Trace Path: Why Net Displacement Is the Wrong Check

The Trace Path problem on GeeksforGeeks asks whether a given sequence of moves can be completed within the bounds of an n x m grid, starting from any cell. At first glance, the solution might seem to involve simulating the path from every possible starting position, but this brute-force approach quickly becomes infeasible for large grids.

The key insight is that horizontal moves (L and R) do not affect the row, and vertical moves (U and D) do not affect the column. This means the path must be valid both horizontally and vertically independently. The interviewer highlights that the answer for the grid depends only on the net displacement along each axis, not the order of moves.

To determine if a start position is valid, the interviewer proposes checking whether the path's span (the difference between the farthest offset to the left and right, or the farthest offset up and down) fits within the grid dimensions. For example, if the maximum horizontal displacement is greater than the grid width, no valid start exists.

Rather than attempting to brute-force every possible start, the interviewer suggests a more efficient approach. By considering all starting positions simultaneously, the problem can be reduced to tracking the minimum and maximum offsets along each axis as the path progresses. The valid start positions correspond to those whose offsets remain within the grid bounds throughout the entire sequence.

The interviewer provides a compact solution in Java that implements this approach. The `isPossible` function checks the validity of the path on both axes by maintaining a shrinking interval for each axis. As it iterates through the move string, it updates the interval boundaries based on the move type. If at any point the minimum offset exceeds the maximum, the interval becomes invalid, indicating that no valid start position exists.

This elegant solution leverages the independence of horizontal and vertical movements to reduce the problem to two one-dimensional checks, each operating in linear time relative to the length of the move string. The final result is the logical conjunction of the two checks, yielding a boolean indicating whether the move sequence can be completed within the grid boundaries.