Tree Queries
Contest: Codeforces Round 629 (Div. 3)
Difficulty rating: 1900
Tags: dfs and similar, graphs, trees

You are given a rooted tree consisting of $$$n$$$ vertices numbered from $$$1$$$ to $$$n$$$. The root of the tree is a vertex number $$$1$$$.

A tree is a connected undirected graph with $$$n-1$$$ edges.

You are given $$$m$$$ queries. The $$$i$$$-th query consists of the set of $$$k_i$$$ distinct vertices $$$v_i[1], v_i[2], \dots, v_i[k_i]$$$. Your task is to say if there is a path from the root to some vertex $$$u$$$ such that each of the given $$$k$$$ vertices is either belongs to this path or has the distance $$$1$$$ to some vertex of this path.

Input format:
The first line of the input contains two integers $$$n$$$ and $$$m$$$ ($$$2 \le n \le 2 \cdot 10^5$$$, $$$1 \le m \le 2 \cdot 10^5$$$) — the number of vertices in the tree and the number of queries.

Each of the next $$$n-1$$$ lines describes an edge of the tree. Edge $$$i$$$ is denoted by two integers $$$u_i$$$ and $$$v_i$$$, the labels of vertices it connects $$$(1 \le u_i, v_i \le n, u_i \ne v_i$$$).

It is guaranteed that the given edges form a tree.

The next $$$m$$$ lines describe queries. The $$$i$$$-th line describes the $$$i$$$-th query and starts with the integer $$$k_i$$$ ($$$1 \le k_i \le n$$$) — the number of vertices in the current query. Then $$$k_i$$$ integers follow: $$$v_i[1], v_i[2], \dots, v_i[k_i]$$$ ($$$1 \le v_i[j] \le n$$$), where $$$v_i[j]$$$ is the $$$j$$$-th vertex of the $$$i$$$-th query.

It is guaranteed that all vertices in a single query are distinct.

It is guaranteed that the sum of $$$k_i$$$ does not exceed $$$2 \cdot 10^5$$$ ($$$\sum\limits_{i=1}^{m} k_i \le 2 \cdot 10^5$$$).

Output format:
For each query, print the answer — "YES", if there is a path from the root to some vertex $$$u$$$ such that each of the given $$$k$$$ vertices is either belongs to this path or has the distance $$$1$$$ to some vertex of this path and "NO" otherwise.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
The picture corresponding to the example:

Consider the queries.

The first query is $$$[3, 8, 9, 10]$$$. The answer is "YES" as you can choose the path from the root $$$1$$$ to the vertex $$$u=10$$$. Then vertices $$$[3, 9, 10]$$$ belong to the path from $$$1$$$ to $$$10$$$ and the vertex $$$8$$$ has distance $$$1$$$ to the vertex $$$7$$$ which also belongs to this path.

The second query is $$$[2, 4, 6]$$$. The answer is "YES" as you can choose the path to the vertex $$$u=2$$$. Then the vertex $$$4$$$ has distance $$$1$$$ to the vertex $$$1$$$ which belongs to this path and the vertex $$$6$$$ has distance $$$1$$$ to the vertex $$$2$$$ which belongs to this path.

The third query is $$$[2, 1, 5]$$$. The answer is "YES" as you can choose the path to the vertex $$$u=5$$$ and all vertices of the query belong to this path.

The fourth query is $$$[4, 8, 2]$$$. The answer is "YES" as you can choose the path to the vertex $$$u=9$$$ so vertices $$$2$$$ and $$$4$$$ both have distance $$$1$$$ to the vertex $$$1$$$ which belongs to this path and the vertex $$$8$$$ has distance $$$1$$$ to the vertex $$$7$$$ which belongs to this path.

The fifth and the sixth queries both have answer "NO" because you cannot choose suitable vertex $$$u$$$.

Example input:
10 6
1 2
1 3
1 4
2 5
2 6
3 7
7 8
7 9
9 10
4 3 8 9 10
3 2 4 6
3 2 1 5
3 4 8 2
2 6 10
3 5 4 7

Example output:
YES
YES
YES
YES
NO
NO

Editorial:
Firstly, let's choose some deepest (farthest from the root) vertex $$$fv$$$ in the query (among all such vertices we can choose any). It is obvious that every vertex in the query should either belong to the path from the root to $$$fv$$$ or the distance to some vertex of this path should be at most one. Now there are two ways: write some LCA algorithms and other hard stuff which is unnecessary in this problem or write about $$$15$$$ lines of code and solve the problem.
Let's take every non-root vertex (except $$$fv$$$) and replace it with its parent. So, what's next? Now the answer is "YES" if each vertex (after transformation) belongs to the path from root to $$$fv$$$. Now we just need to check if it is true.
We can do this using the very standard technique: firstly, let's run dfs from the root and calculate for each vertex the first time we visited it ($$$tin$$$) and the last time we visited it ($$$tout$$$).
We can do this using the following code:
Initially, $$$T$$$ equals zero. Now we have a beautiful structure giving us so much information about the tree. Consider all segments $$$[tin_v; tout_v]$$$. We can see that there is no pair of intersecting segments. The pair of segments $$$[tin_v; tout_v]$$$ and $$$[tin_u; tout_u]$$$ is either non-intersecting at all or one segment lies inside the other one. The second beautiful fact is that for each vertex $$$u$$$ in the subtree of $$$v$$$ the segment $$$[tin_u; tout_u]$$$ lies inside the segment $$$[tin_v; tout_v]$$$.
So, we can check if one vertex is the parent of the other: the vertex $$$v$$$ is the parent of the vertex $$$u$$$ if and only if $$$tin_v \le tin_u$$$ and $$$tout_u \le tout_v$$$ (the vertex is the parent of itself).
How do we check if the vertex $$$u$$$ lies on the path from the root to the vertex $$$fv$$$? It lies on this path if the root is the parent of $$$u$$$ (it is always true) and $$$u$$$ is the parent of $$$fv$$$. This approach can be used for each vertical path (such a path from $$$x$$$ to $$$y$$$ that $$$lca(x, y)$$$ is either $$$x$$$ or $$$y$$$).
Time complexity: $$$O(n + m)$$$.