Smooth Sailing (Hard Version)
Contest: Codeforces Round 919 (Div. 2)
Difficulty rating: 3000
Tags: binary search, data structures, dsu, geometry, graphs, trees

The only difference between the two versions of this problem is the constraint on $$$q$$$. You can make hacks only if both versions of the problem are solved.

Thomas is sailing around an island surrounded by the ocean. The ocean and island can be represented by a grid with $$$n$$$ rows and $$$m$$$ columns. The rows are numbered from $$$1$$$ to $$$n$$$ from top to bottom, and the columns are numbered from $$$1$$$ to $$$m$$$ from left to right. The position of a cell at row $$$r$$$ and column $$$c$$$ can be represented as $$$(r, c)$$$. Below is an example of a valid grid.

Example of a valid grid

There are three types of cells: island, ocean and underwater volcano. Cells representing the island are marked with a '#', cells representing the ocean are marked with a '.', and cells representing an underwater volcano are marked with a 'v'. It is guaranteed that there is at least one island cell and at least one underwater volcano cell. It is also guaranteed that the set of all island cells forms a single connected component$$$^{\dagger}$$$ and the set of all ocean cells and underwater volcano cells forms a single connected component. Additionally, it is guaranteed that there are no island cells at the edge of the grid (that is, at row $$$1$$$, at row $$$n$$$, at column $$$1$$$, and at column $$$m$$$).

Define a round trip starting from cell $$$(x, y)$$$ as a path Thomas takes which satisfies the following conditions:

- The path starts and ends at $$$(x, y)$$$.
- If Thomas is at cell $$$(i, j)$$$, he can go to cells $$$(i+1, j)$$$, $$$(i-1, j)$$$, $$$(i, j-1)$$$, and $$$(i, j+1)$$$ as long as the destination cell is an ocean cell or an underwater volcano cell and is still inside the grid. Note that it is allowed for Thomas to visit the same cell multiple times in the same round trip.
- The path must go around the island and fully encircle it. Some path $$$p$$$ fully encircles the island if it is impossible to go from an island cell to a cell on the grid border by only traveling to adjacent on a side or diagonal cells without visiting a cell on path $$$p$$$. In the image below, the path starting from $$$(2, 2)$$$, going to $$$(1, 3)$$$, and going back to $$$(2, 2)$$$ the other way does not fully encircle the island and is not considered a round trip.

Example of a path that does not fully encircle the island

The safety of a round trip is the minimum Manhattan distance$$$^{\ddagger}$$$ from a cell on the round trip to an underwater volcano (note that the presence of island cells does not impact this distance).

You have $$$q$$$ queries. A query can be represented as $$$(x, y)$$$ and for every query, you want to find the maximum safety of a round trip starting from $$$(x, y)$$$. It is guaranteed that $$$(x, y)$$$ is an ocean cell or an underwater volcano cell.

$$$^{\dagger}$$$A set of cells forms a single connected component if from any cell of this set it is possible to reach any other cell of this set by moving only through the cells of this set, each time going to a cell with a common side.

$$$^{\ddagger}$$$Manhattan distance between cells $$$(r_1, c_1)$$$ and $$$(r_2, c_2)$$$ is equal to $$$|r_1 - r_2| + |c_1 - c_2|$$$.

Input format:
The first line contains three integers $$$n$$$, $$$m$$$, and $$$q$$$ ($$$3 \leq n, m \leq 10^5$$$, $$$9 \leq n \cdot m \leq 3 \cdot 10^5$$$, $$$1 \leq q \leq 3 \cdot 10^5$$$) — the number of rows and columns of the grid and the number of queries.

Each of the following $$$n$$$ lines contains $$$m$$$ characters describing the cells of the grid. The character '#' denotes an island cell, '.' denotes an ocean cell, and 'v' denotes an underwater volcano cell.

It is guaranteed that there is at least one island cell and at least one underwater volcano cell. It is guaranteed that the set of all island cells forms a single connected component and the set of all ocean cells and underwater volcano cells forms a single connected component. Also, it is guaranteed that there are no island cells at the edge of the grid (that is, at the row $$$1$$$, at the row $$$n$$$, at the column $$$1$$$, and at the column $$$m$$$).

The following $$$q$$$ lines describe the queries. Each of these lines contains two integers $$$x$$$ and $$$y$$$ ($$$1 \leq x \leq n$$$, $$$1 \leq y \leq m$$$) denoting a round trip starting from $$$(x, y)$$$.

It is guaranteed that $$$(x, y)$$$ is an ocean cell or an underwater volcano cell.

Output format:
For each query, output a single integer — the maximum safety of a round trip starting from the specified position.

Constraints:
Time limit: 5.0 s
Memory limit: 1024.0 MB

Note:
For the first example, the image below shows an optimal round trip starting from $$$(1, 1)$$$. The round trip has a safety of $$$3$$$ as the minimum Manhattan distance from a cell on the round trip to an underwater volcano is $$$3$$$.

Example of an optimal round trip

For the fourth example, remember that it is allowed for Thomas to visit the same cell multiple times in the same round trip. For example, doing so is necessary for the round trip starting from $$$(7, 6)$$$.

Example input:
9 9 3
.........
.........
....###..
...v#....
..###....
...##...v
...##....
.........
v........
1 1
9 1
5 7

Example output:
3
0
3

Example input:
3 3 5
..v
.#.
...
1 2
1 3
2 3
2 1
3 2

Example output:
0
0
0
0
0

Editorial:
For each non-island cell $$$(i, j)$$$, let $$$d_{i,j}$$$ be the minimum Manhattan distance of cell $$$(i, j)$$$ to an underwater volcano. We can find all $$$d_{i,j}$$$ with a multisource BFS from all underwater volcanoes at once. The danger of a round trip is the smallest value of $$$d_{u,v}$$$ over all $$$(u, v)$$$ in the path.
Consider any island cell. Draw an imaginary line along the top border of the cell and extend it all the way to the right of the grid. We can make two observations:
• Any path that starts and ends at the same cell and crosses the line an odd number of times will fully encircle the island and be a round trip.
• An optimal round trip will always cross the line an odd number of times.
This allows us to simplify things. For each $$$(x, y)$$$ we only need to look at paths that start and end at $$$(x, y)$$$ and crosses the line an odd number of times. And we want the one that maximizes the smallest $$$d_{u,v}$$$ over all $$$(u, v)$$$ on the path.
We can solve each query $$$(x, y)$$$ naively by binary searching on the answer — we only consider all $$$(u, v)$$$ such that $$$d_{u,v} \geq \text{limit}$$$. Then we can do a BFS with state $$$(\text{row}, \, \text{column}, \, \text{parity of the number of times we crossed the line})$$$ and check if $$$(x, y, 0)$$$ and $$$(x, y, 1)$$$ are connected.
To fully solve this problem, we can add states (and their corresponding edges to already added states) one at a time from highest $$$d$$$ to lowest $$$d$$$. For each query $$$(x, y)$$$, we want to find the first time when $$$(x, y, 0)$$$ and $$$(x, y, 1)$$$ become connected. This is a classic DSU with small to large merging problem. In short, we drop a token labeled with the index of the query at both $$$(x, y, 0)$$$ and $$$(x, y, 1)$$$. Each time we merge, we also merge the sets of tokens small to large and check if merging has caused two tokens of the same label to be in the same component. The time complexity of our solution is $$$O(nm \, \alpha(nm)+q\log^2 q)$$$.