2097. Valid Arrangement of Pairs

Difficulty: Hard

Topics: Depth-First Search, Graph, Eulerian Circuit

You are given a 0-indexed 2D integer array pairs where pairs[i] = [starti, endi]. An arrangement of pairs is valid if for every index i where 1 <= i < pairs.length, we have endi-1 == starti.

Return any valid arrangement of pairs.

Note: The inputs will be generated such that there exists a valid arrangement of pairs.

Example 1:

• Input: pairs = [[5,1],[4,5],[11,9],[9,4]]
• Output: [[11,9],[9,4],[4,5],[5,1]]
• Explanation: This is a valid arrangement since end_i-1 always equals start_i.
  • end₀ = 9 == 9 = start₁
  • end₁ = 4 == 4 = start₂
  • end₂ = 5 == 5 = start₃

Example 2:

• Input: pairs = [[1,3],[3,2],[2,1]]
• Output: [[1,3],[3,2],[2,1]]
• Explanation: This is a valid arrangement since end_i-1 always equals start_i.
  • end₀ = 3 == 3 = start₁
  • end₁ = 2 == 2 = start₂
  • The arrangements [[2,1],[1,3],[3,2]] and [[3,2],[2,1],[1,3]] are also valid.

Example 3:

• Input: pairs = [[1,2],[1,3],[2,1]]
• Output: [[1,2],[2,1],[1,3]]
• Explanation: This is a valid arrangement since end_i-1 always equals start_i.
  • end₀ = 2 == 2 = start₁
  • end₁ = 1 == 1 = start₂

Constraints:

• 1 <= pairs.length <= 105
• pairs[i].length == 2
• 0 <= starti, endi <= 109
• starti != endi
• No two pairs are exactly the same.
• There exists a valid arrangement of pairs.

Hint:

1. Could you convert this into a graph problem?
2. Consider the pairs as edges and each number as a node.
3. We have to find an Eulerian path of this graph. Hierholzer’s algorithm can be used.

Solution:

We can approach it as an Eulerian Path problem in graph theory. In this case, the pairs can be treated as edges, and the values within the pairs (the start and end) can be treated as nodes. We need to find an Eulerian path, which is a path that uses every edge exactly once, and the end of one edge must match the start of the next edge.

Key Steps:

1. Graph Representation: Each unique number in the pairs will be a node, and each pair will be an edge from start[i] to end[i].
2. Eulerian Path Criteria:
   • An Eulerian path exists if there are exactly two nodes with odd degrees, and the rest must have even degrees.
   • We need to make sure that the graph is connected (though this is guaranteed by the problem statement).
3. Hierholzer's Algorithm: This algorithm can be used to find the Eulerian path. It involves:
   • Starting at a node with an odd degree (if any).
   • Traversing through the edges, marking them as visited.
   • If a node is reached with unused edges, continue traversing until all edges are used.

Plan:

• Build a graph using a hash map to store the adjacency list (each node and its connected nodes).
• Track the degree (in-degree and out-degree) of each node.
• Use Hierholzer's algorithm to find the Eulerian path.

Let's implement this solution in PHP: 2097. Valid Arrangement of Pairs

<?php
/**
 * @param Integer[][] $pairs
 * @return Integer[][]
 */
function validArrangement($pairs) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Example usage:
$pairs1 = [[5, 1], [4, 5], [11, 9], [9, 4]];
$pairs2 = [[1, 3], [3, 2], [2, 1]];
$pairs3 = [[1, 2], [1, 3], [2, 1]];

print_r(validArrangement($pairs1)); // Output: [[11, 9], [9, 4], [4, 5], [5, 1]]
print_r(validArrangement($pairs2)); // Output: [[1, 3], [3, 2], [2, 1]]
print_r(validArrangement($pairs3)); // Output: [[1, 2], [2, 1], [1, 3]]
?>

Explanation:

1. Graph Construction:

   • We build the graph using an adjacency list where each key is a start node, and the value is a list of end nodes.
   • We also maintain the out-degree and in-degree for each node, which will help us find the start node for the Eulerian path.

2. Finding the Start Node:

   • An Eulerian path starts at a node where the out-degree is greater than the in-degree by 1 (if such a node exists).
   • If no such node exists, the graph is balanced, and we can start at any node.

3. Hierholzer's Algorithm:

   • We start from the startNode and repeatedly follow edges, marking them as visited by removing them from the adjacency list.
   • Once we reach a node with no more outgoing edges, we backtrack and build the result.

4. Return the Result:

   • The result is constructed in reverse order because of the way we backtrack, so we reverse it at the end.

Example Output:

Array
(
    [0] => Array
        (
            [0] => 11
            [1] => 9
        )

    [1] => Array
        (
            [0] => 9
            [1] => 4
        )

    [2] => Array
        (
            [0] => 4
            [1] => 5
        )

    [3] => Array
        (
            [0] => 5
            [1] => 1
        )
)

Time Complexity:

• Building the graph: O(n), where n is the number of pairs.
• Hierholzer's Algorithm: O(n), because each edge is visited once.
• Overall Time Complexity: O(n).

This approach efficiently finds a valid arrangement of pairs by treating the problem as an Eulerian path problem in a directed graph.

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