Problem Description
You are given a 0-indexed m x n integer matrix grid. Your initial position is at the top-left cell (0, 0). Starting from the cell (i, j), you can move to one of the following cells: Cells (i, k) with j < k <= grid[i][j] + j (rightward movement), or Cells (k, j) with i < k <= grid[i][j] + i (downward movement). Return the minimum number of cells you need to visit to reach the bottom-right cell (m - 1, n - 1). If there is no valid path, return -1.
Key Insights
- The movement is determined by the value in the current cell, which dictates how far you can move right or down.
- A breadth-first search (BFS) approach is suitable for finding the shortest path in an unweighted grid.
- Keeping track of visited cells is necessary to avoid cycles and redundant paths.
- The problem can be viewed as navigating a dynamically defined graph where nodes are matrix cells and edges are defined by allowable moves.
Space and Time Complexity
Time Complexity: O(m * n), as each cell may be processed once in the BFS.
Space Complexity: O(m * n), for the queue used in BFS and the visited array.
Solution
To solve this problem, we will use a breadth-first search (BFS) approach. We will initialize a queue starting from the top-left cell (0, 0) and explore all possible cells we can reach based on the movement rules defined by the grid values. For each cell, we will check the valid cells we can move to, add them to the queue if they haven't been visited, and keep track of the number of cells visited. If we reach the bottom-right cell, we return the count of cells visited. If the queue is exhausted and we haven't reached the target cell, we return -1.