Matching Balancing Parenthesis

1
Applications of Stacks

• Matching Balancing Parenthesis
• Evaluating an postfix expression
• Infix to postfix expression

2
Matching Balancing Parenthesis

• Given an expression of ( and ), a ( must
  match with a ), or else it is illegal.
• ( )( ), ( ( ( ) ) ), ( ( ( ) ) ( ) ) are all
  legal, while
• ( ( ) (, ) ( ) ( are all illegal.
• Obviously, counting the numbers of ( and ) in
  the expression is not enough.

3
Matching Balancing Parenthesis

• Insights ? By convention, the expression is
  reading from left-to-right. A ) is matched
  with the closest, unmatched ( on its left.
• For example,
• ( ( ( ) ) ( ) )

4
Matching Balancing Parenthesis

• Suppose we have a ). How do we know which (
  is closest and unmatched?
• If we read the expression from left-to-right, the
  MOST RECENTLY UNMATCHED ( is cancelled with
  ).
• How can we keep track of the MOST RECENTLY READ
  (LAST) ( ? (keep in mind there are many pending
  unmatched ().
• Which data structure is keeping track of the most
  recent item ?

Stack ? LIFO structure
5
Matching Balancing Parenthesis

• When we see a (, we push it into a stack.
• When we see a ), we pop ( from the stack.
  This ( matches with current ).
• What happen if the stack is empty (meaning there
  is no such () at that moment ?
• What happen if we have finished reading the
  expression, but the stack is not empty ?
• It means there are more ( than ) in the
  expression.

It means the input is illegal.
It means the input is illegal.
6
Infix to Postfix Expressions

• Infix expression
• Suppose the expression only has operators and
  and has a higher precedence than .
• So, 523 10, and 1249, etc.
• The expression may also have parenthesis to
  complicate the problem, i.e.,
• (12)412
• (1251)336.
• (12(51))339.

7
Infix to Postfix Expressions

• Postfix expression
• 1 3
• 1 2 4
• 1 2 4
• 6 5 2 3 8 3
• are all postfix expressions.
• No (, ) is in the expression.
• To evaluate a postfix expression, we need a stack.

8
Infix to Postfix Expressions

• Evaluate a postfix expression.
• Read the expression from left-to-right.
• When a number is seen, it is pushed onto the
  stack.
• When an operator is seen, the operator is applied
  to the two numbers that are popped from the
  stack. The result is pushed on the stack.

9
Infix to Postfix Expressions

• Example 6 5 2 3 8 3

10
Infix to Postfix Expressions

• How to convert an infix expression to a postfix
  expression? Use a stack.
• Read the infix expression from left-to-right.
• When an operand (number) is read, output it.
• If an operator (other than (, )) is read, pop
  the stack (and output the operator) until a lower
  precedence operator or ( is on the top of
  stack. Then push the current operator on the
  stack.
• If ( is read, push it on the stack.
• If a ) is read, pop the stacks (and output the
  operators), until we meet (. Pop that ( (do
  no output it).
• Finally, if we read the end of the expression,
  pop the stack (and output the operators) until
  the stack is empty.

11
Infix to Postfix Expressions

• Example (12(51))3
• postfix expression ? 1 2 5 1 3