Polish notation (prefix Notation)

 

Polish notation, also called prefix notation.

 

When calculating a polish expression, you only need to find the operator of the first operation without remembering the operation hierarchy. Take the binary operation as an example. Read the expression from left to right. When an operator follows two operands, it is calculated and the result is replaced by the operator and two operands as the operand; repeat this step until all operators are processed. In the correct prefix expression, the number of operations must be one more than the operator, so an operator must be found to meet the operation conditions. In the replacement, two operations and one operator must be replaced with one, therefore, each operator and operand are reduced, and the operation can still be iterated until the entire formula is calculated. The same is true for multivariate operations. When there are enough operands, an operation is generated until iteration is completed. The condition for iteration end is guaranteed by the correctness of the expression. The following is an example of the operation sequence of each step:

The operator is placed before the operand, so it is also called prefix notation. If the number of elements (Arity) of the operator is fixed, the operators can still be parsed without parentheses.

 

Calculation method:

 

When calculating a polish expression, you only need to find the operator of the first operation without remembering the operation hierarchy. Take the binary operation as an example. Read the expression from left to right. When an operator follows two operands, it is calculated and the result is replaced by the operator and two operands as the operand; repeat this step until all operators are processed. In the correct prefix expression, the number of operations must be one more than the operator, so an operator must be found to meet the operation conditions. In the replacement, two operations and one operator must be replaced with one, therefore, each operator and operand are reduced, and the operation can still be iterated until the entire formula is calculated. The same is true for multivariate operations. When there are enough operands, an operation is generated until iteration is completed. The condition for iteration end is guaranteed by the correctness of the expression. The following is an example of the operation sequence of each step:

− × Listen 15 − 7 + 1 1 3 + 2 + 1 1 =

− × Listen 15 − 7 2 3 + 2 + 1 1 =

−×000015 5 3 + 2 + 1 1 =

−×3 3 + 2 + 1 1 =

− 9 + 2 + 1 1 =

− 9 + 2 2 =

− 9 4 =

5

Equivalent infix expression: (15 Gbit/s (7−( 1 + 1) × 3) −( 2 + (1 + 1) = 5

The following pseudocode uses a stack to obtain the prefix value. Note: Unlike the algorithms processed from left to right, the algorithms scan from right to left, but the values calculated by the two algorithms are the same. (In fact, this algorithm is equivalent to traversing the right subtree in the next traversal)

Scan the given prefix expression from right to left

For each symbol

{

If operand then

Push onto Stack

If operator then

{

Operand1 = pop Stack

Operand2 = pop Stack

Compute operand1 operator operand2

Push result onto Stack

}

}

Return top of stack as result

 

Applying this algorithm to the example abve yields the following:

− × Listen 15 − 7 + 1 1 3 + 2 + 1 1 =

− × Listen 15 − 7 + 1 1 3 + 2 2 =

− × Listen 15 − 7 + 1 1 3 4 =

−×000015 −7 2 3 4 =

−×000015 5 3 4 =

−×3 3 4 =

− 9 4 =

5

This uses the same expression as before and the algorithm above.

−×000015 −7 + 1 1 3 + 2 + 1 1

 

Token	Action	Stack	Notes
1	Operand	1	Push onto stack.
1	Operand	1 1	Push onto stack.
+	Operator	2	Pop the two operands (1, 1), calculate (1 + 1 = 2) and push onto stack.
2	Operand	2 2	Push onto stack.
+	Operator	4	Pop the two operands (2, 2), calculate (2 + 2 = 4) and push onto stack.
3	Operand	3 4	Push onto stack.
1	Operand	1 3 4	Push onto stack.
1	Operand	1 1 3 4	Push onto stack.
+	Operator	2 3 4	Pop the two operands (1, 1), calculate (1 + 1 = 2) and push onto stack.
7	Operand	7 2 3 4	Push onto stack.
−	Operator	5 3 4	Pop the two operands (7, 2), calculate (7−2 = 5) and push onto stack.
15	Operand	15 5 3 4	Push onto stack.
Bytes	Operator	3 3 4	Pop the two operands (15, 5), calculate (15 minutes 5 = 3) and push onto stack.
×	Operator	9 4	Pop the two operands (3, 3), calculate (3 × 3 = 9) and push onto stack.
−	Operator	5	Pop the two operands (9, 4), calculate (9 −4 = 5) and push onto stack.

The result is at the top of the stack.

 

Refer:

Http://en.wikipedia.org/wiki/Polish_notation and its Chinese Link

 

 

 

Polish notation (prefix Notation)