20511번 - Specijacija 서브태스크다국어

You are given a positive integer $n$ and a sequence $a_1, a_2, \dots , a_n$ of positive integers, such that $\frac{i(i−1)}{2} < a_i \le \frac{i(i+1)}{2}$.

The sequence parameterizes a tree with $\frac{(n+1)(n+2)}{2}$ vertices, consisting of $n + 1$ levels with $1, 2, \dots , n + 1$ vertices, in the following way:

The tree parameterized by $a = (1, 2, 6)$.

The $i$-th level contains vertices $\frac{i(i−1)}{2} + 1, \dots , \frac{i(i+1)}{2}$. The vertex $a_i$ has two children, and the rest of the vertices on the level have one child each.

We want to answer $q$ queries of the form “what is the largest common ancestor of $x$ and $y$”, i.e. the vertex with the largest label which is an ancestor of both $x$ and $y$.

The first line contains integers $n$, $q$ and $t$ ($1 \le n, q \le 200 000, t \in \{0, 1\}$), the number of parameters, the number of queries, and a value which will be used to determine the labels of vertices in the queries.

The second line contains a sequence of $n$ integers $a_i$ ($\frac{i(i−1)}{2} \le a_i \le \frac{i(i+1)}{2}$) which parameterize the tree.

The $i$-th of the following $q$ lines contains two integers $\tilde{x}_i$ and $\tilde{y}_i$ ($1 ≤ \tilde{x}_i, \tilde{y}_i ≤ \frac{(n+1)(n+2)}{2}$) which will be used to determine the labels of vertices in the queries.

Let $z_i$ be the answer to the $i$-th query, and let $z_0 = 0$. The labels in the $i$-th query $x_i$ and $y_i$ are:

$x_i = \left(\left(\tilde{x}_i - 1 + t \cdot z_{i-1}\right) \mod \frac{(n+1)(n+2)}{2}\right) + 1 \text{,}$

$y_i = \left(\left(\tilde{y}_i - 1 + t \cdot z_{i-1}\right) \mod \frac{(n+1)(n+2)}{2}\right) + 1 \text{,}$

where $\text{mod}$ is the remainder of integer divison.

Remark: Note that if $t = 0$, it holds $x_i = \tilde{x}_i$ and $y_i = \tilde{y}_i$, so all queries are known from input. If $t = 1$, the queries are not known in advance, but are determined using answers to previous queries.

Output $q$ lines. In the $i$-th line, output the largest common ancestor of $x_i$ and $y_i$.

Clarification of the examples: The tree from both examples is shown on the figure in the statement. Labels of verticies in queries in the second example are: $x_1 = 7, y_1 = 10, \\ x_2 = 9, y_2 = 6,\\ x_3 = 2, y_3 = 8,\\ x_4 = 1, y_4 = 2,\\ x_5 = 3, y_5 = 4$.