedence of operators is:

Illustration: RPN expression will produce the sum of 2 and 3, namely 5: 2 3 +

Input: (3+4)*5
Output: 34+5*

This is the postfix notation of the above infix notation

Concepts Involved:

Examples: 

Infix Notation:   a+b*(c^d-e)^(f+g*h)-i 
Postfix Notation: abcd^e-fgh*+^*+i- 

Algorithm: AE is the arithmetic expression written in infix notation PE will be the postfix expression of AE 

1. Push “(“ onto Stack, and add “)” to the end of AE.
2. Scan AE from left to right and repeat Step 3 to 6 for each element of AE until the Stack is empty.
3. If an operand is encountered, append it to PE.
4. If a left parenthesis is encountered, push it onto Stack.
5. If an operator is encountered, then: Repeatedly pop from Stack and append to PE each operator which has the same precedence as or higher precedence than the operator. Add an operator to Stack. [End of if]
6. If a right parenthesis is encountered, then: Repeatedly pop from Stack and append to PE each operator until a left parenthesis is encountered.     Remove the left Parenthesis.  [End of If] [End of If]

Applying the same above algorithms for two examples given below:

Example 1 : Applying Shunting yard algorithm on the expression "1 + 2" 

Step 1: Input “1 + 2”
Step 2: Push 1 to the output queue
Step 3: Push + to the operator stack, because + is an operator.
Step 4: Push 2 to the output queue
Step 5: After reading the input expression, the output queue and operator stack pop the expression and then add them to the output.
Step 6: Output “1 2 +”

Example 2: Applying the Shunting yard algorithm on the expression 5 + 2 / (3- 8) ^ 5 ^ 2 

Token	Action	Stack	output
5	"5" add token to output		5
+	Push token to stack	+	5
2	"2" add token to output	+	5 2
/	Push token to stack	+/	5 2
(	Push token to stack	+/(	5 2
3	"3" add token to output	+/(	5 2 3
-	Push token to stack	+/(-	5 2 3
8	"8" add token to output	+/(-	5 2 3 8
)	Pop stack to output	+/	5 2 3 8 -
^	Push token to stack	+/^	5 2 3 8 -
5	"5" add token to output	+/^	5 2 3 8 – 5
^	Push token to stack	+/^	5 2 3 8 - 5^
2	"2" add token to output	+/^	5 2 3 8 - 5 ^ 2
End	Pop whole stack		5238-5^2^/+

Implementing: Shunting Yard Algorithm 

// Java Implementation of Shunting Yard Algorithm

// Importing stack class for stacks DS
import java.util.Stack;
// Importing specific character class as
// dealing with only operators and operands
import java.lang.Character;

class GFG {

    // Method is used to get the precedence of operators
    private static boolean letterOrDigit(char c)
    {
        // boolean check
        if (Character.isLetterOrDigit(c))
            return true;
        else
            return false;
    }

    // Operator having higher precedence
    // value will be returned
    static int getPrecedence(char ch)
    {

        if (ch == '+' || ch == '-')
            return 1;
        else if (ch == '*' || ch == '/')
            return 2;
        else if (ch == '^')
            return 3;
        else
            return -1;
    }
    
      // Operator has Left --> Right associativity
      static boolean hasLeftAssociativity(char ch) {
        if (ch == '+' || ch == '-' || ch == '/' || ch == '*') {
            return true;
        } else {
            return false;
        }
    }
  
    // Method converts  given infixto postfix expression
    // to illustrate shunting yard algorithm
    static String infixToRpn(String expression)
    {
        // Initialising an empty String
        // (for output) and an empty stack
        Stack<Character> stack = new Stack<>();

        // Initially empty string taken
        String output = new String("");

        // Iterating over tokens using inbuilt
        // .length() function
        for (int i = 0; i < expression.length(); ++i) {
            // Finding character at 'i'th index
            char c = expression.charAt(i);

            // If the scanned Token is an
            // operand, add it to output
            if (letterOrDigit(c))
                output += c;

            // If the scanned Token is an '('
            // push it to the stack
            else if (c == '(')
                stack.push(c);

