Find the number of distinct islands in a 2D matrix

Given a boolean 2D matrix. The task is to find the number of distinct islands where a group of connected 1s (horizontally or vertically) forms an island. Two islands are considered to be distinct if and only if one island is equal to another (not rotated or reflected).

Examples:

Input: grid[][] = 
{{1, 1, 0, 0, 0}, 
1, 1, 0, 0, 0}, 
0, 0, 0, 1, 1}, 
0, 0, 0, 1, 1}}
Output: 1 
Island 1, 1 at the top left corner is same as island 1, 1 at the bottom right corner
Input: grid[][] = 
{{1, 1, 0, 1, 1}, 
1, 0, 0, 0, 0}, 
0, 0, 0, 0, 1}, 
1, 1, 0, 1, 1}}
Output: 3 

Distinct islands in the example above are 1, 1 at the top left corner; 1, 1 at the top right corner and 1 at the bottom right corner. We ignore the island 1, 1 at the bottom left corner since 1, 1 it is identical to the top right corner.

Approach: This problem is an extension of the problem Number of Islands.

The core of the question is to know if 2 islands are equal. The primary criteria is that the number of 1’s should be same in both. But this cannot be the only criteria as we have seen in example 2 above. So how do we know? We could use the position/coordinates of the 1’s. 
If we take the first coordinates of any island as a base point and then compute the coordinates of other points from the base point, we can eliminate duplicates to get the distinct count of islands. So, using this approach, the coordinates for the 2 islands in example 1 above can be represented as: [(0, 0), (0, 1), (1, 0), (1, 1)].

Below is the implementation of the above approach: 

C++

// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;

// 2D array for the storing the horizontal and vertical
// directions. (Up, left, down, right}
vector<vector<int> > dirs = { { 0, -1 },
                              { -1, 0 },
                              { 0, 1 },
                              { 1, 0 } };

// Function to perform dfs of the input grid
void dfs(vector<vector<int> >& grid, int x0, int y0,
         int i, int j, vector<pair<int, int> >& v)
{
    int rows = grid.size(), cols = grid[0].size();

    if (i < 0 || i >= rows || j < 0
        || j >= cols || grid[i][j] <= 0)
        return;

    // marking the visited element as -1
    grid[i][j] *= -1;

    // computing coordinates with x0, y0 as base
    v.push_back({ i - x0, j - y0 });

    // repeat dfs for neighbors
    for (auto dir : dirs) {
        dfs(grid, x0, y0, i + dir[0], j + dir[1], v);
    }
}

// Main function that returns distinct count of islands in
// a given boolean 2D matrix
int countDistinctIslands(vector<vector<int> >& grid)
{
    int rows = grid.size();
    if (rows == 0)
        return 0;

    int cols = grid[0].size();
    if (cols == 0)
        return 0;

    set<vector<pair<int, int> > > coordinates;

    for (int i = 0; i < rows; ++i) {
        for (int j = 0; j < cols; ++j) {

            // If a cell is not 1
            // no need to dfs
            if (grid[i][j] != 1)
                continue;

            // vector to hold coordinates
            // of this island
            vector<pair<int, int> > v;
            dfs(grid, i, j, i, j, v);

            // insert the coordinates for
            // this island to set
            coordinates.insert(v);
        }
    }

    return coordinates.size();
}

// Driver code
int main()
{
    vector<vector<int> > grid = { { 1, 1, 0, 1, 1 },
                                  { 1, 0, 0, 0, 0 },
                                  { 0, 0, 0, 0, 1 },
                                  { 1, 1, 0, 1, 1 } };

    cout << "Number of distinct islands is "
         << countDistinctIslands(grid);

    return 0;
}

Java

// Java code for the above approach
import java.io.*;
import java.util.*;

class GFG {

  // Driver Code
  public static void main(String[] args)
  {

    // Given Inputs
    int[][] grid = { { 1, 1, 0, 1, 1 },
                    { 1, 0, 0, 0, 0 },
                    { 0, 0, 0, 0, 1 },
                    { 1, 1, 0, 1, 1 }
                   };

