Grid Path DP: When to Use It

Start with the idea

A problem is a good match for Grid Path DP when it asks for robot paths, matrix costs, obstacle grids. Here the concrete task is: Return unique paths through a 3 by 4 empty grid moving right or down. The visible input rows=3 cols=4 is an example, not a definition of the whole problem. Ask what operations the solution must support and what the input guarantees. Write the dependency arrows before choosing loop order; a recurrence that reads the future cannot be filled left to right without adjustment. The right choice follows from those guarantees, not from a familiar noun in the prompt.

The decisive invariant is: A cell combines ways or cost from the predecessor directions allowed by the movement rule. Try to say why it applies to the task without quoting the implementation. Which fact becomes known after one update, and which possible answers remain? A multidimensional or structured DP state records the information needed to identify a subproblem: two prefixes, an interval, a bitmask, a tree node, or a resource limit. If that claim cannot be maintained on a legal input, the pattern is wrong or one of its preconditions is missing. A counterexample is more useful than a vague warning: change one property of the sample, trace the first two steps, and identify the exact claim that fails.

Compare costs before committing. The advertised bound is O(rows*cols) time · O(cols) space possible; identify the operation responsible for it and the extra memory it needs.