Program to find minimum total distance between house and nearest mailbox in Python

Suppose we have an array called houses and have another value k. Here houses[i] represents the location of the ith house along a street, we have to allocate k mailboxes in the street, and find the minimum total distance between each house and its nearest mailbox.

So, if the input is like houses = [6,7,9,16,22] and k = 2, then the output will be 9 because if we place mailbox at 7 and 18, then minimum total distance from each house is |6-7|+|7-7|+|9-7|+|16-18|+|22-18| = 1+0+2+2+4 = 9.

Algorithm

To solve this problem, we use a recursive approach with dynamic programming:

• Sort the houses array

• For k=1, place the mailbox at the median position to minimize total distance

• For k>1, try all possible positions for the first mailbox and recursively solve for remaining houses

Example

def solve(houses, k):
    houses.sort()
    
    def util(idx, n, k):
        # Base case: only one mailbox left
        if k == 1:
            # Place mailbox at median position
            core = houses[(n + idx) // 2]
            return sum([abs(houses[i] - core) for i in range(idx, n + 1)])
        
        result = float('inf')
        for i in range(idx, n + 1):
            # Not enough houses left for remaining mailboxes
            if n - i < k - 1:
                break
            # Try placing first mailbox covering houses[idx:i+1]
            # Then solve for remaining houses with k-1 mailboxes
            result = min(result, util(idx, i, 1) + util(i+1, n, k - 1))
        
        return result
    
    return util(0, len(houses) - 1, k)

# Test the solution
houses = [6, 7, 9, 16, 22]
k = 2
result = solve(houses, k)
print(f"Houses: {houses}")
print(f"Number of mailboxes: {k}")
print(f"Minimum total distance: {result}")

Houses: [6, 7, 9, 16, 22]
Number of mailboxes: 2
Minimum total distance: 9

How It Works

The algorithm works by:

• Sorting: Houses are sorted to work with consecutive segments

• Median placement: For a single mailbox, placing it at the median minimizes total distance

• Recursive partitioning: For multiple mailboxes, we try all ways to assign houses to the first mailbox, then solve recursively for the rest

Another Example

# Test with different input
houses = [1, 4, 8, 10, 20]
k = 3
result = solve(houses, k)
print(f"Houses: {houses}")
print(f"Number of mailboxes: {k}")
print(f"Minimum total distance: {result}")

Houses: [1, 4, 8, 10, 20]
Number of mailboxes: 3
Minimum total distance: 5

Conclusion

This recursive solution with optimal mailbox placement finds the minimum total distance by trying all possible ways to partition houses among k mailboxes. The key insight is placing each mailbox at the median of its assigned houses to minimize distance.