<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>

from collections import defaultdict, deque
from typing import List

class Solution:
    def validArrangement(self, pairs: List[List[int]]) -> List[List[int]]:
        graph = defaultdict(deque)
        in_degree = defaultdict(int)
        out_degree = defaultdict(int)
        
        # Build the graph and calculate in-degrees and out-degrees
        for u, v in pairs:
            graph[u].append(v)
            out_degree[u] += 1
            in_degree[v] += 1
        
        # Find the start node for Eulerian path
        start = None
        for node in graph:
            if out_degree[node] - in_degree[node] == 1:
                start = node
                break
        if start is None:
            start = pairs[0][0]  # If no specific start, use any node from pairs
        
        # Hierholzer's algorithm to find Eulerian path
        def findEulerianPath(start):
            stack = [start]
            path = []
            while stack:
                while graph[stack[-1]]:
                    next_node = graph[stack[-1]].popleft()
                    stack.append(next_node)
                path.append(stack.pop())
            return path[::-1]
        
        eulerian_path = findEulerianPath(start)
        
        # Convert the path into pairs
        result = [[eulerian_path[i], eulerian_path[i+1]] for i in range(len(eulerian_path) - 1)]
        return result

# Example usage:
# solution = Solution()
# print(solution.validArrangement([[5,1],[4,5],[11,9],[9,4]]))
103916