Generate Parentheses

本博客介绍了一种使用回溯法解决给定括号数生成所有合法括号组合的方法,包括核心解题思路、约束条件和具体实现代码。

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题目:

Given n pairs of parentheses, write a function to generate all combinations of well-formed parentheses.

For example, given n = 3, a solution set is:

"((()))", "(()())", "(())()", "()(())", "()()()"

题目大意:给定n个括号求解括号配对的所有可能情况。

解题思路:回溯法,三要素见下。

回溯约束:匹配过程中左括号数目必定大于或等于右括号数目而且左括号的数目必定小于等于总数的一半。

回溯策略:解空间如下

'(',')'
'(',')'
'(',')'
'(',')'
...
逐次深度搜索,当row == 2n则输出结果;当cols[row] == 2 列数大于解空间的列数,回溯;当不满足约束函数,回溯;否则深度搜索。代码如下:
class Solution {
public:
	bool valid(vector<int> &cols, int &row, vector<int> &signal){
		int total = 0;
		int left = 0;
		for(int i = 0; i <= row; ++i){
			total += signal[cols[i]];
			if(signal[cols[i]] > 0)
				left++;
			if(left > cols.size() / 2)
				return false;
		}
		if(total >= 0)
			return true;
		else
			return false;
	}

	vector<string> generateParenthesis(int n) {
		vector<string> result;
		vector<int> cols(n*2, 0);
		vector<int> signal{1,-1};
		int row = 0;

		while(1){
			if(row == 2*n){
				string temp(2*n, '(');
				for(int i = 0; i < 2*n; ++i)
					if(signal[cols[i]] < 0)
						temp[i] = ')';
				result.push_back(temp);
				cols[--row]++;
			}else if(cols[row] == 2){
				cols[row--] = 0;
				if(row == -1)
					break;
				cols[row]++;
			}else if(!valid(cols, row, signal)){
				cols[row]++;
			}else{
				row++;
			}
		}
		return result;
	}
};

#include <cassert> /// for assert #include <iostream> /// for I/O operation #include <vector> /// for vector container /** * @brief Backtracking algorithms * @namespace backtracking */ namespace backtracking { /** * @brief generate_parentheses class */ class generate_parentheses { private: std::vector<std::string> res; ///< Contains all possible valid patterns void makeStrings(std::string str, int n, int closed, int open); public: std::vector<std::string> generate(int n); }; /** * @brief function that adds parenthesis to the string. * * @param str string build during backtracking * @param n number of pairs of parentheses * @param closed number of closed parentheses * @param open number of open parentheses */ void generate_parentheses::makeStrings(std::string str, int n, int closed, int open) { if (closed > open) // We can never have more closed than open return; if ((str.length() == 2 * n) && (closed != open)) { // closed and open must be the same return; } if (str.length() == 2 * n) { res.push_back(str); return; } makeStrings(str + ')', n, closed + 1, open); makeStrings(str + '(', n, closed, open + 1); } /** * @brief wrapper interface * * @param n number of pairs of parentheses * @return all well-formed pattern of parentheses */ std::vector<std::string> generate_parentheses::generate(int n) { backtracking::generate_parentheses::res.clear(); std::string str = "("; generate_parentheses::makeStrings(str, n, 0, 1); return res; } } // namespace backtracking /** * @brief Self-test implementations * @returns void */ static void test() { int n = 0; std::vector<std::string> patterns; backtracking::generate_parentheses p; n = 1; patterns = {{"()"}}; assert(p.generate(n) == patterns); n = 3; patterns = {{"()()()"}, {"()(())"}, {"(())()"}, {"(()())"}, {"((()))"}}; assert(p.generate(n) == patterns); n = 4; patterns = {{"()()()()"}, {"()()(())"}, {"()(())()"}, {"()(()())"}, {"()((()))"}, {"(())()()"}, {"(())(())"}, {"(()())()"}, {"(()()())"}, {"(()(()))"}, {"((()))()"}, {"((())())"}, {"((()()))"}, {"(((())))"}}; assert(p.generate(n) == patterns); std::cout << "All tests passed\n"; } /** * @brief Main function * @returns 0 on exit */ int main() { test(); // run self-test implementations return 0; } 解释一下这段代码?
最新发布
03-08
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