    // Function Call
    System.out.println("Number of distinct islands is " + countDistinctIslands(grid));
  }

  // 2D array for the storing the horizontal and vertical directions. (Up, left, down, right}
  static int[][] dirs = { { 0, -1 },
                         { -1, 0 },
                         { 0, 1 },
                         { 1, 0 }
                        };

  private static String toString(int r, int c) {
    return Integer.toString(r) + " " + Integer.toString(c);
  }

  // Function to perform dfs of the input grid
  private static void dfs(int[][] grid, int x0, int y0, int i, int j, ArrayList<String>v) {

    int rows = grid.length, cols = grid[0].length;

    if (i < 0 || i >= rows || j < 0 || j >= cols || grid[i][j] <= 0)
      return;

    // marking the visited element as -1
    grid[i][j] *= -1;

    // computing coordinates with x0, y0 as base
    v.add(toString(i - x0, j - y0));

    // repeat dfs for neighbors
    for (int k = 0; k < 4; k++) {
      dfs(grid, x0, y0, i + dirs[k][0], j + dirs[k][1], v);
    }
  }

  // Main function that returns distinct count of islands in a given boolean 2D matrix
  public static int countDistinctIslands(int[][] grid) {
    int rows = grid.length;
    if (rows == 0)
      return 0;

    int cols = grid[0].length;
    if (cols == 0)
      return 0;

    HashSet<ArrayList<String>> coordinates = new HashSet<> ();

    for (int i = 0; i < rows; ++i) {
      for (int j = 0; j < cols; ++j) {

        // If a cell is not 1 no need to dfs
        if (grid[i][j] != 1)
          continue;

        // vector to hold coordinates of this island
        ArrayList<String> v = new ArrayList<>();
        dfs(grid, i, j, i, j, v);

        // insert the coordinates for
        // this island to set
        coordinates.add(v);
      }
    }

    return coordinates.size();
  }
}

// This code is contributed by ajaymakvana.

Python3

# Python implementation of above approach

# 2D array for the storing the horizontal and vertical
# directions. (Up, left, down, right
dirs = [ [ 0, -1 ],
        [ -1, 0 ],
        [ 0, 1 ],
        [ 1, 0 ] ]

# Function to perform dfs of the input grid
def dfs(grid, x0, y0, i, j, v):
    rows = len(grid)
    cols = len(grid[0])

    if i < 0 or i >= rows or j < 0 or j >= cols or grid[i][j] <= 0:
        return
    # marking the visited element as -1
    grid[i][j] *= -1

    # computing coordinates with x0, y0 as base
    v.append( (i - x0, j - y0) )

    # repeat dfs for neighbors
    for dir in dirs:
        dfs(grid, x0, y0, i + dir[0], j + dir[1], v)



# Main function that returns distinct count of islands in
# a given boolean 2D matrix
def countDistinctIslands(grid):
    rows = len(grid)
    if rows == 0:
        return 0

    cols = len(grid[0])
    if cols == 0:
        return 0

    coordinates = set()

    for i in range(rows):
        for j in range(cols):

            # If a cell is not 1
            # no need to dfs
            if grid[i][j] != 1:
                continue

            # to hold coordinates
            # of this island
            v = []
            dfs(grid, i, j, i, j, v)

            # insert the coordinates for
            # this island to set
            coordinates.add(tuple(v))

    return len(coordinates)

# Driver code
grid = [[ 1, 1, 0, 1, 1 ],
[ 1, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 1 ],
[ 1, 1, 0, 1, 1 ] ]

print("Number of distinct islands is", countDistinctIslands(grid))

# This code is contributed by ankush_953

Output:

Number of distinct islands is 3

Time complexity: O(rows * cols * log(rows * cols))

Where rows is the number of rows and cols is the number of columns in the matrix, here we visit every cell so O(row * col) for that and for every cell we need to add atmost (row * col) pairs in set which will cost us O(log(rows*cols)) so overall time complexity will be O(rows * cols * log(rows * cols)) 

Auxiliary Space: O(rows * cols) 

In set we need to ad atmost rows*cols entry so space complexity will be O(rows * cols)