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 Problem Statement for ParenthesisRemoval

Problem Statement


Correct parentheses sequences can be defined recursively as follows:

  • The empty string "" is a correct sequence.
  • If "X" and "Y" are correct sequences, then "XY" (the concatenation of X and Y) is a correct sequence.
  • If "X" is a correct sequence, then "(X)" is a correct sequence.
  • Each correct parentheses sequence can be derived using the above rules.

Examples of correct parentheses sequences include "", "()", "()()()", "(()())", and "(((())))".

You are given a String s that is guaranteed to be a correct parentheses sequence. A removal is an action that consists of two steps:

  1. Remove the first opening parenthesis in s.
  2. Remove one closing parenthesis in s. After you do so, s must again be a correct parentheses sequence.

Compute and return the number of distinct ways in which s can be reduced to an empty string by performing consecutive removals, modulo 10^9+7. Two ways are considered distinct if there is a step in which you remove a different closing parenthesis. (See Example 1 for clarification.)



Method signature:int countWays(String s)
(be sure your method is public)


-s will have between 2 and 2,500 characters, inclusive.
-s will be a correct parentheses sequence.


Returns: 1
In each removal we have to choose the leftmost closing parenthesis.
Returns: 24
In each removal we can choose any closing parenthesis we want. Note that these count as distint choices, even though all choices lead to the same string. Thus, there are 4*3*2*1 = 24 different sequences of removals that change s into an empty string.
Returns: 54
Below is one of the 54 possible sequences of removals. Remember that in each step we also remove the first opening parenthesis.
  • Remove the fourth closing parenthesis: "(()()())"
  • Remove the second closing parenthesis: "()(())"
  • Remove the first closing parenthesis: "(())"
  • Remove the second closing parenthesis: "()"
  • Remove the first closing parenthesis: ""
Returns: 8
Returns: 948334170
Don't forget about the mod.

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This problem was used for:
       Single Round Match 714 Round 1 - Division I, Level One
       2017 TCO Algorithm Russia Regional Round - Division I, Level Two