Connected Component Labeling - Implementation

I'll first give you the code and then explain it a bit:

// direction vectors
const int dx[] = {+1, 0, -1, 0};
const int dy[] = {0, +1, 0, -1};

// matrix dimensions
int row_count;
int col_count;

// the input matrix
int m[MAX][MAX];

// the labels, 0 means unlabeled
int label[MAX][MAX];

void dfs(int x, int y, int current_label) {
  if (x < 0 || x == row_count) return; // out of bounds
  if (y < 0 || y == col_count) return; // out of bounds
  if (label[x][y] || !m[x][y]) return; // already labeled or not marked with 1 in m

  // mark the current cell
  label[x][y] = current_label;

  // recursively mark the neighbors
  for (int direction = 0; direction < 4; ++direction)
    dfs(x + dx[direction], y + dy[direction], current_label);
}

void find_components() {
  int component = 0;
  for (int i = 0; i < row_count; ++i) 
    for (int j = 0; j < col_count; ++j) 
      if (!label[i][j] && m[i][j]) dfs(i, j, ++component);
}

This is a common way of solving this problem.

The direction vectors are just a nice way to find the neighboring cells (in each of the four directions).

The dfs function performs a depth-first-search of the grid. That simply means it will visit all the cells reachable from the starting cell. Each cell will be marked with current_label

The find_components function goes through all the cells of the grid and starts a component labeling if it finds an unlabeled cell (marked with 1).

This can also be done iteratively using a stack. If you replace the stack with a queue, you obtain the bfs or breadth-first-search.

This can be solved with union find (although DFS, as shown in the other answer, is probably a bit simpler).

The basic idea behind this data structure is to repeatedly merge elements in the same component. This is done by representing each component as a tree (with nodes keeping track of their own parent, instead of the other way around), you can check whether 2 elements are in the same component by traversing to the root node and you can merge nodes by simply making the one root the parent of the other root.

A short code sample demonstrating this:

const int w = 5, h = 5;
int input[w][h] =  {{1,0,0,0,1},
                    {1,1,0,1,1},
                    {0,1,0,0,1},
                    {1,1,1,1,0},
                    {0,0,0,1,0}};
int component[w*h];

void doUnion(int a, int b)
{
    // get the root component of a and b, and set the one's parent to the other
    while (component[a] != a)
        a = component[a];
    while (component[b] != b)
        b = component[b];
    component[b] = a;
}

void unionCoords(int x, int y, int x2, int y2)
{
    if (y2 < h && x2 < w && input[x][y] && input[x2][y2])
        doUnion(x*h + y, x2*h + y2);
}

int main()
{
    for (int i = 0; i < w*h; i++)
        component[i] = i;
    for (int x = 0; x < w; x++)
    for (int y = 0; y < h; y++)
    {
        unionCoords(x, y, x+1, y);
        unionCoords(x, y, x, y+1);
    }

    // print the array
    for (int x = 0; x < w; x++)
    {
        for (int y = 0; y < h; y++)
        {
            if (input[x][y] == 0)
            {
                cout << ' ';
                continue;
            }
            int c = x*h + y;
            while (component[c] != c) c = component[c];
            cout << (char)('a'+c);
        }
        cout << "\n";
    }
}

Live demo.

The above will show each group of ones using a different letter of the alphabet.

p   i
pp ii
 p  i
pppp 
   p 

It should be easy to modify this to get the components separately or get a list of elements corresponding to each component. One idea is to replace cout << (char)('a'+c); above with componentMap[c].add(Point(x,y)) with componentMap being a map<int, list<Point>> - each entry in this map will then correspond to a component and give a list of points.

There are various optimisations to improve the efficiency of union find, the above is just a basic implementation.