Question

I’ve been trying to find solution to my problem for more than a week and I couldn’t find out anything better than a milion iterations prog, so I think it’s time to ask someone to help me.

I’ve got a 3D array. Let’s say, we’re talking about the ground and the first layer is a surface.
Another layers are floors below the ground. I have to find deepest path’s length, count of isolated caves underground and the size of the biggest cave.

Here’s the visualisation of my problem.

Input:
5 5 5 // x, y, z
xxxxx
oxxxx
xxxxx
xoxxo
ooxxx

xxxxx
xxoxx

and so...

Output:
5 // deepest path - starting from the surface
22 // size of the biggest cave
3 // number of izolated caves (red ones) (izolated - cave that doesn't reach the surface)

Note, that even though red cell on the 2nd floor is placed next to green one, It’s not the same cave because it’s placed diagonally and that doesn’t count.
I’ve been told that the best way to do this, might be using recursive algorithm “divide and rule” however I don’t really know how could it look like.

Answer 1

I think you should be able to do it in O(N).
When you parse your input, assign each node a ‘caveNumber’ initialized to 0. Set it to a valid number whenever you visit a cave:

CaveCount = 0, IsolatedCaveCount=0
AllSizes = new Vector.
For each node, 
   ProcessNode(size:0,depth:0);

ProcessNode(size,depth):
   If node.isCave and !node.caveNumber 
       if (size==0) ++CaveCount
       if (size==0 and depth!=0) IsolatedCaveCount++
       node.caveNumber = CaveCount
       AllSizes[CaveCount]++
       For each neighbor of node, 
            if (goingDeeper) depth++
            ProcessNode(size+1, depth).

You will visit each node 7 times at worst case: once from the outer loop, and possibly once from each of its six neighbors. But you’ll only work on each one once, since after that the caveNumber is set, and you ignore it.

You can do the depth tracking by adding a depth parameter to the recursive ProcessNode call, and only incrementing it when visiting a lower neighbor.