<p>You are given an array of variable pairs <code>equations</code> and an array of real numbers <code>values</code>, where <code>equations[i] = [A<sub>i</sub>, B<sub>i</sub>]</code> and <code>values[i]</code> represent the equation <code>A<sub>i</sub> / B<sub>i</sub> = values[i]</code>. Each <code>A<sub>i</sub></code> or <code>B<sub>i</sub></code> is a string that represents a single variable.</p>

<p>You are also given some <code>queries</code>, where <code>queries[j] = [C<sub>j</sub>, D<sub>j</sub>]</code> represents the <code>j<sup>th</sup></code> query where you must find the answer for <code>C<sub>j</sub> / D<sub>j</sub> = ?</code>.</p>

<p>Return <em>the answers to all queries</em>. If a single answer cannot be determined, return <code>-1.0</code>.</p>

<p><strong>Note:</strong> The input is always valid. You may assume that evaluating the queries will not result in division by zero and that there is no contradiction.</p>

<p><strong>Note:&nbsp;</strong>The variables that do not occur in the list of equations are undefined, so the answer cannot be determined for them.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> equations = [[&quot;a&quot;,&quot;b&quot;],[&quot;b&quot;,&quot;c&quot;]], values = [2.0,3.0], queries = [[&quot;a&quot;,&quot;c&quot;],[&quot;b&quot;,&quot;a&quot;],[&quot;a&quot;,&quot;e&quot;],[&quot;a&quot;,&quot;a&quot;],[&quot;x&quot;,&quot;x&quot;]]
<strong>Output:</strong> [6.00000,0.50000,-1.00000,1.00000,-1.00000]
<strong>Explanation:</strong> 
Given: <em>a / b = 2.0</em>, <em>b / c = 3.0</em>
queries are: <em>a / c = ?</em>, <em>b / a = ?</em>, <em>a / e = ?</em>, <em>a / a = ?</em>, <em>x / x = ? </em>
return: [6.0, 0.5, -1.0, 1.0, -1.0 ]
note: x is undefined =&gt; -1.0</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> equations = [[&quot;a&quot;,&quot;b&quot;],[&quot;b&quot;,&quot;c&quot;],[&quot;bc&quot;,&quot;cd&quot;]], values = [1.5,2.5,5.0], queries = [[&quot;a&quot;,&quot;c&quot;],[&quot;c&quot;,&quot;b&quot;],[&quot;bc&quot;,&quot;cd&quot;],[&quot;cd&quot;,&quot;bc&quot;]]
<strong>Output:</strong> [3.75000,0.40000,5.00000,0.20000]
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> equations = [[&quot;a&quot;,&quot;b&quot;]], values = [0.5], queries = [[&quot;a&quot;,&quot;b&quot;],[&quot;b&quot;,&quot;a&quot;],[&quot;a&quot;,&quot;c&quot;],[&quot;x&quot;,&quot;y&quot;]]
<strong>Output:</strong> [0.50000,2.00000,-1.00000,-1.00000]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= equations.length &lt;= 20</code></li>
	<li><code>equations[i].length == 2</code></li>
	<li><code>1 &lt;= A<sub>i</sub>.length, B<sub>i</sub>.length &lt;= 5</code></li>
	<li><code>values.length == equations.length</code></li>
	<li><code>0.0 &lt; values[i] &lt;= 20.0</code></li>
	<li><code>1 &lt;= queries.length &lt;= 20</code></li>
	<li><code>queries[i].length == 2</code></li>
	<li><code>1 &lt;= C<sub>j</sub>.length, D<sub>j</sub>.length &lt;= 5</code></li>
	<li><code>A<sub>i</sub>, B<sub>i</sub>, C<sub>j</sub>, D<sub>j</sub></code> consist of lower case English letters and digits.</li>
</ul>

class Solution {
public:
    std::vector<double> calcEquation(std::vector<std::vector<std::string>>& equations, std::vector<double>& values,
                                     std::vector<std::vector<std::string>>& queries) {

        // Step 1: Build the graph
        std::unordered_map<std::string, std::unordered_map<std::string, double>> graph;
        for (int i = 0; i < equations.size(); i++) {
            std::string dividend = equations[i][0];
            std::string divisor = equations[i][1];
            double quotient = values[i];

            graph[dividend][divisor] = quotient;
            graph[divisor][dividend] = 1.0 / quotient;
        }

        // Step 2: Evaluate each query using DFS
        std::vector<double> results;
        for (const auto& query : queries) {
            std::string dividend = query[0];
            std::string divisor = query[1];

            if (!graph.count(dividend) || !graph.count(divisor)) {
                results.push_back(-1.0);  // if either variable doesn't exist in the graph
            } else if (dividend == divisor) {
                results.push_back(1.0);  // if the variables are the same
            } else {
                std::unordered_set<std::string> visited;
                double result = backtrackEvaluate(graph, dividend, divisor, 1.0, visited);
                results.push_back(result);
            }
        }

        return results;
    }

private:
    double backtrackEvaluate(std::unordered_map<std::string, std::unordered_map<std::string, double>>& graph,
                             const std::string& currNode, const std::string& targetNode, double accProduct,
                             std::unordered_set<std::string>& visited) {
        // Mark the current node as visited
        visited.insert(currNode);

        if (graph[currNode].count(targetNode)) {
            // If there is a direct connection between currNode and targetNode
            return accProduct * graph[currNode][targetNode];
        }

        // Otherwise, perform DFS on the neighbors
        for (const auto& neighbor : graph[currNode]) {
            const std::string& nextNode = neighbor.first;
            if (visited.count(nextNode)) continue;  // Skip visited nodes

            double result = backtrackEvaluate(graph, nextNode, targetNode, accProduct * neighbor.second, visited);
            if (result != -1.0) {
                return result;  // If a valid path is found, return the result
            }
        }

        // Unmark the current node (backtrack)
        visited.erase(currNode);
        return -1.0;  // Return -1 if no path is found
    }
};

10600