### PROBLEM LINK:

**Author:** Ke Bi

**Tester:** Kevin Atienza

**Translators:** Vasya Antoniuk (Russian), Team VNOI (Vietnamese) and Hu Zecong (Mandarin)

**Editorialist:** Kevin Atienza

### DIFFICULTY:

Easy-Medium

### PREREQUISITES:

BFS, DFS, Union find, dual graph

### PROBLEM:

Given an R\times C grid with W coated walls between some pairs of adjacent cells, what is the minimum number of walls you have to coat so that the grid becomes disconnected?

### QUICK EXPLANATION:

The answer can only be 2, 1, 0, or â€śChefland is doomedâ€ť.

If R\times C = 1, the answer is â€śChefland is doomedâ€ť.

Consider the endpoints of coated walls as nodes of a graph, and the walls themselves as edges. Also, consider the boundary of the grid as a single node.

If there is a cycle in this graph, then the answer is 0.

If you can add an edge which makes a cycle, the answer is 1. This can be done efficiently using union find or BFS.

Otherwise, the answer is 2.

### EXPLANATION:

The only time that the grid canâ€™t be disconnected is when R = C = 1, which means thereâ€™s only one bomb. Otherwise, regardless of the configuration, thereâ€™s always a way to disconnect the grid using only 2 coatings: simply coat off the walls of one of the corner bombs! (If some of those walls have already been coated, or if the grid has a width or height of 1, then all the better; we will need less than 2 coatings.) This gives us an upper bound of 2 for the answer, and the only problem now is to figure out whether the answer is 2, 1 or 0.

When is the answer 0? This happens when the grid is already disconnected. So given a grid, how do you know if itâ€™s already disconnected? Note that we canâ€™t do a BFS/DFS on the grid because the grid can be very large. Instead, we take a look at some cases of disconnected grids:

What makes these disconnected is that there are â€śwalled-offâ€ť sections of the grid. Such a section exists if and only if **there is a cycle along the walls!** See the following image:

More formally, if we consider the *corners* of the cells as â€śnodesâ€ť of a graph, and the walls themselves as the edges, then the answer is 0 if and only if there is a cycle in this graph!

Now, you might say there is already a cycle all this time if you consider the **boundary of the grid** as walls:

But we donâ€™t consider that cycle. To do this in code, simply consider the whole boundary of the grid as a single node.

To find this cycle, we just construct this graph and perform a BFS/DFS on it. So now, we have a fast algorithm to compute whether the answer is 0!

By the way, the graph we constructed is also known as the **dual graph** of the original graph, in the context of planar graphs.

But if the answer is not 0, then the answer can only be 1 or 2 at this point, and we need to figure out which case it is. So when is the answer equal to 1? The answer is 1 if you can **create a cycle along the walls by coating just one wall.** For example, in the following images, we can coat one wall each to create a cycle:

So how do we code this? With BFS also! Hereâ€™s one way:

- Perform a BFS on the dual graph, and remember the connected component of each node.
- For each node, try checking each of its four neighbors. If thereâ€™s a neighbor in the same connected component but is not already adjacent to the current node, then the answer is 1.
- If we canâ€™t find any, then the answer is 2.

But be careful with implementation! If you implemented it like I mentioned above, where you turn the whole boundary of the grid into a single node, then itâ€™s possible you might get the following case wrong:

To guard against this test case, just remember the original walls as given in the input.

Another possible case you might get wrong is the following:

In this case, the answer is 1 because you can coat any vertical wall to disconnect the graph, even though *the boundary is represented as a single node.*

So now we have an algorithm that runs in O(W \log W) time (because of the usage of maps/sets)! As a final note, you can also use **union find** to implement the things we described above.

### Time Complexity:

O(W \log W)