Problem Statement
Consider a square matrix of size n x n, where each cell of the matrix contains either a positive integer or the value -1 which indicates that the cell is blocked. The challenge involves calculating the "remoteness" for each cell in the grid. Here, the remoteness, denoted as R[i][j], follows certain rules:
- For a non-blocked cell
(i, j), the value ofR[i][j]is the sum of values in all those non-blocked cells from which(i, j)is not accessible. - If a cell
(i, j)is blocked, thenR[i][j]is directly set to 0.
The task is to compute the sum of R[i][j] for all cells in the matrix. This involves understanding which cells are isolated from others due to the blockages and adding up their values as described.
Examples
Example 1
Input:
grid = [[-1,1,-1],[5,-1,4],[-1,3,-1]]
Output:
39
Explanation:
In the picture above, there are four grids. The top-left grid contains the initial values in the grid. Blocked cells are colored black, and other cells get their values as it is in the input. In the top-right grid, you can see the value of R[i][j] for all cells. So the answer would be the sum of them. That is: 0 + 12 + 0 + 8 + 0 + 9 + 0 + 10 + 0 = 39. Let's jump on the bottom-left grid in the above picture and calculate R[0][1] (the target cell is colored green). We should sum up the value of cells that can't be reached by the cell (0, 1). These cells are colored yellow in this grid. So R[0][1] = 5 + 4 + 3 = 12. Now let's jump on the bottom-right grid in the above picture and calculate R[1][2] (the target cell is colored green). We should sum up the value of cells that can't be reached by the cell (1, 2). These cells are colored yellow in this grid. So R[1][2] = 1 + 5 + 3 = 9.
Example 2
Input:
grid = [[-1,3,4],[-1,-1,-1],[3,-1,-1]]
Output:
13
Explanation:
In the picture above, there are four grids. The top-left grid contains the initial values in the grid. Blocked cells are colored black, and other cells get their values as it is in the input. In the top-right grid, you can see the value of R[i][j] for all cells. So the answer would be the sum of them. That is: 3 + 3 + 0 + 0 + 0 + 0 + 7 + 0 + 0 = 13. Let's jump on the bottom-left grid in the above picture and calculate R[0][2] (the target cell is colored green). We should sum up the value of cells that can't be reached by the cell (0, 2). This cell is colored yellow in this grid. So R[0][2] = 3. Now let's jump on the bottom-right grid in the above picture and calculate R[2][0] (the target cell is colored green). We should sum up the value of cells that can't be reached by the cell (2, 0). These cells are colored yellow in this grid. So R[2][0] = 3 + 4 = 7.
Example 3
Input:
grid = [[1]]
Output:
0
Explanation:
Since there are no other cells than (0, 0), R[0][0] is equal to 0. So the sum of R[i][j] over all cells would be 0.
Constraints
1 <= n <= 3001 <= grid[i][j] <= 106orgrid[i][j] == -1
Approach and Intuition
To solve this problem, let's examine the provided examples and focus on the necessary computations based on the constraints. Given that each cell can only be reached via its direct neighbors (up, down, left, right), the challenge emphasizes efficient navigation and isolation detection within a matrix environment.
Analyzing the Examples:
Example 1:
- The grid provided is
[[ -1, 1, -1], [5, -1, 4], [-1, 3, -1]] - For R[0][1], the unaccessible cells are: 5, 4, 3 (because they are separated by blocked cells). The calculation becomes
5 + 4 + 3 = 12 - In the complete summary across the grid, every blocked cell contextually isolates groups of cells, thereby making their values contributory to the remoteness of accessible neighbor cells.
- The grid provided is
Example 2:
- The remoteness calculations take into account which clusters of cells are severed from the others due to blockages and tally their values.
- For any directly non-blocked cells that are unaccessed in traversing sequences due to barriers, they are summed up for corresponding accessible cells' remoteness.
Example 3:
- A single cell provides a trivial case where no other cell is there to compute remoteness with, hence resulting in zero.
Proposed Method:
To implement a solution, a logical approach might involve:
- Utilizing a graph traversal technique such as BFS (Breadth-First Search) or DFS (Depth-First Search) to identify regions (clusters of cells) that are reachable amongst themselves.
- Mapping out these clusters will help to understand which groups of cells add to a particular cell's remoteness.
- For every cell
(i, j), one needs to compute the sum of cells' values not reachable from(i, j)considering all non-blocked cells. - An optimized pathfinding or connectivity analysis within the grid is integral due to the allowable matrix maximum size constraint (up to 300 x 300), leading to potential for a high complexity operation.
By creating a clear mapping of connectivity and using efficient algorithms for cluster detection, we can significantly streamline the computation of remoteness across the grid cells.
