如何确定数学表达式的求值顺序？

Question

Given a mathematical expression, I would like to find out which are the parts that can be evaluated simultaneously. For example, given the following expression,

(a + b) * c - (d + e) / f

(a + b) and (d + e) could be evaluated simultaneously because they are independent of each other.

Is there any algorithm for this purpose? Or is there even any library that implements this functionality?

Answer 1

This is a short and simple step by step manual for creating math parser. After you have parsed the expression, and you hold a tree representing parsed expression, you can iterate over it, and in every iteration each pairs leaves of the tree will represent an independent expression (that can be evaluated simultaneously). [in every iteration replace the evaluated expressions with their result]

Building Expression Evaluator with Expression Trees in C#

Answer 2

The order of evaluation depends on two factor :

• Precedence : in each language usually there is a table that indicate the precedence of each operator
• Associativity : when an expression involves several operators that have the same precedence, the operator associativity governs the order in which the operations are performed.

If there could be some optimization that involve simultaneous evaluation of parts of an expression, i think could be done only at JVM/CLR level and not with a library ...

Answer 3

I believe you should use the Reversed Polish Notation .
Read the following for more details:

• http://www.dreamincode.net/forums/topic/35320-reverse-polish-notation-in-c%23/
• http://www.calculator.org/rpn.aspx

This is a function written in C++ that converts normal notation to reversed polish notation. someone might help you in converting it to C#:

Answer 4

Ok so let's see the example as a tree:

          -
   (*           /)
(c    +)     (+    f)
   (a   b) (d   e)

(brackets pair the nodes that are both children of the same parent)

Getting that tree if simple, for example with the Shunting Yard algorithm.

Now observe that what you need in order to evaluate a node, are only the children of that node (but recursively so). Therefore (a + b) and (d + e) do not depend on each other, as you note. Also (c * (a + b)) does not depend on (d + e) or on ((d + e) / f) , and ((d + e) / f) does not depend on (a + b) .

In general, taking a node n , any node that's neither a descendant of n not a ancestor, can be evaluated simultaneously. If you're working with a schedule, you'd have to add "if that node can be evaluated now" - clearly you can not evaluate a node before you've evaluated its descendants.

I'm not sure what "this purpose" is what you refer to. What do you want to calculate?