103214: [Atcoder]ABC321 E - Complete Binary Tree

Problem Statement

There is a tree with $N$ vertices numbered $1$ to $N$. For each $i\ (2 \leq i \leq N)$, there is an edge connecting vertex $i$ and vertex $\lfloor \frac{i}{2} \rfloor$. There are no other edges.

In this tree, find the number of vertices whose distance from vertex $X$ is $K$. Here, the distance between two vertices $u$ and $v$ is defined as the number of edges in the simple path connecting vertices $u$ and $v$.

You have $T$ test cases to solve.

Constraints

• $1\leq T \leq 10^5$
• $1\leq N \leq 10^{18}$
• $1\leq X \leq N$
• $0\leq K \leq N-1$
• All input values are integers.

Input

The input is given from Standard Input in the following format, where $\mathrm{test}_i$ represents the $i$-th test case:

$T$
$\mathrm{test}_1$
$\mathrm{test}_2$
$\vdots$
$\mathrm{test}_T$

Each test case is given in the following format:

$N$ $X$ $K$ 

Output

Print $T$ lines.

The $i$-th line $(1 \leq i \leq T)$ should contain the answer to the $i$-th test case as an integer.

Sample Input 1

5
10 2 0
10 2 1
10 2 2
10 2 3
10 2 4

Sample Output 1

1
3
4
2
0

The tree for $N=10$ is shown in the following figure.

Here,

• There is $1$ vertex, $2$, whose distance from vertex $2$ is $0$.
• There are $3$ vertices, $1,4,5$, whose distance from vertex $2$ is $1$.
• There are $4$ vertices, $3,8,9,10$, whose distance from vertex $2$ is $2$.
• There are $2$ vertices, $6,7$, whose distance from vertex $2$ is $3$.
• There are no vertices whose distance from vertex $2$ is $4$.

Sample Input 2

10
822981260158260522 52 20
760713016476190629 2314654 57
1312150450968417 1132551176249851 7
1000000000000000000 1083770654 79
234122432773361868 170290518806790 23
536187734191890310 61862 14
594688604155374934 53288633578 39
1000000000000000000 120160810 78
89013034180999835 14853481725739 94
463213054346948152 825589 73

Sample Output 2

1556480
140703128616960
8
17732923532771328
65536
24576
2147483640
33776997205278720
7881299347898368
27021597764222976

问题描述

有一个有N个顶点的树，编号为1到N。 对于每个$i\ (2 \leq i \leq N)$，存在一条连接顶点i和顶点$\lfloor \frac{i}{2} \rfloor$的边。 没有其他边。

在这个树中，找出与顶点X的距离为K的顶点的数量。 这里，两个顶点$u$和$v$之间的距离定义为连接顶点$u$和$v$的简单路径上的边的数量。

你需要解决T个测试用例。

约束

• $1\leq T \leq 10^5$
• $1\leq N \leq 10^{18}$
• $1\leq X \leq N$
• $0\leq K \leq N-1$
• 所有输入值都是整数。

输入

输入从标准输入给出以下格式，其中$\mathrm{test}_i$表示第i个测试用例：

$T$
$\mathrm{test}_1$
$\mathrm{test}_2$
$\vdots$
$\mathrm{test}_T$

每个测试用例给出以下格式：

$N$ $X$ $K$ 

输出

打印T行。

第i行$(1 \leq i \leq T)$应包含第i个测试用例的答案，即整数。

样例输入1

5
10 2 0
10 2 1
10 2 2
10 2 3
10 2 4

样例输出1

1
3
4
2
0

当N=10时，树如下图所示。

在这里，

• 与顶点2的距离为0的顶点有1个，即2。
• 与顶点2的距离为1的顶点有3个，即1,4,5。
• 与顶点2的距离为2的顶点有4个，即3,8,9,10。
• 与顶点2的距离为3的顶点有2个，即6,7。
• 没有与顶点2的距离为4的顶点。

样例输入2

10
822981260158260522 52 20
760713016476190629 2314654 57
1312150450968417 1132551176249851 7
1000000000000000000 1083770654 79
234122432773361868 170290518806790 23
536187734191890310 61862 14
594688604155374934 53288633578 39
1000000000000000000 120160810 78
89013034180999835 14853481725739 94
463213054346948152 825589 73

样例输出2

1556480
140703128616960
8
17732923532771328
65536
24576
2147483640
33776997205278720
7881299347898368
27021597764222976