Convert a Memoization Solution to Bottom-Up DP on Grids

Memoized recursion is a great start, but interviewers often ask you to convert it to bottom-up DP for clearer time analysis. Grid problems make this translation approachable if you map the same state to table coordinates. This guide shows how to do it with a minimal checklist and visual table.

TL;DR

• Goal: reuse your memoized dp(r, c) definition by laying it into a 2D table with explicit fill order.
• Interviews value this because bottom-up reasoning proves you understand dependencies instead of trusting recursion magic.
• Core steps: freeze the state shape, define base cells, decide traversal order, and copy the recurrence into loop form.
• Beginners often skip boundary initialization or iterate in the wrong direction, breaking dependencies.
• You will learn a table diagram, a loop template, and safety checks for indices and memory.

Beginner-Friendly Explanation: Same State, Different Execution

A memoized function dp(r, c) caches answers after exploring neighbors. Bottom-up uses the exact state but fills a 2D array in an order that respects dependencies. The logic stays the same; only the evaluation order changes. You already mapped state and overlap if you followed How to Spot Overlapping Subproblems Before Writing DP.

Step-by-Step Learning Guidance

1) Write the recurrence clearly

Example (unique paths with obstacles): dp[r][c] = dp[r-1][c] + dp[r][c-1] when the cell is open.

2) Identify base cases

• dp[0][0] = 1 if the start is open.
• First row depends only on left cells; first column depends only on top cells. Reinforce with Dynamic Programming Base Cases for Grid Problems.

3) Choose fill order

Iterate rows top to bottom and columns left to right so parents are ready before children. This mirrors the traversal advice in Dynamic Programming Tabulation Checklist for Beginners.

4) Translate memo calls to loops

Replace recursive calls with direct table lookups. Carry over obstacle checks or bounds guards exactly as in the memoized version.

5) Sanity-check constraints

Before allocating dp, confirm rows * cols fits memory, echoing the constraint-first habit from How to Analyze LeetCode Constraints Before Coding: A Beginner Playbook.

6) Narrate dependencies

Explain which neighbors feed the current cell. Tools like LeetCopilot can render a quick heatmap of fill order so you see if any cell reads uninitialized values.

Visualizable Example: Fill Order Grid

Cell order (3x3, row-major):
(0,0) → (0,1) → (0,2)
(1,0) → (1,1) → (1,2)
(2,0) → (2,1) → (2,2)
Parents of (r,c): (r-1,c) and (r,c-1) when inside bounds.

This simple arrow map keeps your loops aligned with dependencies.

Code Example: Converting Memoized Paths to Tabulation

def unique_paths_with_obstacles(grid):
    rows, cols = len(grid), len(grid[0])
    dp = [[0] * cols for _ in range(rows)]
    if grid[0][0] == 1:
        return 0
    dp[0][0] = 1

    for r in range(rows):
        for c in range(cols):
            if grid[r][c] == 1:
                dp[r][c] = 0
                continue
            if r == 0 and c == 0:
                continue
            from_top = dp[r-1][c] if r > 0 else 0
            from_left = dp[r][c-1] if c > 0 else 0
            dp[r][c] = from_top + from_left
    return dp[-1][-1]

Compare this to the memoized recurrence; only evaluation order changed.

Practical Preparation Strategies

• Rewrite your last memo solution: Pick a grid DP you solved with recursion and convert it line by line.
• Annotate dependencies: Draw arrows on a 3x3 board before coding bigger cases.
• Test boundary rows first: Run cases with one row or one column to confirm base handling.
• Use guided validation: Let LeetCopilot highlight cells read before write during dry runs to catch order mistakes early.

Common Mistakes to Avoid

Mistake 1: Skipping obstacle checks

Failing to zero out blocked cells allows illegal paths to accumulate.

Mistake 2: Wrong iteration order

Iterating bottom-up for top-left–anchored recurrences reads uninitialized parents.

Mistake 3: Missing base cells

Forgetting dp[0][0] or first-row/column seeds makes the entire table zero.

Mistake 4: Memory blowups

Allocating dp without checking rows * cols can exceed limits; compress to 1D when needed.

FAQ

• How do I know my order is correct? Each cell must read only from indices already filled; trace one row manually.
• What should I practice first? Start with obstacle-free unique paths, then add walls and diagonals.
• Is bottom-up required in interviews? Often yes; it demonstrates you understand dependencies and complexity.
• When should I keep memoization? If the state graph is sparse or irregular, memoization may be clearer than forcing a dense table.

Conclusion

Converting memoization to bottom-up DP is about honoring the same recurrence with explicit order and base cases. By mapping state to table coordinates, drawing a tiny fill diagram, and double-checking dependencies, you can present a clean iterative solution with confidence.