# 711 Number of Distinct Islands II

## 711. Number of Distinct Islands II

## 1. Question

Given a non-empty 2D array`grid`of 0's and 1's, an **island** is a group of`1`'s (representing land) connected 4-directionally (horizontal or vertical.) You may assume all four edges of the grid are surrounded by water.

Count the number of **distinct** islands. An island is considered to be the same as another if they have the same shape, or have the same shape after **rotation** (90, 180, or 270 degrees only) or **reflection** (left/right direction or up/down direction).

**Example 1:**

```
11000
10000
00001
00011
```

Given the above grid map, return`1`.

Notice that:

```
11
1
```

and

```
 1
11
```

are considered **same** island shapes. Because if we make a 180 degrees clockwise rotation on the first island, then two islands will have the same shapes.

**Example 2:**

```
11100
10001
01001
01110
```

Given the above grid map, return`2`.

Here are the two distinct islands:

```
111
1
```

and

```
1
1
```

Notice that:

```
111
1
```

and

```
1
111
```

are considered **same** island shapes. Because if we flip the first array in the up/down direction, then they have the same shapes.

**Note:**The length of each dimension in the given`grid`does not exceed 50.

## 2. Implementation

**(1) DFS**

思路: 由于这里的island只要通过旋转，对称变换后形状一样，就被认为是一样的island，所以我们可以通过以下步骤得到解

1.首先通过DFS寻找island，并在这个过程中通过一个ArrayList存DFS找到的一个island的所有坐标

2.通过normalize(),得到当前island的shape key。normalize会将island的每个坐标转换成8种变化，然后通过排序得到取第一个作为表示该island形状的key，将key放入set中

3.最后set的size就代表有多少个distinct island

```java
class Solution {
    public int numDistinctIslands2(int[][] grid) {
        if (grid == null || grid.length == 0) {
            return 0;
        }

        Set<String> set = new HashSet();

        int[][] directions = {{-1, 0}, {1, 0}, {0, -1}, {0, 1}};
        int[][] transformations = {{-1, -1}, {-1, 1}, {1, -1}, {1, 1}};

        int m = grid.length, n = grid[0].length;

        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == 1) {
                    List<int[]> cells = new ArrayList();
                    getIslandsByDFS(i, j, grid, cells, directions);
                    String key = normalize(cells, transformations);
                    set.add(key);
                }
            }
        }
        return set.size();
    }

    public void getIslandsByDFS(int row, int col, int[][] grid, List<int[]> cells, int[][] directions) {
        cells.add(new int[] {row, col});
        grid[row][col] = 0;

        for (int[] direction : directions) {
            int nextRow = row + direction[0];
            int nextCol = col + direction[1];

            if (isValid(nextRow, nextCol, grid)) {
                getIslandsByDFS(nextRow, nextCol, grid, cells, directions);
            }
        }
    }

    public boolean isValid(int row, int col, int[][] grid) {
        return row >= 0 && row < grid.length && col >= 0 && col < grid[0].length && grid[row][col] == 1;
    }

    public String normalize(List<int[]> cells, int[][] transformations) {
        List<String> forms = new ArrayList();

        for (int[] tran : transformations) {
            List<int[]> list1 = new ArrayList();
            List<int[]> list2 = new ArrayList();

            for (int[] cell : cells) {
                int x = cell[0];
                int y = cell[1];

                // Generate the 8 different transformations
                // (x, y), (x, -y), (-x, y), (-x, -y)
                // (y, x), (-y, x), (y, -x), (-y, -x)
                list1.add(new int[] {x * tran[0], y * tran[1]});
                list2.add(new int[] {y * tran[1], x * tran[0]});
            }
            forms.add(getKey(list1));
            forms.add(getKey(list2));
        }

        // Take the first one the represent the shape of the island
        Collections.sort(forms);
        return forms.get(0);
    }

    public String getKey(List<int[]> cells) {
        // Sort the cells and the take the first cell in sorted order as the origin
        Collections.sort(cells, new Comparator<int[]>(){
            @Override
            public int compare(int[] a, int[] b) {
                return a[0] == b[0] ? a[1] - b[1] : a[0] - b[0];
            }
        });

        StringBuilder key = new StringBuilder();

        int originX = cells.get(0)[0], originY = cells.get(0)[1];
        // Generate the key 
        for (int[] cell : cells) {
            key.append((cell[0] - originX) + ":" + (cell[1] - originY));
        }
        return key.toString();
    }
}
```

## 3. Time & Space Complexity

DFS: 时间复杂度O(mn \* log(mn)), 空间复杂度O(mn)


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