Make maze connected by removing internal walls

Question

Recently I've stumbled upon a strange graph problem. Here is a brief description.

Given $n\times m$ matrix with $2n + 1$ rows such that each row contains $2m + 1$ characters "+", "-", "|", "."

• First, last row, and first, last column are the borders of the maze.
• Even coordinates, e.g. (0, 0), (0, 2), are filled with "+", which you can't step on.
• Odd coordinates are filled with ".", which is used normally for the traversal.
• The rest of the coordinates share "|", "-", "." characters. "|" defines a horizontal wall, whereas "-" defines vertical.

The problem asks to delete a minimal number of walls, i.e. "|" and "-", in order to make the whole maze connected.

The example is given below

Input

2 3
+-+-+-+
|.|...|
+-+-+-+
|.|...|
+-+-+-+

Output

2 3
+-+-+-+
|.....|
+.+-+-+
|.|...|
+-+-+-+

As you can see two walls are deleted, the vertical wall having the coordinate $(1, 2) $ and the horizontal wall with the coordinates $(2, 1)$.

This problem seems like a perfect application of Disjoint Set Union, but I am not convinced that this is any right approach to this problem. Any help is much appreciated!

Answer 1

You can use a graph traversal algorithm such as BFS/DFS to detect the connected components, and to construct a graph in which the vertices are connected components and the edges are walls adjacent to two connected components. A spanning tree in this graph corresponds to the walls that need to be removed.