Problem solving with algorithm and data structure

Problem Solving withalgorithm and Data Structure
Infix, Prefix and Postfix Expressions:
When you write an arithmetic expression such as B * C, the form of the expression provides you
with information so that you can interpret it correctly. In this case we know that the variable B
is being multiplied by the variable C since the multiplication operator * appears between them
in the expression. This type of notation is referred to as infix since the operator is in
between the two operands that it is working on.
Consider another infix example, A + B * C. The operators + and * still appear between the
operands, but there is a problem. Which operands do they work on? Does the + work on A and
B or does the * take B and C? The expression seems ambiguous.
In fact, you have been reading and writing these types of expressions for a long time and they
do not cause you any problem. The reason for this is that you know something about the
operators + and *. Each operator has a precedence level. Operators of higher precedence are
used before operators of lower precedence. The only thing that can change that order is the
presence of parentheses. The precedence order for arithmetic operators places multiplication
and division above addition and subtraction. If two operators of equal precedence appear, then
a left-to-right ordering or associatively is used.
Let’s interpret the troublesome expression A + B * C using operator precedence. B and C are
multiplied first, and A is then added to that result. (A + B) * C would force the addition of A and
B to be done first before the multiplication. In expression A + B + C, by precedence (via
associatively), the leftmost + would be done first.
Although all this may be obvious to you, remember that computers need to know exactly what
operators to perform and in what order. One way to write an expression that guarantees there
will be no confusion with respect to the order of operations is to create what is called a fully
parenthesized expression. This type of expression uses one pair of parentheses for each
operator. The parentheses dictate the order of operations; there is no ambiguity. There is also
no need to remember any precedence rules.
The expression A + B * C + D can be rewritten as ((A + (B * C)) + D) to show that the
multiplication happens first, followed by the leftmost addition. A + B + C + D can be written as
(((A + B) + C) + D) since the addition operations associate from left to right.
There are two other very important expression formats that may not seem obvious to you at
first. Consider the infix expression A + B. What would happen if we moved the operator before
the two operands? The resulting expression would be + A B. Likewise, we could move the
operator to the end. We would get A B +. These look a bit strange.
These changes to the position of the operator with respect to the operands create two new
expression formats, prefix and postfix. Prefix expression notation requires that all operators
precede the two operands that they work on. Postfix, on the other hand, requires that its

operators come after the corresponding operands. A few more examples should help to make
this a bit clearer (see Table 2).
A + B * C would be written as + A * B C in prefix. The multiplication operator comes
immediately before the operands B and C, denoting that * has precedence over +. The addition
operator then appears before the A and the result of the multiplication.
In postfix, the expression would be A B C * +. Again, the order of operations is preserved since
the * appears immediately after the B and the C, denoting that * has precedence, with +
coming after. Although the operators moved and now appear either before or after their
respective operands, the order of the operands stayed exactly the same relative to one
another.
Table 2: Examples of Infix, Prefix, and Postfix
Infix Expression Prefix Expression Postfix Expression
A + B + A B A B +
A + B * C + A * B C A B C * +
Now consider the infix expression (A + B) * C. Recall that in this case, infix requires the parentheses
to force the performance of the addition before the multiplication. However, when A + B was written
in prefix, the addition operator was simply moved before the operands, + A B. The result of this
operation becomes the first operand for the multiplication. The multiplication operator is moved in
front of the entire expression, giving us * + A B C. Likewise, in postfix A B + forces the addition to
happen first. The multiplication can be done to that result and the remaining operand C. The proper
postfix expression is then A B + C *.
Consider these three expressions again (see Table 3). Something very important has happened.
Where did the parentheses go? Why don’t we need them in prefix and postfix? The answer is that
the operators are no longer ambiguous with respect to the operands that they work on. Only infix
notation requires the additional symbols. The order of operations within prefix and postfix
expressions is completely determined by the position of the operator and nothing else. In many
ways, this makes infix the least desirable notation to use.
Table 3: An Expression with Parentheses

Table 3: An Expression with Parentheses
(A + B) * C * + A B C A B + C *
Table 4 shows some additional examples of infix expressions and the equivalent prefix and postfix
expressions. Be sure that you understand how they are equivalent in terms of the order of the
operations being performed.
Table 4: Additional Examples of Infix, Prefix, and Postfix
A + B * C + D + + A * B C D A B C * + D +
(A + B) * (C + D) * + A B + C D A B + C D + *
A * B + C * D + * A B * C D A B * C D * +
A + B + C + D + + + A B C D A B + C + D +
Conversion of InfixExpressionsto Prefixand Postfix
So far, we have used ad hoc methods to convert between infix expressions and the equivalent
prefix and postfix expression notations. As you might expect, there are algorithmic ways to
perform the conversion that allow any expression of any complexity to be correctly
transformed.
The first technique that we will consider uses the notion of a fully parenthesized expression
that was discussed earlier. Recall that A + B * C can be written as (A + (B * C)) to show explicitly
that the multiplication has precedence over the addition. On closer observation, however, you
can see that each parenthesis pair also denotes the beginning and the end of an operand pair
with the corresponding operator in the middle.
Look at the right parenthesis in the sub expression (B * C) above. If we were to move the
multiplication symbol to that position and remove the matching left parenthesis, giving us B C

*, we would in effect have converted the sub expression to postfix notation. If the addition
operator were also moved to its corresponding right parenthesis position and the matching left
parenthesis were removed, the complete postfix expression would result (see Figure 6).
Figure 6: Moving Operators to the Right for Postfix Notation
If we do the same thing but instead of moving the symbol to the position of the right
parenthesis, we move it to the left, we get prefix notation (see Figure 7). The position of the
parenthesis pair is actually a clue to the final position of the enclosed operator.
Figure 7: Moving Operators to the Left for Prefix Notation
So in order to convert an expression, no matter how complex, to either prefix or postfix
notation, fully parenthesize the expression using the order of operations. Then move the
enclosed operator to the position of either the left or the right parenthesis depending on
whether you want prefix or postfix notation.
Here is a more complex expression: (A + B) * C - (D - E) * (F + G). Figure 8 shows the conversion
to postfix and prefix notations.
Figure 8: Converting a Complex Expression to Prefix and Postfix Notations

Problem solving with algorithm and data structure

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