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Shunting yard algorithm

Class	Parsing
Data structure	Stack
Worst-caseperformance	${\displaystyle O(n)}$
Worst-casespace complexity	${\displaystyle O(n)}$

Incomputer science, theshunting yard algorithm is a method for parsing arithmetical or logical expressions, or a combination of both, specified ininfix notation. It can produce either a postfix notation string, also known asreverse Polish notation (RPN), or anabstract syntax tree (AST).^[1] Thealgorithm was invented byEdsger Dijkstra, first published in November 1961,^[2] and named because its operation resembles that of arailroad shunting yard.

Like the evaluation of RPN, the shunting yard algorithm isstack-based. Infix expressions are the form of mathematical notation most people are used to, for instance"3 + 4" or"3 + 4 × (2 − 1)". For the conversion there are two textvariables (strings), the input and the output. There is also astack that holds operators not yet added to the output queue. To convert, the program reads each symbol in order and does something based on that symbol. The result for the above examples would be (inreverse Polish notation)"3 4 +" and"3 4 2 1 − × +", respectively.

The shunting yard algorithm will correctly parse all valid infix expressions, but does not reject all invalid expressions. For example,"1 2 +" is not a valid infix expression, but would be parsed as"1 + 2". The algorithm can however reject expressions with mismatched parentheses.

The shunting yard algorithm was later generalized intooperator-precedence parsing.

1. Input:3 + 4
2. Push 3 to the outputqueue (whenever a number is read it is pushed to the output)
3. Push + (or its ID) onto the operatorstack
4. Push 4 to the output queue
5. After reading the expression,pop the operators off the stack and add them to the output.

   In this case there is only one, "+".

6. Output:3 4 +

This already shows a couple of rules:

• All numbers are pushed to the output when they are read.
• At the end of reading the expression, pop all operators off the stack and onto the output.

Graphical illustration of algorithm, using athree-way railroad junction. The input is processed one symbol at a time: if a variable or number is found, it is copied directly to the output a), c), e), h). If the symbol is an operator, it is pushed onto the operator stack b), d), f). If the operator's precedence is lower than that of the operators at the top of the stack or the precedences are equal and the operator is left associative, then that operator is popped off the stack and added to the output g). Finally, any remaining operators are popped off the stack and added to the output i).

/* The functions referred to in this algorithm are simple single argument functions such as sine, inverse or factorial. *//* This implementation does not implement composite functions, functions with a variable number of arguments, or unary operators. */while there aretokens to be read:    read a tokenif the token is:    - anumber:        put it into the output queue    - afunction:        push it onto the operator stack     - anoperatoro1:while (            there is an operatoro2 at the top of the operator stack which is not a left parenthesis,and (o2 has greaterprecedence thano1or (o1 ando2 have the same precedenceando1 is left-associative))        ):            popo2 from the operator stack into the output queue        pusho1 onto the operator stack    - a",":while the operator at the top of the operator stack is not a left parenthesis:             pop the operator from the operator stack into the output queue    - aleft parenthesis (i.e. "("):        push it onto the operator stack    - aright parenthesis (i.e. ")"):while the operator at the top of the operator stack is not a left parenthesis:            {assert the operator stack is not empty}/* If the stack runs out without finding a left parenthesis, then there are mismatched parentheses. */            pop the operator from the operator stack into the output queue        {assert there is a left parenthesis at the top of the operator stack}        pop the left parenthesis from the operator stack and discard itif there is a function token at the top of the operator stack,then:            pop the function from the operator stack into the output queue/* After the while loop, pop the remaining items from the operator stack into the output queue. */while there are tokens on the operator stack:/* If the operator token on the top of the stack is a parenthesis, then there are mismatched parentheses. */    {assert the operator on top of the stack is not a (left) parenthesis}    pop the operator from the operator stack onto the output queue

To analyze the running time complexity of this algorithm, one has only to note that each token will be read once, each number, function, or operator will be printed once, and each function, operator, or parenthesis will be pushed onto the stack and popped off the stack once—therefore, there are at most a constant number of operations executed per token, and the running time is thus O(n) — linear in the size of the input.

