Make Biconnected

AtCoder

IDarc160_e

Time3000ms

Memory256MB

Difficulty—

You are given an undirected tree $G$ with $N$ vertices. **The degree of every vertex in $G$ is at most $3$.** The vertices are numbered $1$ to $N$. The edges are numbered $1$ to $N-1$, and edge $i$ connects vertex $u_i$ and vertex $v_i$. Each vertex has a fixed weight, and the weight of vertex $i$ is $W_i$. You will add zero or more edges to $G$. The cost of adding an edge between vertex $i$ and vertex $j$ is $W_i + W_j$. Print one way to add edges to satisfy the following condition for the minimum possible total cost. * $G$ is $2$\-vertex-connected. In other words, for every vertex $v$ in $G$, removing $v$ and its incident edges from $G$ would not disconnect $G$. You have $T$ test cases to solve. ## Constraints * $1 \leq T \leq 2 \times 10^5$ * $3 \leq N \leq 2 \times 10^5$ * $1 \leq u_i, v_i \leq N$ * The given graph is a tree. * The degree of every vertex in the given graph is at most $3$. * $1 \leq W_i \leq 10^9$ * $W_i$ is an integer. * The sum of $N$ across the test cases is at most $2 \times 10^5$. ## Input The input is given from Standard Input in the following format, where $\mathrm{case}_i$ represents the $i$\-th test case: $T$ $\mathrm{case}_1$ $\mathrm{case}_2$ $\vdots$ $\mathrm{case}_N$ Each test case is in the following format: $N$ $W_1$ $W_2$ $\dots$ $W_N$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_{N-1}$ $v_{N-1}$ [samples]

Samples

Input #1

2
3
2 3 5
1 2
2 3
7
1 10 100 1000 10000 100000 1000000
1 2
2 3
2 4
3 5
3 6
4 7

Output #1

1
1 3
2
7 6
1 5

In the first test case, adding an edge connecting vertex $1$ and vertex $3$ makes $G$ satisfy the condition in the problem statement.  
The cost of this is $W_1 + W_3 = 2 + 5 = 7$. There is no way to add edges to satisfy the condition for a cost of less than $7$, so this is a valid solution.
In the second test case, the solution above has a total cost of $(W_7 + W_6) + (W_1 + W_5) = 1100000 + 10001 = 1110001$, which is the minimum possible.

API Response (JSON)

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