Delivering Boxes from Storage to Ports

Delivering Boxes from Storage to Ports

1. What is this problem about?

The Delivering Boxes from Storage to Ports interview question is a complex optimization problem. You have a truck with weight and box-count limits and a sequence of boxes destined for different ports. You must deliver them in the given order, making multiple trips if necessary. Each trip starts and ends at the storage. A trip to a port costs 1, and moving between different ports costs 1. You need to minimize the total number of trips (trips between locations).

2. Why is this asked in interviews?

Companies like Nutanix ask this Delivering Boxes from Storage to Ports coding problem to test a candidate's mastery of Dynamic Programming and optimization techniques. It's a "Hard" problem because a standard $O(N^2)$ DP will time out. Candidates must use a Sliding Window or Monotonic Queue to optimize the transitions to $O(N)$.

3. Algorithmic pattern used

This problem uses Dynamic Programming with Monotonic Queue Optimization.

• dp[i] is the min trips to deliver the first i boxes.
• dp[i] = min(dp[j] + cost_to_deliver_from_j_to_i) where the window [j, i] satisfies truck limits.
• Pre-calculate costs and weights using Prefix Sums.
• Use a deque to maintain candidate j indices that potentially minimize the DP expression.

4. Example explanation

Boxes for ports: [1, 1, 2]. Truck limits: max 2 boxes, max 10kg.

1. Trip 1: Boxes [1, 1]. Cost: Storage -> Port 1 (1) -> Storage (1) = 2.
2. Trip 2: Box [2]. Cost: Storage -> Port 2 (1) -> Storage (1) = 2. Total: 4. Alternatively, if the truck could take all 3: Storage -> Port 1 (1) -> Port 2 (1) -> Storage (1) = 3.

5. Common mistakes candidates make

• Brute Force DP: Writing a simple loop for j from 0 to i-1, resulting in $O(N^2)$ which is too slow for $N=10^5$.
• Cost Logic Errors: Miscalculating the trips between ports (e.g., forgetting that consecutive boxes for the same port don't add trip cost).
• Truck Limits: Failing to strictly enforce both weight and box-count constraints within the sliding window.

6. Interview preparation tip

This is an advanced variation of the "Sliding Window Maximum" pattern. If you see a DP where the transition depends on a range that "slides" forward, look into Monotonic Queue Optimization to bring the complexity down from quadratic to linear.