The shunting yard algorithm can also be applied to produce prefix notation (also known asPolish notation). To do this one would simply start from the end of a string of tokens to be parsed and work backwards, reverse the output queue (therefore making the output queue an output stack), and flip the left and right parenthesis behavior (remembering that the now-left parenthesis behavior should pop until it finds a now-right parenthesis), while making sure to change theassociativity condition to right.

Input:3 + 4 × 2 ÷ ( 1 − 5 ) ^ 2 ^ 3

Operator	Precedence	Associativity
^	4	Right
×	3	Left
÷	3	Left
+	2	Left
−	2	Left

The symbol ^ represents thepower operator.

Token	Action	Output (inRPN)	Operator stack	Notes
3	Add token to output	3
+	Push token to stack	3	+
4	Add token to output	3 4	+
×	Push token to stack	3 4	× +	× has higher precedence than +
2	Add token to output	3 4 2	× +
÷	Pop stack to output	3 4 2 ×	+	÷ and × have same precedence
	Push token to stack	3 4 2 ×	÷ +	÷ has higher precedence than +
(	Push token to stack	3 4 2 ×	( ÷ +
1	Add token to output	3 4 2 × 1	( ÷ +
−	Push token to stack	3 4 2 × 1	− ( ÷ +
5	Add token to output	3 4 2 × 1 5	− ( ÷ +
)	Pop stack to output	3 4 2 × 1 5 −	( ÷ +	Repeated until "(" found
	Pop stack	3 4 2 × 1 5 −	÷ +	Discard matching parenthesis
^	Push token to stack	3 4 2 × 1 5 −	^ ÷ +	^ has higher precedence than ÷
2	Add token to output	3 4 2 × 1 5 − 2	^ ÷ +
^	Push token to stack	3 4 2 × 1 5 − 2	^ ^ ÷ +	^ is evaluated right-to-left
3	Add token to output	3 4 2 × 1 5 − 2 3	^ ^ ÷ +
end	Pop entire stack to output	3 4 2 × 1 5 − 2 3 ^ ^ ÷ +

Input:sin ( max ( 2, 3 ) ÷ 3 ×π )

Token	Action	Output (inRPN)	Operator stack	Notes
sin	Push token to stack		sin
(	Push token to stack		( sin
max	Push token to stack		max ( sin
(	Push token to stack		( max ( sin
2	Add token to output	2	( max ( sin
,	Ignore	2	( max ( sin	The operator at the top of the stack is a left parenthesis
3	Add token to output	2 3	( max ( sin
)	Pop stack to output	2 3	( max ( sin	Repeated until "(" is at the top of the stack
Pop stack	2 3	max ( sin	Discarding matching parentheses
Pop stack to output	2 3 max	( sin	Function at top of the stack
÷	Push token to stack	2 3 max	÷ ( sin
3	Add token to output	2 3 max 3	÷ ( sin
×	Pop stack to output	2 3 max 3 ÷	( sin
	Push token to stack	2 3 max 3 ÷	× ( sin
π	Add token to output	2 3 max 3 ÷π	× ( sin
)	Pop stack to output	2 3 max 3 ÷π ×	( sin	Repeated until "(" is at the top of the stack
	Pop stack	2 3 max 3 ÷π ×	sin	Discarding matching parentheses
	Pop stack to output	2 3 max 3 ÷π × sin		Function at top of the stack
end	Pop entire stack to output	2 3 max 3 ÷π × sin

• Operator-precedence parser
• Stack-sortable permutation

• Dijkstra's original description of the Shunting yard algorithm
• Literate Programs implementation in C
• Demonstration of Shunting yard algorithm in Rust
• Java Applet demonstrating the Shunting yard algorithm
• Parsing Expressions by Recursive Descent Theodore Norvell © 1999–2001. Access date September 14, 2006.
• Matlab code, evaluation of arithmetic expressions using the shunting yard algorithm

• Parsing algorithms
• Dutch inventions

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