1-Trees and Queries

4000ms

524288K

Description:

Gildong was hiking a mountain, walking by millions of trees. Inspired by them, he suddenly came up with an interesting idea for trees in data structures: What if we add another edge in a tree?

Then he found that such tree-like graphs are called 1-trees. Since Gildong was bored of solving too many tree problems, he wanted to see if similar techniques in trees can be used in 1-trees as well. Instead of solving it by himself, he's going to test you by providing queries on 1-trees.

First, he'll provide you a tree (not 1-tree) with $$$n$$$ vertices, then he will ask you $$$q$$$ queries. Each query contains $$$5$$$ integers: $$$x$$$, $$$y$$$, $$$a$$$, $$$b$$$, and $$$k$$$. This means you're asked to determine if there exists a path from vertex $$$a$$$ to $$$b$$$ that contains exactly $$$k$$$ edges after adding a bidirectional edge between vertices $$$x$$$ and $$$y$$$. A path can contain the same vertices and same edges multiple times. All queries are independent of each other; i.e. the added edge in a query is removed in the next query.

Input:

The first line contains an integer $$$n$$$ ($$$3 \le n \le 10^5$$$), the number of vertices of the tree.

Next $$$n-1$$$ lines contain two integers $$$u$$$ and $$$v$$$ ($$$1 \le u,v \le n$$$, $$$u \ne v$$$) each, which means there is an edge between vertex $$$u$$$ and $$$v$$$. All edges are bidirectional and distinct.

Next line contains an integer $$$q$$$ ($$$1 \le q \le 10^5$$$), the number of queries Gildong wants to ask.

Next $$$q$$$ lines contain five integers $$$x$$$, $$$y$$$, $$$a$$$, $$$b$$$, and $$$k$$$ each ($$$1 \le x,y,a,b \le n$$$, $$$x \ne y$$$, $$$1 \le k \le 10^9$$$) – the integers explained in the description. It is guaranteed that the edge between $$$x$$$ and $$$y$$$ does not exist in the original tree.

Output:

For each query, print "YES" if there exists a path that contains exactly $$$k$$$ edges from vertex $$$a$$$ to $$$b$$$ after adding an edge between vertices $$$x$$$ and $$$y$$$. Otherwise, print "NO".

You can print each letter in any case (upper or lower).

Sample Input:

Sample Output:

YES
YES
NO
YES
NO

Note:

The image below describes the tree (circles and solid lines) and the added edges for each query (dotted lines).

$\text{[math]}$

Possible paths for the queries with "YES" answers are:

$$$1$$$-st query: $$$1$$$ – $$$3$$$ – $$$2$$$
$$$2$$$-nd query: $$$1$$$ – $$$2$$$ – $$$3$$$
$$$4$$$-th query: $$$3$$$ – $$$4$$$ – $$$2$$$ – $$$3$$$ – $$$4$$$ – $$$2$$$ – $$$3$$$ – $$$4$$$ – $$$2$$$ – $$$3$$$

Informação

Provedor Codeforces

Código CF1304E