Allocate Mailboxes

Allocate Mailboxes

What is this problem about?

The "Allocate Mailboxes interview question" is a difficult optimization problem. You are given the positions of several houses on a street (represented by an array of integers) and a number of mailboxes k. Your goal is to place these k mailboxes in such a way that the total distance from each house to its nearest mailbox is minimized. This is essentially a clustering problem where you need to partition houses into groups and find an optimal "center" for each group.

Why is this asked in interviews?

Companies like Goldman Sachs and Amazon use the "Allocate Mailboxes coding problem" to test a candidate's mastery of Dynamic Programming (DP) and mathematical optimization. It requires the candidate to break down a complex global problem into smaller sub-problems: "What is the minimum cost to place $i$ mailboxes for the first $j$ houses?"

Algorithmic pattern used

This problem follows the Dynamic Programming with Pre-calculation pattern.

1. Median Property: For a group of houses in a range [i, j], the optimal place to put a single mailbox is at the median house position. The total distance to this median is the minimum possible cost for that specific range.
2. Pre-calculation: Create a 2D array cost[i][j] that stores the minimum distance for a single mailbox covering houses from index i to j.
3. DP State: dp[k][n] represents the minimum distance to cover the first n houses using k mailboxes.
4. Transition: To find dp[k][n], you iterate through possible split points m, taking the best of dp[k-1][m] + cost[m+1][n].

Example explanation

Houses at: [1, 4, 8, 10, 20], $k = 2$.

• One possible split: Mailbox 1 for [1, 4] (Median at 2.5, total dist: |1-2.5|+|4-2.5|=3). Mailbox 2 for [8, 10, 20] (Median at 10, total dist: |8-10|+|10-10|+|20-10|=12). Total: 15.
• Another split: Mailbox 1 for [1, 4, 8] (Median 4, dist: 3+0+4=7). Mailbox 2 for [10, 20] (Median 15, dist: 5+5=10). Total: 17. The DP algorithm explores all such splits to find the minimum.

Common mistakes candidates make

• Not sorting houses: The median property only works if the house positions are in order.
• Inefficient calculation: Re-calculating the median distance inside the DP loop instead of pre-calculating it, leading to $O(N^4)$ instead of $O(N^3)$ or $O(N^2 \cdot K)$.
• Wrong DP transition: Forgetting to handle the base case where $k=1$.

Interview preparation tip

This is a variation of the "K-Median" problem. When you see "minimize the sum of distances," think of the median. When you see "split an array into K segments," think of Dynamic Programming. Practice similar problems like "Palindrome Partitioning III" to get used to the segment-based DP pattern.