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You are given a collection of directed segments represented as a 0-indexed 2D integer array segments, where segments[i] = [origin_i, destination_i] represents a directed connection from the origin point to the destination point.
Your task is to find a valid chain arrangement of all the segments. An arrangement is considered valid if and only if for every consecutive pair of segments in the arrangement (at positions i and i+1 where 0 ≤ i < segments.length - 1), the destination of segment i equals the origin of segment i+1. In other words, the ending point of each segment must match the starting point of the next segment in the sequence.
Return any valid chain arrangement of the segments.
Note: The input is guaranteed to have at least one valid arrangement. Multiple valid arrangements may exist—returning any one of them is acceptable.
This problem is fundamentally about finding an Eulerian Path in a directed graph. Each segment [origin, destination] represents a directed edge from node origin to node destination.
Key Observations:
Graph Representation: Model the segments as edges in a directed graph. Each unique value appearing as an origin or destination becomes a node.
Eulerian Path Conditions: An Eulerian path exists in a directed graph if and only if:
(out-degree) - (in-degree) = 1 (this is the starting vertex)(in-degree) - (out-degree) = 1 (this is the ending vertex)Finding the Start Node:
out-degree - in-degree = 1, start from that vertexHierholzer's Algorithm: Use this classic algorithm to construct the Eulerian path:
Time Complexity: O(E) where E is the number of segments Space Complexity: O(V + E) for the adjacency list and degree tracking
segments = [[5,1],[4,5],[11,9],[9,4]][[11,9],[9,4],[4,5],[5,1]]This forms a valid chain because each segment's destination matches the next segment's origin: 11→9 (end=9), 9→4 (start=9), 4→5 (start=4 matches previous end), 5→1 (start=5 matches previous end). We verify: end₀=9=start₁, end₁=4=start₂, end₂=5=start₃.
segments = [[1,3],[3,2],[2,1]][[1,3],[3,2],[2,1]]These segments form a cycle: 1→3→2→1. The arrangement is valid since end₀=3=start₁ and end₁=2=start₂. Note that [[3,2],[2,1],[1,3]] and [[2,1],[1,3],[3,2]] are also valid arrangements since the segments form a complete cycle.
segments = [[1,2],[1,3],[2,1]][[1,2],[2,1],[1,3]]Starting from 1, we trace: 1→2, then 2→1 (returning to 1), then 1→3. This uses all segments exactly once while maintaining the chain property. The verification shows: end₀=2=start₁, end₁=1=start₂.
Constraints