Hard: Maximum score of a path in a grid with dynamic constraints

The stored title “Hard: Maximum score of a path in a grid with dynamic constraints” is not enough to identify a unique algorithm safely. Begin by restating what is known and ask for the input shape, desired output, constraints, and whether correctness, optimization, or counting is required. Then propose a conditional plan: if the relation is local, consider scanning or dynamic programming; if it involves connectivity, model a graph; if it needs repeated best-element access, consider a heap; if it needs fast lookup, consider a map. Work through a small example and state an invariant before committing to code. Explain edge cases such as empty input, duplicates, overflow, invalid values, and ties. Give complexity only once the data model and operation are agreed. This is a strong interview response because it prevents fabricated assumptions while showing that you can turn an ambiguous request into an executable, testable design.

Complexity: If the dynamic constraints still permit one state per cell, an R by C grid DP is O(RC) time and O(RC) space; extra state dimensions must be counted explicitly.