I went straight for DFS with memoization like it was a standard decreasing-path problem and then realized mid-explanation that the transition rule is weird.
Clarify the problem constraints and edge cases first, then propose a dynamic programming solution with memoization where the state includes the current cell and the previous cell's value. Discuss how to handle the non-standard constraint that the next cell must be ≤ current OR ≤ the cell before that, and analyze time/space complexity.
Pro tip: Demonstrate strong problem-solving by first restating the problem in your own words and asking clarifying questions about matrix dimensions, value ranges, and whether the path can revisit cells. This shows you think before coding and helps avoid misunderstandings.
Ask questions to understand constraints: matrix size, value range, whether diagonal moves are allowed, if paths can revisit cells, and if the path must be simple. Confirm the exact condition for the next cell.
Define DP state as (row, col, prev_value) where prev_value is the value of the cell before the current one. The recurrence considers moving to a neighbor if its value ≤ current cell's value OR ≤ prev_value.
Decide between top-down memoization (easier to implement) or bottom-up DP (more efficient). Discuss how to handle cycles if revisiting is allowed, possibly using DFS with memoization and a visited set.
Analyze time and space complexity: O(R*C*V) where V is the number of distinct values, if using value-based state. Discuss potential optimizations like coordinate compression or using a segment tree for range queries.
Walk through a small example to verify the logic, including edge cases like a single cell, all equal values, or strictly decreasing paths. Mention how to handle large inputs.
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