3391. Design a 3D Binary Matrix with Efficient Layer Tracking 🔒

Description

You are given a n x n x n binary 3D array matrix.

Implement the Matrix3D class:

• Matrix3D(int n) Initializes the object with the 3D binary array matrix, where all elements are initially set to 0.
• void setCell(int x, int y, int z) Sets the value at matrix[x][y][z] to 1.
• void unsetCell(int x, int y, int z) Sets the value at matrix[x][y][z] to 0.
• int largestMatrix() Returns the index x where matrix[x] contains the most number of 1's. If there are multiple such indices, return the largest x.

 

Example 1:

Input:
["Matrix3D", "setCell", "largestMatrix", "setCell", "largestMatrix", "setCell", "largestMatrix"]
[[3], [0, 0, 0], [], [1, 1, 2], [], [0, 0, 1], []]

Output:
[null, null, 0, null, 1, null, 0]

Explanation

Matrix3D matrix3D = new Matrix3D(3); // Initializes a 3 x 3 x 3 3D array matrix, filled with all 0's.
matrix3D.setCell(0, 0, 0); // Sets matrix[0][0][0] to 1.
matrix3D.largestMatrix(); // Returns 0. matrix[0] has the most number of 1's.
matrix3D.setCell(1, 1, 2); // Sets matrix[1][1][2] to 1.
matrix3D.largestMatrix(); // Returns 1. matrix[0] and matrix[1] tie with the most number of 1's, but index 1 is bigger.
matrix3D.setCell(0, 0, 1); // Sets matrix[0][0][1] to 1.
matrix3D.largestMatrix(); // Returns 0. matrix[0] has the most number of 1's.

Example 2:

Input:
["Matrix3D", "setCell", "largestMatrix", "unsetCell", "largestMatrix"]
[[4], [2, 1, 1], [], [2, 1, 1], []]

Output:
[null, null, 2, null, 3]

Explanation

Matrix3D matrix3D = new Matrix3D(4); // Initializes a 4 x 4 x 4 3D array matrix, filled with all 0's.
matrix3D.setCell(2, 1, 1); // Sets matrix[2][1][1] to 1.
matrix3D.largestMatrix(); // Returns 2. matrix[2] has the most number of 1's.
matrix3D.unsetCell(2, 1, 1); // Sets matrix[2][1][1] to 0.
matrix3D.largestMatrix(); // Returns 3. All indices from 0 to 3 tie with the same number of 1's, but index 3 is the biggest.

 

Constraints:

• 1 <= n <= 100
• 0 <= x, y, z < n
• At most 105 calls are made in total to setCell and unsetCell.
• At most 104 calls are made to largestMatrix.

Solutions

Solution 1: Counting + Ordered Set

We use a three-dimensional array \(\textit{g}\) to represent the matrix, where \(\textit{g}[x][y][z]\) represents the value at coordinate \((x, y, z)\) in the matrix. We use an array \(\textit{cnt}\) of length \(n\) to record the number of 1s in each layer. We use an ordered set \(\textit{sl}\) to maintain the number of 1s and the layer number for each layer. The elements in \(\textit{sl}\) are \((\textit{cnt}[x], x)\), so \(\textit{sl}\) can be sorted in descending order by the number of 1s, and in descending order by layer number if the number of 1s is the same.

When calling the setCell method, we first check if \((x, y, z)\) has already been set to 1. If it has, we return directly. Otherwise, we set \(\textit{g}[x][y][z]\) to 1, remove \((\textit{cnt}[x], x)\) from \(\textit{sl}\), increment \(\textit{cnt}[x]\) by 1, and add \((\textit{cnt}[x], x)\) to \(\textit{sl}\).

When calling the unsetCell method, we first check if \((x, y, z)\) has already been set to 0. If it has, we return directly. Otherwise, we set \(\textit{g}[x][y][z]\) to 0, remove \((\textit{cnt}[x], x)\) from \(\textit{sl}\), decrement \(\textit{cnt}[x]\) by 1, and if \(\textit{cnt}[x]\) is greater than 0, add \((\textit{cnt}[x], x)\) to \(\textit{sl}\).

