## 749. Contain Virus

A virus is spreading rapidly, and your task is to quarantine the infected area by installing walls.

The world is modeled as a 2-D array of cells, where 0 represents uninfected cells, and 1 represents cells contaminated with the virus. A wall (and only one wall) can be installed between any two 4-directionally adjacent cells, on the shared boundary.

Every night, the virus spreads to all neighboring cells in all four directions unless blocked by a wall. Resources are limited. Each day, you can install walls around only one region -- the affected area (continuous block of infected cells) that threatens the most uninfected cells the following night. There will never be a tie.

Can you save the day? If so, what is the number of walls required? If not, and the world becomes fully infected, return the number of walls used.

Example 1:

Input: grid =
[[0,1,0,0,0,0,0,1],
[0,1,0,0,0,0,0,1],
[0,0,0,0,0,0,0,1],
[0,0,0,0,0,0,0,0]]
Output: 10
Explanation:
There are 2 contaminated regions.
On the first day, add 5 walls to quarantine the viral region on the left. The board after the virus spreads is:

[[0,1,0,0,0,0,1,1],
[0,1,0,0,0,0,1,1],
[0,0,0,0,0,0,1,1],
[0,0,0,0,0,0,0,1]]

On the second day, add 5 walls to quarantine the viral region on the right. The virus is fully contained.


Example 2:

Input: grid =
[[1,1,1],
[1,0,1],
[1,1,1]]
Output: 4
Explanation: Even though there is only one cell saved, there are 4 walls built.
Notice that walls are only built on the shared boundary of two different cells.


Example 3:

Input: grid =
[[1,1,1,0,0,0,0,0,0],
[1,0,1,0,1,1,1,1,1],
[1,1,1,0,0,0,0,0,0]]
Output: 13
Explanation: The region on the left only builds two new walls.


Note:

1. The number of rows and columns of grid will each be in the range [1, 50].
2. Each grid[i][j] will be either 0 or 1.
3. Throughout the described process, there is always a contiguous viral region that will infect strictly more uncontaminated squares in the next round.

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#### Approach #1: Simulation [Accepted]

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Intuition

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Let\'s work on simulating one turn of the process. We can repeat this as necessary while there are still infected regions.

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Algorithm

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Though the implementation is long, the algorithm is straightforward. We perform the following steps:

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Find all viral regions (connected components), additionally for each region keeping track of the frontier (neighboring uncontaminated cells), and the perimeter of the region.

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Spread the virus in the remaining regions outward by 1 square.

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Complexity Analysis

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Time Complexity: where is the number of rows and columns. After time , viral regions that are alive must have size at least , so the total number removed across all time is .

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