Vultr DocsLatest Content

Minimum Cost to Make at Least One Valid Path in a Grid

Updated on 13 June, 2025
Minimum Cost to Make at Least One Valid Path in a Grid header image

Problem Statement

In this problem, you are given an m x n grid where each cell contains a directional sign. This sign indicates which adjoining cell to visit next:

  • 1 points to the right cell.
  • 2 points to the left cell.
  • 3 points downward.
  • 4 points upward.

These directions may lead towards the next cell or, if on the boundary, might point outside the grid itself which is not a valid path. The goal is to trace a path starting from the top-left corner cell (0, 0) to the bottom-right corner cell (m - 1, n - 1).

If necessary, you're allowed a single alteration (change of direction) on any cell, which counts as having a cost of 1. This modification to the grid should help in building at least one valid path from start to end. You are required to determine the minimum cost required to establish such a path using the signs within the grid.

Examples

Example 1

Input:

grid = [[1,1,1,1],[2,2,2,2],[1,1,1,1],[2,2,2,2]]

Output:

3

Explanation:

You will start at point (0, 0).
The path to (3, 3) is as follows. (0, 0) --> (0, 1) --> (0, 2) --> (0, 3) change the arrow to down with cost = 1 --> (1, 3) --> (1, 2) --> (1, 1) --> (1, 0) change the arrow to down with cost = 1 --> (2, 0) --> (2, 1) --> (2, 2) --> (2, 3) change the arrow to down with cost = 1 --> (3, 3)
The total cost = 3.

Example 2

Input:

grid = [[1,1,3],[3,2,2],[1,1,4]]

Output:

0

Explanation:

You can follow the path from (0, 0) to (2, 2).

Example 3

Input:

grid = [[1,2],[4,3]]

Output:

1

Constraints

  • m == grid.length
  • n == grid[i].length
  • 1 <= m, n <= 100
  • 1 <= grid[i][j] <= 4

Approach and Intuition

Based on the provided examples and constraints, constructing a solution will primarily revolve around path-finding with potential modifications. Here's the intuitive approach to solve this:

  1. Initial Grid Validation: First, check if a valid path exists without any modifications. This can be done using BFS or DFS starting from (0, 0) and attempting to reach (m - 1, n - 1) following the inherent directions.

  2. Modification Necessity: If step 1 is unsuccessful, analyze which modification creates a valid path with the minimum cost:

    • Simulate the effect of changing the direction of each cell (with permissible directions, considering the cell's position in the grid to avoid pointing out of bounds).
    • After each prospective modification, attempt to traverse from start to end again. Record the successful paths and their associated costs (which will be 1 due to a single modification).

  3. Priority Queue for Pathfinding: Utilizing a min-heap (or priority queue in some algorithms) could effectively manage multiple potential paths in terms of their cost and feasibility, especially when integrating direction modifications dynamically during path traversal.

  4. Accounting for Edge Cases: Ensure robust handling for cases like very small grids (e.g., 1x1), grids with existing direct paths, or grids where excessive numbers of direction changes make traversal impractical under the constraint of a single modification.

This strategic approach allows blending of direct traversal methods with strategic modifications, providing a comprehensive solution to find the minimal modification cost required for a valid path from the grid's beginning to its end.

Solutions

  • C++
  • Java
  • Python
cpp 
class PathFinder {
private:
    // Movement vectors: right, left, down, up matching to grid value indices (1-4).
    const vector<vector<int>> movement = {{0, 1}, {0, -1}, {1, 0}, {-1, 0}};

public:
    int calculateMinimumCost(vector<vector<int>>& grid) {
        int rows = grid.size(), cols = grid[0].size(), pathCost = 0;

        // Minimum cost for each cell initialization.
        vector<vector<int>> costs(rows, vector<int>(cols, INT_MAX));

        // Queue for BFS - cells to explore at incremental costs.
        queue<pair<int, int>> explorationQueue;

        // Start with the top-left corner.
        explore(grid, 0, 0, costs, pathCost, explorationQueue);

        // BFS - incrementally explore cells.
        while (!explorationQueue.empty()) {
            pathCost++;
            int currentLevel = explorationQueue.size();
            while (currentLevel-- > 0) {
                auto [currentRow, currentCol] = explorationQueue.front();
                explorationQueue.pop();

