Recount Cable Commitments

An election office must connect its recount stations using a set of secure cables. Stations are numbered from 0 to n - 1.

Each record in cables is [u, v, cost], describing an undirected cable between two different stations. Different records may connect the same pair of stations; they are distinct cable choices. Costs may be negative because some installations receive subsidies.

A valid installation selects exactly n - 1 cable records and connects every station. Among valid installations, the office will choose one with the minimum possible total cost. The supplied network is guaranteed to allow a valid installation.

For every cable record, determine its status across all minimum-cost valid installations:

• 0: it appears in none of them;
• 1: it appears in some, but not all, of them;
• 2: it appears in every one of them.

Return an integer list in the original order of cables. For a single station with no cables, return an empty list.

Input: n = 3, cables = [[0,1,1],[1,2,1],[0,2,2]]
Output: [2,2,0]

Input: n = 4, cables = [[0,1,1],[1,2,1],[2,0,1],[2,3,5]]
Output: [1,1,1,2]

Input: n = 2, cables = [[0,1,-2],[1,0,-2],[0,1,0]]
Output: [1,1,0]

An iterative bridge traversal avoids recursion-depth limits. The intended complexity is O(n + m log m) time and O(n + m) auxiliary space, where m is the number of cables. Bridge detection must skip the incoming edge record, not every edge to the parent vertex.