201472: [AtCoder]ARC147 C - Min Diff Sum

Problem Statement

$N$ people, numbered $1,2,\ldots ,N$, are going to stand on the number line. Let's denote by $x_i$ the coordinate the Person $i$ stands at. Then, $x_i$ should be an integer satisfying $L_i \leq x_i \leq R_i$. Multiple people can occupy the same coordinate.

We define the dissatisfaction level as the following formula:

$\displaystyle\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}|x_j-x_i|$

Find the minimum possible value of the dissatisfaction level.

Constraints

• $2 \leq N \leq 3 \times 10^5$
• $1 \leq L_i \leq R_i \leq 10^7 \,(1 \leq i \leq N)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

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

Output

Print the answer.

Sample Input 1

3
1 3
2 4
5 6

Sample Output 1

4

If we let $x_1=3,x_2=4,x_3=5$, we get the dissatisfaction level of $4$. We cannot make it $3$ or less, so the answer is $4$.

Sample Input 2

3
1 1
1 1
1 1

Sample Output 2

0

Sample Input 3

6
1 5
2 4
1 1
4 4
3 6
3 3

Sample Output 3

15