Solutions
- C++
class Solution {
private:
const vector<vector<int>> directions = {{0, 1}, {0, -1}, {1, 0}, {-1, 0}};
public:
long long calculateRemoteness(vector<vector<int>>& matrix) {
int n = matrix.size();
UnionFind uf(n);
for (int r = 0; r < n; r++) {
for (int c = 0; c < n; c++) {
if (matrix[r][c] == -1) continue;
for (auto& d : directions) {
int newRow = d[0] + r;
int newCol = d[1] + c;
if (cellValid(matrix, newRow, newCol)) {
uf.connect(convertToIndex(n, r, c),
convertToIndex(n, newRow, newCol));
}
}
}
}
long long totalSum = 0;
unordered_map<int, long long> componentSum;
for (int r = 0; r < n; r++) {
for (int c = 0; c < n; c++) {
if (matrix[r][c] == -1) continue;
int root = uf.getRoot(convertToIndex(n, r, c));
componentSum[root] += matrix[r][c];
totalSum += matrix[r][c];
}
}
long long result = 0;
for (int r = 0; r < n; r++) {
for (int c = 0; c < n; c++) {
if (matrix[r][c] == -1) continue;
long long currComponentSum = componentSum[uf.getRoot(convertToIndex(n, r, c))];
result += totalSum - currComponentSum;
}
}
return result;
}
class UnionFind {
private:
vector<int> parent;
vector<int> rank;
public:
UnionFind(int n) {
parent.resize(n * n, -1);
rank.resize(n * n, 1);
}
int getRoot(int idx) {
if (parent[idx] == -1) return idx;
return parent[idx] = getRoot(parent[idx]);
}
void connect(int i, int j) {
int root1 = getRoot(i);
int root2 = getRoot(j);
if (root1 == root2) return;
if (rank[root1] < rank[root2]) {
parent[root1] = root2;
} else if (rank[root1] > rank[root2]) {
parent[root2] = root1;
} else {
parent[root2] = root1;
rank[root1]++;
}
}
};
int convertToIndex(int n, int row, int col) { return row * n + col; }
bool cellValid(vector<vector<int>>& matrix, int row, int col) {
int n = matrix.size();
return row >= 0 && col >= 0 && row < n && col < n &&
matrix[row][col] != -1;
}
};
The provided C++ solution aims to calculate the "Sum of Remoteness of All Cells" in a given matrix. The matrix cells contain either an integer value indicating openness or a sentinel value (-1) indicating blocked cells. Below is an explanation of how the solution works.
Start by defining a helper class
UnionFindto aid in grouping connected cells. This class tracks each cell's parent and rank, providing methods for path compression (getRoot) and union operation (connect).The main method
calculateRemotenessprocesses the matrix to connect adjacent open cells using the Union-Find data structure. The connectivity here is defined in horizontal and vertical directions only, given by thedirectionsvector.For each cell, use nested for loops to navigate through the matrix. If the current cell is open, iterate over the possible directions and compute new row and column indices. Then, using the helper functions
cellValidandconvertToIndex, connect current cells to its neighbors if the neighboring cells are open.Create a mapping (
componentSum) to track the sum of values for each connected component. Sum up the values in each connected component as you continue connecting cells. Keep a running total (totalSum) of all cell values as well.Finally, calculate the remoteness for each cell. For cells that are part of the same component, calculate the difference between the total sum of all cells and the sum for the component that the current cell belongs to. Accumulate these differences in
result.The function returns the total calculated remoteness across all cells.
This program efficiently handles connectedness using the Union-Find data structure and provides a direct way to compute remoteness by leveraging the sums of connected components. It assumes accurate integer handling without overflowing and operates within the constraints of matrix traversal and connectivity logic.
- Java
class Solution {
// Movement directions for adjacent cells
private final int[][] movements = { { 0, 1 }, { 0, -1 }, { 1, 0 }, { -1, 0 } };
public long sumDistances(int[][] matrix) {
int size = matrix.length;
// Creating an instance of a union-find structure
UnionFind unionFind = new UnionFind(size);
// Linking adjacent open cells
for (int i = 0; i < size; i++) {
for (int j = 0; j < size; j++) {
if (matrix[i][j] == -1) continue;
for (int[] move : movements) {
int newI = move[0] + i;
int newJ = move[1] + j;
if (isValidGrid(matrix, newI, newJ)) {
unionFind.union(
toIndex(size, i, j),
toIndex(size, newI, newJ)
);
}
}
}
}
// Summing values in components
long remoteTotal = 0;
Map<Integer, Long> componentSums = new HashMap<>();
for (int i = 0; i < size; i++) {
for (int j = 0; j < size; j++) {
if (matrix[i][j] == -1) continue;
int root = unionFind.find(toIndex(size, i, j));
componentSums.put(
root,
componentSums.getOrDefault(root, 0L) + matrix[i][j]
);
remoteTotal += matrix[i][j];
}
}
// Calculating total remoteness
long remotenessSum = 0;
for (int i = 0; i < size; i++) {
for (int j = 0; j < size; j++) {
if (matrix[i][j] == -1) continue;
long componentSum = componentSums.get(unionFind.find(toIndex(size, i, j)));
remotenessSum += remoteTotal - componentSum;
}
}
return remotenessSum;
}
private class UnionFind {
int[] parent;
int[] size;
UnionFind(int n) {
parent = new int[n * n];
size = new int[n * n];
Arrays.fill(parent, -1);
Arrays.fill(size, 1);
}
int find(int idx) {
if (parent[idx] == -1) return idx;
return parent[idx] = find(parent[idx]);
}
void union(int x, int y) {
int rootX = find(x);
int rootY = find(y);
if (rootX != rootY) {
if (size[rootX] > size[rootY]) {
parent[rootY] = rootX;
} else if (size[rootX] < size[rootY]) {
parent[rootX] = rootY;
} else {
parent[rootY] = rootX;
size[rootX]++;
}
}
}
}
private int toIndex(int dim, int x, int y) {
return x * dim + y;
}
private boolean isValidGrid(int[][] grid, int x, int y) {
return x >= 0 && y >= 0 && x < grid.length && y < grid.length && grid[x][y] != -1;
}
}
This Java solution calculates the sum of remoteness for all valid cells in a given matrix using a union-find data structure. The algorithm involves several critical steps:
- Initialize a UnionFind instance to handle union and find operations, which help determine connected components in the matrix.
