mirror of
https://github.com/TheAlgorithms/Ruby
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102 lines
4.1 KiB
Ruby
102 lines
4.1 KiB
Ruby
# Given n pairs of parentheses, write a function to generate all combinations
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# of well-formed parentheses.
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#
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# Example 1:
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#
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# Input: n = 3
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# Output: ["((()))","(()())","(())()","()(())","()()()"]
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# Example 2:
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#
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# Input: n = 1
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# Output: ["()"]
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#
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#
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# Constraints:
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#
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# 1 <= n <= 8
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# Approach:
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#
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# Let's only add '(' or ')' when we know it will remain a valid sequence.
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# We can do this by keeping track of the number of opening and closing brackets
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# we have placed so far.
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#
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# We can start an opening bracket if we still have one (of n) left to place.
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# And we could start a closing bracket if it'd not exceed the number of opening
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# brackets.
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# Complexity Analysis
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#
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# Time Complexity: O(4^n/sqrt(n)). Each valid sequence has at most n steps during the backtracking procedure.
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# Space Complexity: O(4^n/sqrt(n)), as described above, and using O(n) space to store the sequence.
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# Refer to the attached diagram for recursion,
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# The numbers next to each node are the counts of left and right parantheses
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# @param {Integer} n
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# @return {String[]}
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def generate_parenthesis(n)
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parenthesis = []
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backtrack(parenthesis, '', 0, 0, n)
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parenthesis
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end
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def backtrack(parenthesis, curr, open, close, max)
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if curr.length == max * 2
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parenthesis.push(curr)
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return
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end
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backtrack(parenthesis, curr + '(', open + 1, close, max) if open < max
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backtrack(parenthesis, curr + ')', open, close + 1, max) if close < open
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end
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n = 3
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print(generate_parenthesis(n))
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# Output: ["((()))","(()())","(())()","()(())","()()()"]
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# *** Example: n = 3 *** Space after each DFS instance
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# backtrack called with 0 0 []
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# backtrack called with ( 1 0 []
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# backtrack called with (( 2 0 []
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# backtrack called with ((( 3 0 []
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# backtrack called with ((() 3 1 []
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# backtrack called with ((()) 3 2 []
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# backtrack called with ((())) 3 3 []
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# backtrack return with ((()) 3 2 ["((()))"]
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# backtrack return with ((() 3 1 ["((()))"]
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# backtrack return with ((( 3 0 ["((()))"]
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# backtrack called with (() 2 1 ["((()))"]
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# backtrack called with (()( 3 1 ["((()))"]
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# backtrack called with (()() 3 2 ["((()))"]
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# backtrack called with (()()) 3 3 ["((()))"]
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# backtrack return with (()() 3 2 ["((()))", "(()())"]
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# backtrack return with (()( 3 1 ["((()))", "(()())"]
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# backtrack called with (()) 2 2 ["((()))", "(()())"]
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# backtrack called with (())( 3 2 ["((()))", "(()())"]
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# backtrack called with (())() 3 3 ["((()))", "(()())"]
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# backtrack return with (())( 3 2 ["((()))", "(()())", "(())()"]
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# backtrack return with (()) 2 2 ["((()))", "(()())", "(())()"]
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# backtrack return with (() 2 1 ["((()))", "(()())", "(())()"]
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# backtrack return with (( 2 0 ["((()))", "(()())", "(())()"]
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# backtrack called with () 1 1 ["((()))", "(()())", "(())()"]
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# backtrack called with ()( 2 1 ["((()))", "(()())", "(())()"]
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# backtrack called with ()(( 3 1 ["((()))", "(()())", "(())()"]
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# backtrack called with ()(() 3 2 ["((()))", "(()())", "(())()"]
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# backtrack called with ()(()) 3 3 ["((()))", "(()())", "(())()"]
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# backtrack return with ()(() 3 2 ["((()))", "(()())", "(())()", "()(())"]
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# backtrack return with ()(( 3 1 ["((()))", "(()())", "(())()", "()(())"]
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# backtrack called with ()() 2 2 ["((()))", "(()())", "(())()", "()(())"]
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# backtrack called with ()()( 3 2 ["((()))", "(()())", "(())()", "()(())"]
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# backtrack called with ()()() 3 3 ["((()))", "(()())", "(())()", "()(())"]
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# backtrack return with ()()( 3 2 ["((()))", "(()())", "(())()", "()(())", "()()()"]
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# backtrack return with ()() 2 2 ["((()))", "(()())", "(())()", "()(())", "()()()"]
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# backtrack return with ()( 2 1 ["((()))", "(()())", "(())()", "()(())", "()()()"]
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# backtrack return with () 1 1 ["((()))", "(()())", "(())()", "()(())", "()()()"]
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# backtrack return with ( 1 0 ["((()))", "(()())", "(())()", "()(())", "()()()"]
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# backtrack return with 0 0 ["((()))", "(()())", "(())()", "()(())", "()()()"]
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n = 1
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print(generate_parenthesis(n))
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# Output: ["()"]
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