                // Explore neighbors in four possible directions.
                for (int direction = 0; direction < 4; ++direction) {
                    explore(grid, currentRow + movement[direction][0], currentCol + movement[direction][1], costs,
                        pathCost, explorationQueue);
                }
            }
        }

        return costs[rows - 1][cols - 1];
    }

private:
    // Explore function with DFS mechanics to reach cells of the same cost.
    void explore(vector<vector<int>>& grid, int currentRow, int currentCol,
                 vector<vector<int>>& costs, int currentCost, 
                 queue<pair<int, int>>& explorationQueue) {
        if (!verifyCell(costs, currentRow, currentCol)) return;

        costs[currentRow][currentCol] = currentCost;
        explorationQueue.push({currentRow, currentCol});

        // Follows predefined grid direction without incurring additional cost.
        int directionToFollow = grid[currentRow][currentCol] - 1;
        explore(grid, currentRow + movement[directionToFollow][0], currentCol + movement[directionToFollow][1], costs, currentCost,
            explorationQueue);
    }

    // Validates if the cell is within grid boundaries and unvisited.
    bool verifyCell(vector<vector<int>>& costs, int row, int col) {
        return row >= 0 && col >= 0 && row < costs.size() &&
               col < costs[0].size() && costs[row][col] == INT_MAX;
    }
};

This C++ solution uses a Breadth-First Search (BFS) approach to solve the problem of finding the minimum cost to make at least one valid path in a grid. Each grid value points to one of four possible movements: right, left, down, or up.

*The PathFinder class provides a calculateMinimumCost method to perform this operation: * The method initially sets up a 2D vector of integer costs to store the minimum cost for each cell, initializing these values to INT_MAX to represent unvisited cells. * A queue is used to hold cells for exploration based on the BFS principle. * The BFS starts from the top-left corner (0,0), checking each direction and adding cells to the queue if they can be visited with a lower cost, given the constraints of movement defined in the grid. * The explore private method is implemented to add neighbors to the BFS queue based on valid movements from the current cell. It employs both BFS and Depth-First Search (DFS) mechanics where needed to propagate through cells that are accessible without additional cost by following the grid's defined directions.

*The verifyCell method serves a crucial role by ensuring that only valid, in-bounds, and unvisited cells are considered for exploration. It reduces unnecessary computations and helps to ensure the BFS remains within grid constraints.

This implementation carefully manages its exploration of the grid space, leveraging BFS for level-wise exploration and selective DFS when iterating through zero-cost pathways as defined by the initial grid setup. It cleverly uses C++ features such as vectors, pairs, and queues to handle grid traversal, state management, and coordinate transformations efficiently.

java 
class PathFinder {

    private final int[][] directions = new int[][] {
        { 0, 1 },
        { 0, -1 },
        { 1, 0 },
        { -1, 0 },
    };

    public int findMinimumCost(int[][] grid) {
        int rows = grid.length, cols = grid[0].length, cost = 0;
        int[][] costGrid = new int[rows][cols];

        for (int r = 0; r < rows; r++) {
            Arrays.fill(costGrid[r], Integer.MAX_VALUE);
        }

        Queue<int[]> queue = new LinkedList<>();
        explore(grid, 0, 0, costGrid, cost, queue);

        while (!queue.isEmpty()) {
            cost++;
            int size = queue.size();

            while (size-- > 0) {
                int[] current = queue.poll();
                int r = current[0], c = current[1];

                for (int d = 0; d < 4; d++) {
                    explore(
                        grid,
                        r + directions[d][0],
                        c + directions[d][1],
                        costGrid,
                        cost,
                        queue
                    );
                }
            }
        }

        return costGrid[rows - 1][cols - 1];
    }

    private void explore(
        int[][] grid,
        int r,
        int c,
        int[][] costGrid,
        int cost,
        Queue<int[]> queue
    ) {
        if (!isValid(costGrid, r, c)) return;

        costGrid[r][c] = cost;
        queue.add(new int[] { r, c });

        int next = grid[r][c] - 1;
        explore(
            grid,
            r + directions[next][0],
            c + directions[next][1],
            costGrid,
            cost,
            queue
        );
    }

    private boolean isValid(int[][] costGrid, int r, int c) {
        return (
            r >= 0 &&
            c >= 0 &&
            r < costGrid.length &&
            c < costGrid[0].length &&
            costGrid[r][c] == Integer.MAX_VALUE
        );
    }
}

The Java solution for finding the minimum cost to ensure at least one valid path in a grid involves several key components, primarily focusing on implementing a breadth-first search (BFS) algorithm with additional checks and updates based on grid conditions.

