Valid Arrangement of Pairs

Problem Description

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

Key Insights

• The problem can be modeled as a directed graph where each pair represents an edge from start to end.
• We need to find an Eulerian path in this directed graph, which requires that all vertices have equal indegree and outdegree, except for at most one vertex which can have one extra outdegree (the starting point) and another with one extra indegree (the endpoint).
• Depth-First Search (DFS) can be used to traverse the graph and construct the valid arrangement.

Space and Time Complexity

Time Complexity: O(n) - where n is the number of pairs, since we process each pair once. Space Complexity: O(n) - for storing the graph and the output arrangement.

Solution

To solve the problem, we can use a graph representation where each pair defines a directed edge. We will create an adjacency list to represent the graph and use Depth-First Search (DFS) to find an Eulerian path. The algorithm works as follows:

1. Build a graph using a dictionary where each key is a start node and the value is a list of end nodes.
2. Use DFS to explore the graph, ensuring we traverse all edges.
3. Backtrack while constructing the result list to maintain the order of edges.
4. Return the result list which represents a valid arrangement of pairs.

Code Solutions