In computer science, the shunting yard algorithm is a method for parsing arithmetical or logical expressions, or a combination of both, specified in infix notation.

This already shows a couple of rules: 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).

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.

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 the associativity condition to right.