Generate Parentheses
Given n pairs of parentheses, generate all combinations of well-formed parentheses.
Output each combination on a separate line, sorted lexicographically.
Example:
3
((())) (()()) (())() ()(()) ()()()
- The problem starts with an input value of
n = 3, representing the number of pairs of parentheses to generate. - The algorithm generates all possible combinations of well-formed parentheses, ensuring that each combination has
npairs of matching parentheses and that no closing parenthesis appears before its corresponding opening parenthesis. - The combinations are generated using a recursive approach, adding open and close parentheses at each step while maintaining a valid sequence: nopen​≥nclose​.
- The resulting combinations are then sorted lexicographically, resulting in the final output: ((())) (()()) (())() ()(()) ()()()
Constraints:
- 1 <= n <= 8
Background Knowledge
The "Generate Parentheses" problem is a classic example of a backtracking problem. Backtracking is an algorithmic technique used to find all possible solutions to a problem by exploring all possible options. It's particularly useful when dealing with problems that have a large solution space, and we need to find all possible solutions. In the context of this problem, we're looking to generate all combinations of well-formed parentheses, which means that every open parenthesis must have a corresponding close parenthesis.
To understand this problem, it's essential to have a solid grasp of recursion, as it's a fundamental concept used in backtracking. Recursion is a programming technique where a function calls itself repeatedly until it reaches a base case that stops the recursion. In the context of backtracking, recursion is used to explore all possible options, and when a dead end is reached, the function backtracks to the previous option and tries the next one. This process continues until all possible solutions have been found.
The concept of well-formed parentheses is also crucial to this problem. A string of parentheses is considered well-formed if it satisfies the following conditions: every open parenthesis has a corresponding close parenthesis, and the string does not contain any unmatched close parentheses. This means that we need to keep track of the number of open and close parentheses as we generate combinations to ensure that they are well-formed.
Algorithm/Approach
The general approach to solving this type of problem is to use a backtracking algorithm with recursion. The algorithm will start by adding an open parenthesis to the current combination, and then it will recursively add more parentheses to the combination. If the combination becomes invalid (i.e., it's not well-formed), the algorithm will backtrack and try a different option. This process will continue until all possible combinations have been generated.
Continue the full explanation
You're reading the free preview. Unlock the complete walkthrough, the code editor, test runner and reference solution with Premium.
Editor locked
The code editor is locked for Pro problems. It is only available for free problems. Please upgrade to gain access to the code editor for all problems.