Remove Boxes - Video Solutions

Problem Overview: You are given an array where each value represents a colored box. Removing k adjacent boxes of the same color gives k * k points. After removal the array collapses. The challenge is choosing the removal order that maximizes total score.

Approach 1: Brute Force Recursion (Exponential Time, O(n) Space)

The naive idea is to try every possible group of equal adjacent boxes, remove it, and recursively solve the remaining array. Each removal changes the structure of the array, so the recursion branches heavily. This explores all valid removal orders and returns the best score. The method quickly becomes infeasible because the number of states grows exponentially with n. It helps build intuition but cannot pass large constraints.

Approach 2: Recursive Dynamic Programming with Memoization (O(n^4) Time, O(n^3) Space)

The key insight is that removing boxes later can produce a larger group if matching colors are preserved. Define a DP state dp(l, r, k) where l and r define the current subarray and k represents how many boxes equal to boxes[r] are attached to the right side. First compress consecutive equal boxes at the end. One option removes the group immediately: dp(l, r-1, 0) + (k+1)^2. Another option merges it with a matching color earlier in the array by splitting at index i where boxes[i] == boxes[r]. Memoizing these states avoids recomputation and turns the exponential search into polynomial time. This approach relies heavily on dynamic programming and memoization to store overlapping subproblems.

Approach 3: Iterative Dynamic Programming (O(n^4) Time, O(n^3) Space)

The same state transition can be implemented bottom‑up. Build a 3D DP table where dp[l][r][k] represents the maximum score for the interval [l, r] with k extra boxes equal to boxes[r]. Iterate over increasing subarray lengths and compute transitions based on removing the last group or merging with a matching color earlier in the interval. The iterative version avoids recursion overhead and works well in languages like C++. The algorithm still performs nested loops over indices and merge positions, giving O(n^4) time complexity.

Recommended for interviews: Interviewers expect the recursive DP with memoization solution. It demonstrates understanding of state compression and interval dynamic programming. Explaining why the extra parameter k is required shows deeper insight. Brute force demonstrates the search space, but the memoized DP shows the optimization skill expected for hard problems.