201562: [AtCoder]ARC156 C - Tree and LCS
Description
Score : $600$ points
Problem Statement
We have a tree $T$ with vertices numbered $1$ to $N$. The $i$-th edge of $T$ connects vertex $u_i$ and vertex $v_i$.
Let us use $T$ to define the similarity of a permutation $P = (P_1,P_2,\ldots,P_N)$ of $(1,2,\ldots,N)$ as follows.
- For a simple path $x=(x_1,x_2,\ldots,x_k)$ in $T$, let $y=(P_{x_1}, P_{x_2},\ldots,P_{x_k})$. The similarity is the maximum possible length of a longest common subsequence of $x$ and $y$.
Construct a permutation $P$ with the minimum similarity.
What is a subsequence?
A subsequence of a sequence is a sequence obtained by removing zero or more elements from that sequence and concatenating the remaining elements without changing the relative order. For instance, $(10,30)$ is a subsequence of $(10,20,30)$, but $(20,10)$ is not.What is a simple path?
For vertices $X$ and $Y$ in a graph $G$, a walk from $X$ to $Y$ is a sequence of vertices $v_1,v_2, \ldots, v_k$ such that $v_1=X$, $v_k=Y$, and there is an edge connecting $v_i$ and $v_{i+1}$. A simple path (or simply a path) is a walk such that $v_1,v_2, \ldots, v_k$ are all different.Constraints
- $2 \leq N \leq 5000$
- $1\leq u_i,v_i\leq N$
- The given graph is a tree.
- All numbers in the input are integers.
Input
The input is given from Standard Input in the following format:
$N$
$u_1$ $v_1$
$u_2$ $v_2$
$\vdots$
$u_{N-1}$ $v_{N-1}$
Output
Print a permutation $P$ with the minimum similarity, separated by spaces. If multiple solutions exist, you may print any of them.
Sample Input 1
3 1 2 2 3
Sample Output 1
3 2 1
This permutation has a similarity of $1$, which can be computed as follows.
-
For $x=(1)$, we have $y=(P_1)=(3)$. The length of a longest common subsequence of $x$ and $y$ is $0$.
-
For $x=(2)$, we have $y=(P_2)=(2)$. The length of a longest common subsequence of $x$ and $y$ is $1$.
-
For $x=(3)$, we have $y=(P_2)=(1)$. The length of a longest common subsequence of $x$ and $y$ is $0$.
-
For $x=(1,2)$, we have $y=(P_1,P_2)=(3,2)$. The length of a longest common subsequence of $x$ and $y$ is $1$. The same goes for $x=(2,1)$, the reversal of $(1,2)$.
-
For $x=(2,3)$, we have $y=(P_2,P_3)=(2,1)$. The length of a longest common subsequence of $x$ and $y$ is $1$. The same goes for $x=(3,2)$, the reversal of $(2,3)$.
-
For $x=(1,2,3)$, we have $y=(P_1,P_2,P_3)=(3,2,1)$. The length of a longest common subsequence of $x$ and $y$ is $1$. The same goes for $x=(3,2,1)$, the reversal of $(1,2,3)$.
We can prove that no permutation has a similarity of $0$ or less, so this permutation is a valid answer.
Sample Input 2
4 2 1 2 3 2 4
Sample Output 2
3 4 1 2
If multiple permutations have the minimum similarity, you may print any of them. For instance, you may also print 4 3 2 1.