Minimum Number of Visited Cells in a Grid

Minimum Number of Visited Cells in a Grid

1. What is this problem about?

You are at the top-left cell (0, 0) of a grid and want to reach the bottom-right cell (m-1, n-1). Each cell (i, j) contains a value k. From (i, j), you can jump to any cell in the same row (i, j+1) to (i, j+k) or any cell in the same column (i+1, j) to (i+k, j). The goal is to find the minimum number of cells you need to visit to reach the target.

2. Why is this asked in interviews?

This is a Hard-level optimization problem that appears in interviews at DE Shaw and WorldQuant. It tests a candidate's ability to optimize a BFS. A standard BFS would check every possible jump, which could be O(M*N * (M+N)), far too slow. It requires using advanced data structures like Segment Trees, Fenwick Trees, or Monotonic Stacks to efficiently find the minimum "steps" in a range.

3. Algorithmic pattern used

The pattern is BFS optimized with a Range Minimum Query (RMQ) structure. You can use a set of Segment Trees (one for each row and one for each column) or a specialized "jump" tracker to store the minimum steps to reach cells in that row/column. As you process a cell, you query the range to find the best next step and update the structure.

4. Example explanation

Grid: [[3, 4, 2, 1], [4, 2, 1, 1], [2, 1, 1, 0]]

• At (0,0), value is 3. You can jump to (0,1), (0,2), (0,3) in the row, or (1,0), (2,0) in the column.
• Each of these destinations now has a "cost" of 2 (starting cell + destination).
• From (0,3), you might be able to reach the target in fewer steps than from (2,0). The optimized BFS ensures we don't re-scan cells we've already "covered" with a better or equal step count.

5. Common mistakes candidates make

The primary mistake is a naive BFS/Dijkstra, which results in a Time Limit Exceeded error. Candidates also struggle with the implementation details of the range-optimization structure—managing 2D updates across rows and columns is mentally taxing. Another error is not correctly handling the "0" value, which means no further jumps are possible from that cell.

6. Interview preparation tip

Master Range Minimum Query (RMQ) techniques. For grid problems with "jumps," you often need to quickly find the best previous result in a 1D range. Segment Trees are powerful but sometimes a simple monotonic queue or a sorted set can suffice depending on the constraints.