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711. Number of Distinct Islands II 👎

  • Time: $O(mn\log(mn))$
  • Space: $O(mn)$
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class Solution {
 public:
  int numDistinctIslands2(vector<vector<int>>& grid) {
    const int m = grid.size();
    const int n = grid[0].size();
    // all the islands with different shapes
    set<vector<pair<int, int>>> islands;
    vector<vector<bool>> seen(m, vector<bool>(n));

    for (int i = 0; i < m; ++i)
      for (int j = 0; j < n; ++j) {
        vector<pair<int, int>> island;
        dfs(grid, i, j, seen, island);
        if (!island.empty())
          islands.insert(normalize(island));
      }

    return islands.size();
  }

 private:
  void dfs(const vector<vector<int>>& grid, int i, int j,
           vector<vector<bool>>& seen, vector<pair<int, int>>& island) {
    if (i < 0 || i == grid.size() || j < 0 || j == grid[0].size())
      return;
    if (grid[i][j] == 0 || seen[i][j])
      return;

    seen[i][j] = true;
    island.emplace_back(i, j);

    dfs(grid, i + 1, j, seen, island);
    dfs(grid, i - 1, j, seen, island);
    dfs(grid, i, j + 1, seen, island);
    dfs(grid, i, j - 1, seen, island);
  }

  vector<pair<int, int>> normalize(const vector<pair<int, int>>& island) {
    // points[i] := 8 different rotations/reflections of an island
    vector<vector<pair<int, int>>> points(8);

    for (const auto& [i, j] : island) {
      points[0].emplace_back(i, j);
      points[1].emplace_back(i, -j);
      points[2].emplace_back(-i, j);
      points[3].emplace_back(-i, -j);
      points[4].emplace_back(j, i);
      points[5].emplace_back(j, -i);
      points[6].emplace_back(-j, i);
      points[7].emplace_back(-j, -i);
    }

    for (vector<pair<int, int>>& p : points)
      ranges::sort(p);

    // Normalize each p by substracting p[1..7] with p[0].
    for (vector<pair<int, int>>& p : points) {
      for (int i = 1; i < island.size(); ++i)
        p[i] = {p[i].first - p[0].first, p[i].second - p[0].second};
      p[0] = {0, 0};
    }

    ranges::sort(points);
    return points[0];
  }
};

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class Solution {
  public int numDistinctIslands2(int[][] grid) {
    final int m = grid.length;
    final int n = grid[0].length;
    // all the islands with different shapes
    Set<List<Pair<Integer, Integer>>> islands = new HashSet<>();
    boolean[][] seen = new boolean[m][n];

    for (int i = 0; i < m; ++i)
      for (int j = 0; j < n; ++j) {
        List<Pair<Integer, Integer>> island = new ArrayList<>();
        dfs(grid, i, j, seen, island);
        if (!island.isEmpty())
          islands.add(normalize(island));
      }

    return islands.size();
  }

  private void dfs(int[][] grid, int i, int j, boolean[][] seen,
                   List<Pair<Integer, Integer>> island) {
    if (i < 0 || i == grid.length || j < 0 || j == grid[0].length)
      return;
    if (grid[i][j] == 0 || seen[i][j])
      return;

    seen[i][j] = true;
    island.add(new Pair<>(i, j));

    dfs(grid, i + 1, j, seen, island);
    dfs(grid, i - 1, j, seen, island);
    dfs(grid, i, j + 1, seen, island);
    dfs(grid, i, j - 1, seen, island);
  }

  private List<Pair<Integer, Integer>> normalize(List<Pair<Integer, Integer>> island) {
    // points[i] := 8 different rotations/reflections of an island
    Pair<Integer, Integer>[][] points = new Pair[8][island.size()];

    for (int k = 0; k < island.size(); ++k) {
      final int i = island.get(k).getKey();
      final int j = island.get(k).getValue();
      points[0][k] = new Pair<>(i, j);
      points[1][k] = new Pair<>(i, -j);
      points[2][k] = new Pair<>(-i, j);
      points[3][k] = new Pair<>(-i, -j);
      points[4][k] = new Pair<>(j, i);
      points[5][k] = new Pair<>(j, -i);
      points[6][k] = new Pair<>(-j, i);
      points[7][k] = new Pair<>(-j, -i);
    }

    for (Pair<Integer, Integer>[] point : points)
      Arrays.sort(point,
                  (a, b)
                      -> a.getKey().equals(b.getKey()) ? a.getValue().compareTo(b.getValue())
                                                       : a.getKey().compareTo(b.getKey()));

    // Normalize each p by substracting p[1..7] with p[0].
    for (Pair<Integer, Integer>[] p : points) {
      for (int i = 1; i < island.size(); ++i)
        p[i] = new Pair<>(p[i].getKey() - p[0].getKey(), p[i].getValue() - p[0].getValue());
      p[0] = new Pair<>(0, 0);
    }

    Arrays.sort(points, new Comparator<Pair<Integer, Integer>[]>() {
      @Override
      public int compare(Pair<Integer, Integer>[] a, Pair<Integer, Integer>[] b) {
        for (int i = 0; i < a.length; ++i) {
          if (a[i].getKey() != b[i].getKey())
            return a[i].getKey() - b[i].getKey();
          if (a[i].getValue() != b[i].getValue())
            return a[i].getValue() - b[i].getValue();
        }
        return 0;
      }
    });

    return Arrays.asList(points[0]);
  }
}

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class Solution:
  def numDistinctIslands2(self, grid: list[list[int]]) -> int:
    seen = set()

    def dfs(i: int, j: int):
      if i < 0 or i == len(grid) or j < 0 or j == len(grid[0]):
        return
      if grid[i][j] == 0 or (i, j) in seen:
        return

      seen.add((i, j))
      island.append((i, j))
      dfs(i + 1, j)
      dfs(i - 1, j)
      dfs(i, j + 1)
      dfs(i, j - 1)

    def normalize(island: list[tuple]) -> list[tuple]:
      # points[i] := 8 different rotations/reflections of an island
      points = [[] for _ in range(8)]

      for i, j in island:
        points[0].append((i, j))
        points[1].append((i, -j))
        points[2].append((-i, j))
        points[3].append((-i, -j))
        points[4].append((j, i))
        points[5].append((j, -i))
        points[6].append((-j, i))
        points[7].append((-j, -i))

      points = [sorted(p) for p in points]

      # Normalize each p by substracting p[1..7] with p[0].
      for p in points:
        for i in range(1, len(island)):
          p[i] = (p[i][0] - p[0][0],
                  p[i][1] - p[0][1])
        p[0] = (0, 0)

      return sorted(points)[0]

    islands = set()  # all the islands with different shapes

    for i in range(len(grid)):
      for j in range(len(grid[0])):
        island = []
        dfs(i, j)
        if island:
          islands.add(frozenset(normalize(island)))

    return len(islands)