102303: [AtCoder]ABC230 D - Destroyer Takahashi

Problem Statement

In a town divided into a grid with $N$ rows and $10^9$ columns, there are $N$ walls, numbered $1$ to $N$.
Wall $i$ ranges from the $L_i$-th column to the $R_i$-th column from the left in the $i$-th row from the top. (See also the figure at Sample Input and Output $1$.)

Takahashi has decided to destroy all $N$ walls.
With his superhuman strength, his one punch can damage consecutive $D$ columns at once.

• More formally, he can choose an integer $x$ between $1$ and $10^9 - D + 1$ (inclusive) to damage all walls that exist (or partly exist) in the $x$-th through $(x + D - 1)$-th columns and are not yet destroyed.

When a part of a wall is damaged, that whole wall will be destroyed by the impact of the punch.
(See also the figure at Sample Input and Output $1$.)

At least how many times does Takahashi need to punch to destroy all walls?

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq D \leq 10^9$
• $1 \leq L_i \leq R_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $D$
$L_1$ $R_1$
$L_2$ $R_2$
$\vdots$
$L_N$ $R_N$

Output

Print the minimum number of punches needed to destroy all walls.

Sample Input 1

3 3
1 2
4 7
5 9

Sample Output 1

2

The figure below describes the arrangements of walls in Sample Input $1$.

He can destroy all walls with two punches, such as the following. (Below, $\lbrack a, b \rbrack$ denotes the range from the $a$-th through $b$-th columns.)

• First, punch $\lbrack 2, 4 \rbrack$. The walls existing in $\lbrack 2, 4 \rbrack$ ― Walls $1$ and $2$ ― are damaged and destroyed.
• Second, punch $\lbrack 5, 7 \rbrack$. The wall existing in $\lbrack 5, 7 \rbrack$ ― Wall $3$ ― is damaged and destroyed.

It is also possible to destroy all walls with two punches in this way:

• First, punch $\lbrack 7, 9 \rbrack$ to destroy Walls $2$ and $3$.
• Second, punch $\lbrack 1, 3 \rbrack$ to destroy Wall $1$.

Sample Input 2

3 3
1 2
4 7
4 9

Sample Output 2

1

The difference from Sample Input/Output $1$ is that Wall $3$ now covers $\lbrack 4, 9 \rbrack$, not $\lbrack 5, 9 \rbrack$.
In this case, he can punch $\lbrack 2, 4 \rbrack$ to destroy all walls with one punch.

Sample Input 3

5 2
1 100
1 1000000000
101 1000
9982 44353
1000000000 1000000000

Sample Output 3

3

问题描述

一个由$N$行和$10^9$列组成的网格中，有$N$堵墙，编号为$1$到$N$。

第$i$堵墙从上数第$i$行的第$L_i$列到第$R_i$列（见样例输入和输出1的图）。

高桥君决定摧毁所有$N$堵墙。

凭借他的超人力量，他的一拳可以同时破坏连续$D$列。

• 更正式地说，他可以选择一个在$1$和$10^9 - D + 1$（包括）之间的整数$x$，来破坏所有存在（或部分存在）在第$x$列到第$(x + D - 1)$列之间的墙，并且这些墙还没有被破坏。

当一部分墙被破坏时，整堵墙都会被拳头的冲击力摧毁。

（见样例输入和输出1的图）。

高桥君至少需要打多少次才能摧毁所有的墙呢？

限制条件

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq D \leq 10^9$
• $1 \leq L_i \leq R_i \leq 10^9$
• 输入中的所有值都是整数。

输入

从标准输入按照以下格式给出输入：

$N$ $D$
$L_1$ $R_1$
$L_2$ $R_2$
$\vdots$
$L_N$ $R_N$

输出

输出摧毁所有墙所需的最少拳数。

样例输入1

3 3
1 2
4 7
5 9

样例输出1

2

下图描述了样例输入1中的墙的布置。

他可以用两次拳摧毁所有墙，例如以下方式：（下面，$\lbrack a, b \rbrack$表示从第$a$列到第$b$列的范围）。

• 首先，打$\lbrack 2, 4 \rbrack$。在$\lbrack 2, 4 \rbrack$中存在的墙——墙1和墙2——被破坏并摧毁。
• 其次，打$\lbrack 5, 7 \rbrack$。在$\lbrack 5, 7 \rbrack$中存在的墙——墙3——被破坏并摧毁。

也可以用这种方式用两次拳摧毁所有墙：

• 首先，打$\lbrack 7, 9 \rbrack$来摧毁墙2和墙3。
• 其次，打$\lbrack 1, 3 \rbrack$来摧毁墙1。

样例输入2

3 3
1 2
4 7
4 9

样例输出2

1

与样例输入/输出1的区别在于，墙3现在覆盖了$\lbrack 4, 9 \rbrack$，而不是$\lbrack 5, 9 \rbrack$。

在这种情况下，他可以打$\lbrack 2, 4 \rbrack$用一次拳摧毁所有墙。

样例输入3

5 2
1 100
1 1000000000
101 1000
9982 44353
1000000000 1000000000

样例输出3

3