When calling the largestMatrix method, we return the second value of the first element in \(\textit{sl}\). If \(\textit{sl}\) is empty, we return \(n - 1\).

In terms of time complexity, the setCell and unsetCell methods both have a time complexity of \(O(\log n)\), and the largestMatrix method has a time complexity of \(O(1)\). The space complexity is \(O(n^3)\).

Python3JavaC++Go

class matrix3D:

    def __init__(self, n: int):
        self.g = [[[0] * n for _ in range(n)] for _ in range(n)]
        self.cnt = [0] * n
        self.sl = SortedList(key=lambda x: (-x[0], -x[1]))

    def setCell(self, x: int, y: int, z: int) -> None:
        if self.g[x][y][z]:
            return
        self.g[x][y][z] = 1
        self.sl.discard((self.cnt[x], x))
        self.cnt[x] += 1
        self.sl.add((self.cnt[x], x))

    def unsetCell(self, x: int, y: int, z: int) -> None:
        if self.g[x][y][z] == 0:
            return
        self.g[x][y][z] = 0
        self.sl.discard((self.cnt[x], x))
        self.cnt[x] -= 1
        if self.cnt[x]:
            self.sl.add((self.cnt[x], x))

    def largestMatrix(self) -> int:
        return self.sl[0][1] if self.sl else len(self.g) - 1


# Your matrix3D object will be instantiated and called as such:
# obj = matrix3D(n)
# obj.setCell(x,y,z)
# obj.unsetCell(x,y,z)
# param_3 = obj.largestMatrix()

class matrix3D {
    private final int[][][] g;
    private final int[] cnt;
    private final TreeSet<int[]> sl
        = new TreeSet<>((a, b) -> a[0] == b[0] ? b[1] - a[1] : b[0] - a[0]);

    public matrix3D(int n) {
        g = new int[n][n][n];
        cnt = new int[n];
    }

    public void setCell(int x, int y, int z) {
        if (g[x][y][z] == 1) {
            return;
        }
        g[x][y][z] = 1;
        sl.remove(new int[] {cnt[x], x});
        cnt[x]++;
        sl.add(new int[] {cnt[x], x});
    }

    public void unsetCell(int x, int y, int z) {
        if (g[x][y][z] == 0) {
            return;
        }
        g[x][y][z] = 0;
        sl.remove(new int[] {cnt[x], x});
        cnt[x]--;
        if (cnt[x] > 0) {
            sl.add(new int[] {cnt[x], x});
        }
    }

    public int largestMatrix() {
        return sl.isEmpty() ? g.length - 1 : sl.first()[1];
    }
}

/**
 * Your matrix3D object will be instantiated and called as such:
 * matrix3D obj = new matrix3D(n);
 * obj.setCell(x,y,z);
 * obj.unsetCell(x,y,z);
 * int param_3 = obj.largestMatrix();
 */

class matrix3D {
private:
    vector<vector<vector<int>>> g;
    vector<int> cnt;
    set<pair<int, int>> sl;

public:
    matrix3D(int n) {
        g.resize(n, vector<vector<int>>(n, vector<int>(n, 0)));
        cnt.resize(n, 0);
    }

    void setCell(int x, int y, int z) {
        if (g[x][y][z] == 1) {
            return;
        }
        g[x][y][z] = 1;
        sl.erase({-cnt[x], -x});
        cnt[x]++;
        sl.insert({-cnt[x], -x});
    }

    void unsetCell(int x, int y, int z) {
        if (g[x][y][z] == 0) {
            return;
        }
        g[x][y][z] = 0;
        sl.erase({-cnt[x], -x});
        cnt[x]--;
        if (cnt[x]) {
            sl.insert({-cnt[x], -x});
        }
    }

    int largestMatrix() {
        return sl.empty() ? g.size() - 1 : -sl.begin()->second;
    }
};

/**
 * Your matrix3D object will be instantiated and called as such:
 * matrix3D* obj = new matrix3D(n);
 * obj->setCell(x,y,z);
 * obj->unsetCell(x,y,z);
 * int param_3 = obj->largestMatrix();
 */

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