<p>You are given a <strong>0-indexed</strong> 2D integer array <code>pairs</code> where <code>pairs[i] = [start<sub>i</sub>, end<sub>i</sub>]</code>. An arrangement of <code>pairs</code> is <strong>valid</strong> if for every index <code>i</code> where <code>1 &lt;= i &lt; pairs.length</code>, we have <code>end<sub>i-1</sub> == start<sub>i</sub></code>.</p>

<p>Return <em><strong>any</strong> valid arrangement of </em><code>pairs</code>.</p>

<p><strong>Note:</strong> The inputs will be generated such that there exists a valid arrangement of <code>pairs</code>.</p>

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

<pre>
<strong>Input:</strong> pairs = [[5,1],[4,5],[11,9],[9,4]]
<strong>Output:</strong> [[11,9],[9,4],[4,5],[5,1]]
<strong>Explanation:
</strong>This is a valid arrangement since end<sub>i-1</sub> always equals start<sub>i</sub>.
end<sub>0</sub> = 9 == 9 = start<sub>1</sub> 
end<sub>1</sub> = 4 == 4 = start<sub>2</sub>
end<sub>2</sub> = 5 == 5 = start<sub>3</sub>
</pre>

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

<pre>
<strong>Input:</strong> pairs = [[1,3],[3,2],[2,1]]
<strong>Output:</strong> [[1,3],[3,2],[2,1]]
<strong>Explanation:</strong>
This is a valid arrangement since end<sub>i-1</sub> always equals start<sub>i</sub>.
end<sub>0</sub> = 3 == 3 = start<sub>1</sub>
end<sub>1</sub> = 2 == 2 = start<sub>2</sub>
The arrangements [[2,1],[1,3],[3,2]] and [[3,2],[2,1],[1,3]] are also valid.
</pre>

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

<pre>
<strong>Input:</strong> pairs = [[1,2],[1,3],[2,1]]
<strong>Output:</strong> [[1,2],[2,1],[1,3]]
<strong>Explanation:</strong>
This is a valid arrangement since end<sub>i-1</sub> always equals start<sub>i</sub>.
end<sub>0</sub> = 2 == 2 = start<sub>1</sub>
end<sub>1</sub> = 1 == 1 = start<sub>2</sub>
</pre>

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

<ul>
	<li><code>1 &lt;= pairs.length &lt;= 10<sup>5</sup></code></li>
	<li><code>pairs[i].length == 2</code></li>
	<li><code>0 &lt;= start<sub>i</sub>, end<sub>i</sub> &lt;= 10<sup>9</sup></code></li>
	<li><code>start<sub>i</sub> != end<sub>i</sub></code></li>
	<li>No two pairs are exactly the same.</li>
	<li>There <strong>exists</strong> a valid arrangement of <code>pairs</code>.</li>
</ul>

class Solution:
    def validArrangement(self, pairs: List[List[int]]) -> List[List[int]]:

        adjacencyMatrix = defaultdict(deque)
        in_degree = defaultdict(int)
        out_degree = defaultdict(int)

        # Populate adjacency matrix and degree counts
        for start, end in pairs:
            adjacencyMatrix[start].append(end)
            out_degree[start] += 1
            in_degree[end] += 1

        # Find the start node (out_degree = in_degree + 1)
        start_node = None
        for node in adjacencyMatrix.keys():
            if out_degree[node] == in_degree[node] + 1:
                start_node = node
                break

        # If no specific start node, start from the first node in the pairs
        if start_node is None:
            start_node = pairs[0][0]

        # Depth-First Search to find the Eulerian path
        result = []
        def visit(node):
            while adjacencyMatrix[node]:
                next_node = adjacencyMatrix[node].popleft()
                visit(next_node)
            result.append(node)

        visit(start_node)

        # Reverse the result to get the correct path
        result.reverse()

        # Convert the result to pairs
        paired_result = [[result[i], result[i+1]] for i in range(len(result) - 1)]

        return paired_result




        #         An Eulerian path in a graph is a trail that visits every edge exactly once. To find an Eulerian path in a directed graph, the following conditions must be met:

# # The graph must be connected (ignoring the direction of edges).
# # There must be exactly one vertex with (in-degree - out-degree) = 1 
# (potential starting point of the path).
# # There must be exactly one vertex with (out-degree - in-degree) = 1 
# (potential ending point of the path).
# # All other vertices must have equal in-degree and out-degree.
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