Closest Pair

Description:

Input:

The first line contains 2 integers $$$n$$$ and $$$q$$$ $$$(2 \leq n \leq 3 \cdot 10^5; 1 \leq q \leq 3 \cdot 10^5)$$$ — the number of points and the number of queries.

Then, $$$n$$$ lines follows, the $$$i$$$-th of them contains two integers $$$x_i$$$ and $$$w_i$$$ $$$(-10^9 \leq x_i \leq 10^9; 1 \leq w_i \leq 10^9)$$$ — the coordinate and the weight of the $$$i$$$-th point.

It is guaranteed that the points are given in the increasing order of $$$x$$$.

Then, $$$q$$$ lines follows, the $$$i$$$-th of them contains two integers $$$l_i$$$ and $$$r_i$$$ $$$(1 \leq l_i < r_i \leq n)$$$ — the given subarray of the $$$i$$$-th query.

Output:

For each query output one integer, the minimum weighted distance among all pair of distinct points in the given subarray.

Sample Input:

5 5
-2 2
0 10
1 1
9 2
12 7
1 3
2 3
1 5
3 5
2 4

Sample Output:

9
11
9
24
11

Note:

For the first query, the minimum weighted distance is between points $$$1$$$ and $$$3$$$, which is equal to $$$|x_1 - x_3| \cdot (w_1 + w_3) = |-2 - 1| \cdot (2 + 1) = 9$$$.

For the second query, the minimum weighted distance is between points $$$2$$$ and $$$3$$$, which is equal to $$$|x_2 - x_3| \cdot (w_2 + w_3) = |0 - 1| \cdot (10 + 1) = 11$$$.

For the fourth query, the minimum weighted distance is between points $$$3$$$ and $$$4$$$, which is equal to $$$|x_3 - x_4| \cdot (w_3 + w_4) = |1 - 9| \cdot (1 + 2) = 24$$$.