            // If the scanned Token is an ')' pop and append
            // it to output from the stack until an '(' is
            // encountered
            else if (c == ')') {
                while (!stack.isEmpty()
                       && stack.peek() != '(')
                    output += stack.pop();

                stack.pop();
            }

            // If an operator is encountered then taken the
            // further action based on the precedence of the
            // operator

            else {
                while (
                    !stack.isEmpty()
                    && getPrecedence(c)
                           <= getPrecedence(stack.peek()) 
                    && hasLeftAssociativity(c)) {
                    // peek() inbuilt stack function to
                    // fetch the top element(token)

                    output += stack.pop();
                }
                stack.push(c);
            }
        }

        // pop all the remaining operators from
        // the stack and append them to output
        while (!stack.isEmpty()) {
            if (stack.peek() == '(')
                return "This expression is invalid";
            output += stack.pop();
        }
        return output;
    }

    // Main driver code
    public static void main(String[] args)
    {
        // Considering random infix string notation
        String expression = "5+2/(3-8)^5^2";

        // Printing RPN for the above infix notation
        // Illustrating shunting yard algorithm
        System.out.println(infixToRpn(expression));
    }
}

// Java Implementation of Shunting Yard Algorithm

​

// Importing stack class for stacks DS

import java.util.Stack;

// Importing specific character class as

// dealing with only operators and operands

import java.lang.Character;

​

class GFG {

​

    // Method is used to get the precedence of operators

    private static boolean letterOrDigit(char c)

    {

        // boolean check

        if (Character.isLetterOrDigit(c))

            return true;

        else

            return false;

    }

​

    // Operator having higher precedence

    // value will be returned

    static int getPrecedence(char ch)

    {

​

        if (ch == '+' || ch == '-')

            return 1;

        else if (ch == '*' || ch == '/')

            return 2;

        else if (ch == '^')

            return 3;

        else

            return -1;

    }

      // Operator has Left --> Right associativity

      static boolean hasLeftAssociativity(char ch) {

        if (ch == '+' || ch == '-' || ch == '/' || ch == '*') {

            return true;

        } else {

            return false;

        }

    }

    // Method converts  given infixto postfix expression

    // to illustrate shunting yard algorithm

    static String infixToRpn(String expression)

    {

        // Initialising an empty String

        // (for output) and an empty stack

        Stack<Character> stack = new Stack<>();

​

        // Initially empty string taken

        String output = new String("");

​

        // Iterating over tokens using inbuilt

        // .length() function

        for (int i = 0; i < expression.length(); ++i) {

            // Finding character at 'i'th index

            char c = expression.charAt(i);

​

            // If the scanned Token is an

            // operand, add it to output

            if (letterOrDigit(c))

                output += c;

​

            // If the scanned Token is an '('

            // push it to the stack

            else if (c == '(')

                stack.push(c);

​

            // If the scanned Token is an ')' pop and append

            // it to output from the stack until an '(' is

            // encountered

            else if (c == ')') {

                while (!stack.isEmpty()

                       && stack.peek() != '(')

                    output += stack.pop();

​

                stack.pop();

            }

​

            // If an operator is encountered then taken the

            // further action based on the precedence of the

            // operator

​

            else {

                while (

                    !stack.isEmpty()

                    && getPrecedence(c)

                           <= getPrecedence(stack.peek()) 

                    && hasLeftAssociativity(c)) {

                    // peek() inbuilt stack function to

                    // fetch the top element(token)

​

                    output += stack.pop();

                }

                stack.push(c);

            }

        }

​

        // pop all the remaining operators from

        // the stack and append them to output

        while (!stack.isEmpty()) {

            if (stack.peek() == '(')

                return "This expression is invalid";

            output += stack.pop();

        }

        return output;

    }

​

    // Main driver code

    public static void main(String[] args)

    {

        // Considering random infix string notation

        String expression = "5+2/(3-8)^5^2";

​

        // Printing RPN for the above infix notation

        // Illustrating shunting yard algorithm

        System.out.println(infixToRpn(expression));

    }

}

5238-52^^/+

Complexitu of the above Method:

• Time Complexity: O(n) This algorithm takes linear time, as we only traverse through the expression once and pop and push only take O(1).
• Space Complexity: O(n) as we use a stack of size n, where n is length given of expression.