tl;dr: Given a string of digits , insert a minimum number of opening and closing parentheses into it such that the resulting string is balanced and each digit is inside exactly pairs of matching parentheses.
Let the nesting of two parentheses within a string be the substring that occurs strictly between them. An opening parenthesis and a closing parenthesis that is further to its right are said to match if their nesting is empty, or if every parenthesis in their nesting matches with another parenthesis in their nesting. The nesting depth of a position is the number of pairs of matching parentheses such that is included in the nesting of .
For example, in the following strings, all digits match their nesting
depth: 0((2)1)
, (((3))1(2))
, ((((4))))
,
((2))((2))(1)
. The first three strings have minimum length among
those that have the same digits in the same order, but the last one does not
since ((22)1)
also has the digits 221
and is
shorter.
Given a string of digits , find another string , comprised of parentheses and digits, such that:
- all parentheses in match some other parenthesis,
- removing any and all parentheses from results in ,
- each digit in is equal to its nesting depth, and
- is of minimum length.
Input Specification
The first line of the input gives the number of test cases, . lines follow. Each line represents a test case and contains only the string .
Output Specification
For each test case, output one line containing Case #x: y
, where
is the test case number (starting from 1) and
is the string defined above.
Limits
Time limit: 20 seconds per test set.
Memory limit: 1GB.
.
.
Test set 1
Each character in is either 0
or 1
.
Test set 2
Each character in is a decimal digit between 0
and
9
, inclusive.
Sample Input
4
0000
101
111000
1
Sample Output
Case #1: 0000
Case #2: (1)0(1)
Case #3: (111)000
Case #4: (1)
The strings ()0000()
, (1)0(((()))1)
and
(1)(11)000
are not valid solutions to Sample Cases #1, #2 and #3,
respectively, only because they are not of minimum length. In addition,
1)(
and )(1
are not valid solutions to Sample Case #4 because they
contain unmatched parentheses and the nesting depth is 0
at the position where there is a 1.
You can create sample inputs that are valid only for Test Set 2 by removing the parentheses from the example strings mentioned in the problem statement.
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