In computer science, the shunting-yard algorithm is a method for parsing mathematical expressions specified in infix notation. It can produce either a postfix notation string, also known as Reverse Polish notation (RPN), or an abstract syntax tree (AST). The algorithm was invented by Edsger Dijkstra and named the "shunting yard" algorithm because its operation resembles that of a railroad shunting yard. Dijkstra first described the Shunting Yard Algorithm in the Mathematisch Centrum report MR 34/61.
Like the evaluation of RPN, the shunting yard algorithm is stack-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 text variables (strings), the input and the output. There is also a stack 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 "3 4 +" or "3 4 2 1 - × +".
The shunting-yard algorithm was later generalized into operator-precedence parsing.
Video Shunting-yard algorithm
A simple conversion
- Input: 3 + 4
- Push 3 to the output queue (whenever a number is read it is pushed to the output)
- Push + (or its ID) onto the operator stack
- Push 4 to the output queue
- After reading the expression, pop the operators off the stack and add them to the output.
- In this case there is only one, "+".
- 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.
Maps Shunting-yard algorithm
Graphical illustration
Graphical illustration of algorithm, using a three-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 less 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 algorithm in detail
Important terms: Token, Function, Operator associativity, Precedence
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 as Polish 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).
Detailed example
Input: 3 + 4 × 2 ÷ ( 1 - 5 ) ^ 2 ^ 3
The symbol ^ represents the power operator.
Input: sin ( max ( 2, 3 ) ÷ 3 × 3.1415 )
See also
- Operator-precedence parser
- Stack-sortable permutation
External links
- Dijkstra's original description of the Shunting yard algorithm
- Literate Programs implementation in C
- Implementation in various languages, including C and Python
- Java Applet demonstrating the Shunting yard algorithm
- Silverlight widget demonstrating the Shunting yard algorithm and evaluation of arithmetic expressions
- Parsing Expressions by Recursive Descent Theodore Norvell © 1999-2001. Access date September 14, 2006.
Source of article : Wikipedia
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