Multi-Dimensional DP State Modeling

When DP state is a vector (i, j, k, maskbit…) complexity multiplies—reduce dimensions by noticing independence or compressing with rolling layers.

Why this shows up in the real world

3D grid path with keys and doors uses position + key bitmask. ML hyper grid search discretized.

Core idea (explained for students)

Enumerate state tuple carefully; use functools.lru_cache on frozen params for prototyping.

Try this in Python

from functools import lru_cache


@lru_cache(maxsize=None)
def grid3(a: int, b: int, c: int) -> int:
    if a < 0 or b < 0 or c < 0:
        return 0
    if a == b == c == 0:
        return 1
    return grid3(a - 1, b, c) + grid3(a, b - 1, c) + grid3(a, b, c - 1)


print(grid3(2, 2, 2))

Common mistakes

• State explosion 2^k.
• Python recursion depth on deep tuples.

Key takeaways

• Prove dimension reduction (e.g., order of pickups irrelevant) before coding.