Allocate Mailboxes

We can use Dynamic Programming (DP) with memoization to efficiently explore different ways of allocating the mailboxes and find the minimal total distance.

Here’s a step-by-step explanation of the solution:

  1. Sort the houses: Sort the houses array so that we have them in increasing order. This will help in calculating the distances easily.

  2. Calculate Pairwise Distances: We will create a precomputed distance table that stores the minimal distance for every pair of houses if we put a mailbox between them.

  3. Use Dynamic Programming with Memoization: We will recursively try to allocate mailboxes to the houses, and we’ll use a DP table to store previously computed results to avoid redundant calculations.

  4. Find the Optimal Solution: We’ll explore different ways of placing the mailboxes and find the one with the minimum total distance.

Here’s the code implementing this solution:

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class Solution:
    def minDistance(self, houses: List[int], k: int) -> int:
        # Sorting houses for easier calculations
        houses.sort()

        # Precompute distances for all possible pairs of houses
        n = len(houses)
        distance = [[0] * n for _ in range(n)]
        for i in range(n):
            for j in range(i, n):
                median = houses[(i + j) // 2]
                distance[i][j] = sum(abs(median - houses[x]) for x in range(i, j + 1))

        # Define the DP function with memoization
        memo = {}
        def dp(i, k):
            if k == 0 and i == n:
                return 0
            if k == 0 or i == n:
                return float('inf')
            if (i, k) not in memo:
                memo[(i, k)] = min(dp(j, k - 1) + distance[i][j - 1] for j in range(i + 1, n + 1))
            return memo[(i, k)]

        return dp(0, k)

The time complexity of this solution is (O(n^2k)), where (n) is the number of houses.

Identifying Problem Isomorphism

“Allocate Mailboxes” is about optimizing the placement of k mailboxes in a street, represented as an array of house locations, so as to minimize the total distance that all houses have from their nearest mailbox. It’s a problem involving dynamic programming, median finding, and a concept of optimization.

A simpler problem that introduces some of these concepts is “Minimum Moves to Equal Array Elements II”. In this problem, you are given an array of integers and you need to find the minimum number of moves required to make all array elements equal. This problem introduces the concept of finding the median to minimize the total distance, although it doesn’t involve dynamic programming.

An approximately isomorphic problem is “Super Egg Drop”. Like “Allocate Mailboxes”, this problem requires dynamic programming to find an optimal strategy under given constraints. The goal in this case is to find the minimum number of moves required to determine the highest floor from which an egg can be dropped without breaking, given a number of eggs and a number of floors.

A more complex problem is “Minimum Number of Refueling Stops”. This problem also involves optimizing a placement to minimize a cost (in this case, the number of refueling stops a car must make to reach its destination). But it is more complex because it adds a further constraint: the fuel tank capacity.

In increasing order of complexity:

  1. “Minimum Moves to Equal Array Elements II” - Minimize the total number of moves required to make all array elements equal.
  2. “Allocate Mailboxes” - Minimize the total distance that all houses have from their nearest mailbox.
  3. “Super Egg Drop” - Minimize the number of moves required to determine the highest floor from which an egg can be dropped without breaking.
  4. “Minimum Number of Refueling Stops” - Minimize the number of refueling stops a car must make to reach its destination.

“1478. Allocate Mailboxes” is a dynamic programming problem that involves understanding of median and distance calculation. Here are some simpler problems to prepare for this:

  1. LeetCode 198. House Robber

    • This problem introduces the concept of dynamic programming for making decisions to maximize a quantity.

  2. LeetCode 64. Minimum Path Sum

    • This problem is about finding the minimum path sum in a grid, similar to finding the minimum distance for mailboxes.

  3. LeetCode 300. Longest Increasing Subsequence

    • This problem will help you understand the concept of dynamic programming over sequences.

  4. LeetCode 1043. Partition Array for Maximum Sum

    • This problem involves partitioning an array for a maximum sum, a similar concept to allocating mailboxes in the “Allocate Mailboxes” problem.

  5. LeetCode 416. Partition Equal Subset Sum

    • This problem is about partitioning an array into two subsets with equal sum, which can help you understand how to partition houses into groups for mailboxes.

  6. LeetCode 931. Minimum Falling Path Sum

    • This problem introduces the concept of finding minimum sums in a matrix, similar to the “Allocate Mailboxes” problem.

  7. LeetCode 221. Maximal Square

    • This problem involves dynamic programming over a grid, similar to “Allocate Mailboxes”.

  8. LeetCode 279. Perfect Squares

    • This problem introduces the concept of dynamic programming for partitioning a number into a sum of squares, similar to partitioning houses into groups for mailboxes.

  9. LeetCode 322. Coin Change

    • This problem is about making choices to minimize cost, which can help you understand how to set up dynamic programming states and transitions.

  10. LeetCode 215. Kth Largest Element in an Array

    • This problem involves finding the Kth largest element in an array, which can help you understand the concept of finding median in the “Allocate Mailboxes” problem.

These cover how to use dynamic programming to solve problems involving partitioning and minimizing a quantity, which are key to solving the “1478. Allocate Mailboxes” problem.

Given the array houses where houses[i] is the location of the ith house along a street and an integer k, allocate k mailboxes in the street.

Return the minimum total distance between each house and its nearest mailbox.

The test cases are generated so that the answer fits in a 32-bit integer.

Example 1:

Input: houses = [1,4,8,10,20], k = 3 Output: 5 Explanation: Allocate mailboxes in position 3, 9 and 20. Minimum total distance from each houses to nearest mailboxes is |3-1| + |4-3| + |9-8| + |10-9| + |20-20| = 5

Example 2:

Input: houses = [2,3,5,12,18], k = 2 Output: 9 Explanation: Allocate mailboxes in position 3 and 14. Minimum total distance from each houses to nearest mailboxes is |2-3| + |3-3| + |5-3| + |12-14| + |18-14| = 9.

Constraints:

1 <= k <= houses.length <= 100 1 <= houses[i] <= 104 All the integers of houses are unique.