Putting Boxes Together

2000ms

262144K

Description:

There is an infinite line consisting of cells. There are $$$n$$$ boxes in some cells of this line. The $$$i$$$-th box stands in the cell $$$a_i$$$ and has weight $$$w_i$$$. All $$$a_i$$$ are distinct, moreover, $$$a_{i - 1} < a_i$$$ holds for all valid $$$i$$$.

You would like to put together some boxes. Putting together boxes with indices in the segment $$$[l, r]$$$ means that you will move some of them in such a way that their positions will form some segment $$$[x, x + (r - l)]$$$.

In one step you can move any box to a neighboring cell if it isn't occupied by another box (i.e. you can choose $$$i$$$ and change $$$a_i$$$ by $$$1$$$, all positions should remain distinct). You spend $$$w_i$$$ units of energy moving the box $$$i$$$ by one cell. You can move any box any number of times, in arbitrary order.

Sometimes weights of some boxes change, so you have queries of two types:

$$$id$$$ $$$nw$$$ — weight $$$w_{id}$$$ of the box $$$id$$$ becomes $$$nw$$$.
$$$l$$$ $$$r$$$ — you should compute the minimum total energy needed to put together boxes with indices in $$$[l, r]$$$. Since the answer can be rather big, print the remainder it gives when divided by $$$1000\,000\,007 = 10^9 + 7$$$. Note that the boxes are not moved during the query, you only should compute the answer.

Note that you should minimize the answer, not its remainder modulo $$$10^9 + 7$$$. So if you have two possible answers $$$2 \cdot 10^9 + 13$$$ and $$$2 \cdot 10^9 + 14$$$, you should choose the first one and print $$$10^9 + 6$$$, even though the remainder of the second answer is $$$0$$$.

Input:

The first line contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le n, q \le 2 \cdot 10^5$$$) — the number of boxes and the number of queries.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots a_n$$$ ($$$1 \le a_i \le 10^9$$$) — the positions of the boxes. All $$$a_i$$$ are distinct, $$$a_{i - 1} < a_i$$$ holds for all valid $$$i$$$.

The third line contains $$$n$$$ integers $$$w_1, w_2, \dots w_n$$$ ($$$1 \le w_i \le 10^9$$$) — the initial weights of the boxes.

Next $$$q$$$ lines describe queries, one query per line.

Each query is described in a single line, containing two integers $$$x$$$ and $$$y$$$. If $$$x < 0$$$, then this query is of the first type, where $$$id = -x$$$, $$$nw = y$$$ ($$$1 \le id \le n$$$, $$$1 \le nw \le 10^9$$$). If $$$x > 0$$$, then the query is of the second type, where $$$l = x$$$ and $$$r = y$$$ ($$$1 \le l_j \le r_j \le n$$$). $$$x$$$ can not be equal to $$$0$$$.

Output:

For each query of the second type print the answer on a separate line. Since answer can be large, print the remainder it gives when divided by $$$1000\,000\,007 = 10^9 + 7$$$.

Sample Input:

Sample Output:

Note:

Let's go through queries of the example:

$$$1\ 1$$$ — there is only one box so we don't need to move anything.
$$$1\ 5$$$ — we can move boxes to segment $$$[4, 8]$$$: $$$1 \cdot |1 - 4| + 1 \cdot |2 - 5| + 1 \cdot |6 - 6| + 1 \cdot |7 - 7| + 2 \cdot |10 - 8| = 10$$$.
$$$1\ 3$$$ — we can move boxes to segment $$$[1, 3]$$$.
$$$3\ 5$$$ — we can move boxes to segment $$$[7, 9]$$$.
$$$-3\ 5$$$ — $$$w_3$$$ is changed from $$$1$$$ to $$$5$$$.
$$$-1\ 10$$$ — $$$w_1$$$ is changed from $$$1$$$ to $$$10$$$. The weights are now equal to $$$w = [10, 1, 5, 1, 2]$$$.
$$$1\ 4$$$ — we can move boxes to segment $$$[1, 4]$$$.
$$$2\ 5$$$ — we can move boxes to segment $$$[5, 8]$$$.

Informação

Provedor Codeforces

Código CF1030F