- Use nested loops to access each cell in the matrix. If a cell contains a negative value, it is skipped.
- For each valid cell, check its four adjacent neighbors (up, down, left, right). If a neighbor is valid and within matrix bounds, perform a union operation to link the current cell with the neighboring cell.
- Create a hash map to maintain the sum of values for each component identified by the union-find structure.
- Calculate the total sum of all values in the matrix for later calculations.
- Compute the remoteness for each cell by subtracting the sum of the values in its component from the total matrix sum.
- Accumulate these values to get the overall remoteness sum.
Utility methods incorporated in this solution:
- isValidGrid: Checks if a specified cell position is valid.
- toIndex: Converts matrix coordinates to a unique index for the union-find structure.
The inner UnionFind class manages connectivity details using the parent and size arrays. The find method uses path compression, and the union method uses union by size to efficiently manage component merging. This approach ensures efficient calculations even for larger matrices, leveraging the data structure to optimize connectivity checks and component summarization.
- Python
class Solution:
def calculateRemoteness(self, matrix: list[list[int]]) -> int:
# Direction vectors for movement
self.directions = [(0, 1), (0, -1), (1, 0), (-1, 0)]
size = len(matrix)
# Initialize Union-Find structure
union_find = self._UnionFind(size)
# Connect adjacent cells
for i in range(size):
for j in range(size):
if matrix[i][j] == -1:
continue
for dx, dy in self.directions:
new_i, new_j = i + dx, j + dy
if self._isValid(matrix, new_i, new_j):
union_find.union(
self._convertToIndex(size, i, j),
self._convertToIndex(size, new_i, new_j)
)
# Calculate component sums and total
component_sums = {}
grand_total = 0
for i in range(size):
for j in range(size):
if matrix[i][j] == -1:
continue
root = union_find.find(self._convertToIndex(size, i, j))
component_sums[root] = component_sums.get(root, 0) + matrix[i][j]
grand_total += matrix[i][j]
# Compute remoteness for each cell
remoteness_total = sum(
grand_total - component_sums[union_find.find(self._convertToIndex(size, i, j))]
for i in range(size)
for j in range(size)
if matrix[i][j] != -1
)
return remoteness_total
class _UnionFind:
def __init__(self, size: int):
self.parent = [-1] * (size * size)
self.rank = [1] * (size * size)
def find(self, x: int) -> int:
if self.parent[x] == -1:
return x
self.parent[x] = self.find(self.parent[x])
return self.parent[x]
def union(self, x: int, y: int):
rootX = self.find(x)
rootY = self.find(y)
if rootX != rootY:
if self.rank[rootX] > self.rank[rootY]:
self.parent[rootY] = rootX
elif self.rank[rootX] < self.rank[rootY]:
self.parent[rootX] = rootY
else:
self.parent[rootY] = rootX
self.rank[rootX] += 1
def _convertToIndex(self, size: int, i: int, j: int) -> int:
return i * size + j
def _isValid(self, matrix: list[list[int]], i: int, j: int) -> bool:
size = len(matrix)
return 0 <= i < size and 0 <= j < size and matrix[i][j] != -1
In this Python solution, calculate the "remoteness" of each cell in a grid by evaluating the sums of the values in connected components. Here's a breakdown:
- Start by defining directions for movement within the grid (up, down, left, right).
- Use a nested loop to iterate over each cell in the matrix.
- Utilize a custom Union-Find data structure to handle the connectivity of cells. Only connect cells if they are valid (not marked by
-1) and adjacent. - For each connected component, determine the sum of values in that component and the total of all matrix values.
- The remoteness of a given cell is calculated by subtracting the sum of its component from the grand total of the matrix.
- Sum the remoteness values for each valid cell to get the total remoteness.
Key functions used:
- _UnionFind: Manages connectivity and rooting of cells.
- _convertToIndex: Converts matrix grid coordinates into a flat index.
- _isValid: Checks if a cell coordinate is valid for considering connectivity (must be within bounds and not marked as
-1).
The solution iteratively builds connectivity with a Union-Find structure and then uses these connections to compute remoteness, leveraging properties of sums within interconnected components.
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