  • An int array (directions) stores possible movement directions in the grid.
  • The findMinimumCost function initializes auxiliary structures and variables, such as a 2D cost array to store minimum costs to reach each cell, and a queue to manage the BFS execution through the grid.
  • Costs are filled initially with Integer.MAX_VALUE to indicate cells that have not been visited yet.
  • The grid exploration starts from the top-left corner, and as BFS proceeds, cells are updated with the current cost if moving to that cell increments the cost, except when the cell's direction is naturally followed based on the grid's value.
  • The explore function handles the logic of checking valid moves within grid boundaries and ensuring that revisits only occur if a better path is found (i.e., a path with a lower cost).
  • Grid cells indicate direction naturally (cost-free move), and any deviation from these directions is penalized with an increased cost.
  • The BFS continues until all possible paths are explored, and finally, the function returns the cost to reach the bottom-right corner of the grid, giving the minimal cost required to ensure at least one valid path.

This implementation ensures that the approach efficiently explores possible paths and updates the paths' costs accordingly, relying on conditions set by the grid's layout and values.

python 
class GridSolver:
    _directions = [(0, 1), (0, -1), (1, 0), (-1, 0)]

    def findMinimumCost(self, grid: List[List[int]]) -> int:
        rows, cols = len(grid), len(grid[0])
        curr_cost = 0

        # Initialize costs to infinity to denote unvisited cells
        costs = [[float('inf')] * cols for _ in range(rows)]

        # Deque for BFS steps - stores cells with updated costs
        queue = collections.deque()

        # Initiate DFS from the top-left corner of the grid
        self.depthFirstSearch(grid, 0, 0, costs, curr_cost, queue)

        # BFS to handle levels of cells with incremental costs
        while queue:
            curr_cost += 1
            for _ in range(len(queue)):
                r, c = queue.popleft()

                # Explore potential movements in four directions
                for idx, (dr, dc) in enumerate(self._directions):
                    self.depthFirstSearch(grid, r + dr, c + dc, costs, curr_cost, queue)

        return costs[rows - 1][cols - 1]

    def depthFirstSearch(
        self, grid, r, c, costs, curr_cost, queue
    ):
        if not self.isValid(costs, r, c):
            return

        costs[r][c] = curr_cost
        queue.append((r, c))

        # Move in the direction indicated by current cell value
        move_dir = grid[r][c] - 1
        dr, dc = self._directions[move_dir]
        self.depthFirstSearch(grid, r + dr, c + dc, costs, curr_cost, queue)

    def isValid(
        self, costs, r, c
    ) -> bool:
        return (
            0 <= r < len(costs)
            and 0 <= c < len(costs[0])
            and costs[r][c] == float("inf")
        )

The provided Python script implements a solution to determine the minimum cost to create at least one valid path through a grid, where each cell in the grid points in one of four cardinal directions. Here’s how the solution operates:

  • The GridSolver class features an internal _directions array for mapping movement based on grid values. Each direction corresponds to right, left, down, and up, respectively.
  • The findMinimumCost function initializes an infinite grid (costs) of the same dimensions as the input grid to represent the cost of reaching each cell. A queue (deque) is used to apply a breadth-first search (BFS) strategy.
  • Using a helper function depthFirstSearch (DFS), the path begins from the top-left corner, recursively following the direction denoted by the cell's value. During this DFS, if a more optimal path to a cell is found, the cell is added to the queue for subsequent BFS processing to explore neighbor cells.
  • As the BFS progresses, it increments the level of cost with each complete exploration of the queue's current elements.

Here are the crucial elements of the script:

  • Using BFS combined with DFS ensures that once a lower cost path is found to a particular cell, it immediately proceeds to explore this path to its fullest potential before moving on to alternative paths.
  • The function returns the cost of reaching the bottom-right corner of the grid, symbolizing the completion of the path based on modified costs.

By using a combination of BFS for exploring all possible paths and DFS for deep diving into promising directions immediately, the algorithm efficiently computes the minimum cost to facilitate at least one valid path across the grid.

Comments

No comments yet.