Antenna Coverage
Description
The mayor of the Central Town wants to modernize Central Street, represented in this problem by the $(Ox)$ axis.
On this street, there are $n$ antennas, numbered from $1$ to $n$. The $i$-th antenna lies on the position $x_i$ and has an initial scope of $s_i$: it covers all integer positions inside the interval $[x_i - s_i; x_i + s_i]$.
It is possible to increment the scope of any antenna by $1$, this operation costs $1$ coin. We can do this operation as much as we want (multiple times on the same antenna if we want).
To modernize the street, we need to make all integer positions from $1$ to $m$ inclusive covered by at least one antenna. Note that it is authorized to cover positions outside $[1; m]$, even if it's not required.
What is the minimum amount of coins needed to achieve this modernization?
The first line contains two integers $n$ and $m$ ($1 \le n \le 80$ and $n \le m \le 100\ 000$).
The $i$-th of the next $n$ lines contains two integers $x_i$ and $s_i$ ($1 \le x_i \le m$ and $0 \le s_i \le m$).
On each position, there is at most one antenna (values $x_i$ are pairwise distinct).
You have to output a single integer: the minimum amount of coins required to make all integer positions from $1$ to $m$ inclusive covered by at least one antenna.
Input
The first line contains two integers $n$ and $m$ ($1 \le n \le 80$ and $n \le m \le 100\ 000$).
The $i$-th of the next $n$ lines contains two integers $x_i$ and $s_i$ ($1 \le x_i \le m$ and $0 \le s_i \le m$).
On each position, there is at most one antenna (values $x_i$ are pairwise distinct).
Output
You have to output a single integer: the minimum amount of coins required to make all integer positions from $1$ to $m$ inclusive covered by at least one antenna.
Samples
3 595
43 2
300 4
554 10
281
1 1
1 1
0
2 50
20 0
3 1
30
5 240
13 0
50 25
60 5
155 70
165 70
26
Note
In the first example, here is a possible strategy:
- Increase the scope of the first antenna by $40$, so that it becomes $2 + 40 = 42$. This antenna will cover interval $[43 - 42; 43 + 42]$ which is $[1; 85]$
- Increase the scope of the second antenna by $210$, so that it becomes $4 + 210 = 214$. This antenna will cover interval $[300 - 214; 300 + 214]$, which is $[86; 514]$
- Increase the scope of the third antenna by $31$, so that it becomes $10 + 31 = 41$. This antenna will cover interval $[554 - 41; 554 + 41]$, which is $[513; 595]$
Total cost is $40 + 210 + 31 = 281$. We can prove that it's the minimum cost required to make all positions from $1$ to $595$ covered by at least one antenna.
Note that positions $513$ and $514$ are in this solution covered by two different antennas, but it's not important.
—
In the second example, the first antenna already covers an interval $[0; 2]$ so we have nothing to do.
Note that the only position that we needed to cover was position $1$; positions $0$ and $2$ are covered, but it's not important.