Solution Explanation for Best Meeting Point

This problem asks to find the minimal total travel distance for friends to meet at a single point, given their home locations on a grid. The distance is calculated using Manhattan distance.

Core Idea: The optimal meeting point's x-coordinate is the median of the x-coordinates of all houses, and similarly for the y-coordinate. This is because choosing the median minimizes the sum of absolute differences between the chosen coordinate and all others.

Algorithm:

1. Collect coordinates: Iterate through the grid and store the row and column indices (x, y coordinates) of all houses (cells with value 1) into separate lists, rows and cols.

2. Find medians: Sort the cols list. The median of the rows list is rows[rows.length / 2] and the median of the cols list is cols[cols.length / 2]. These medians represent the optimal meeting point coordinates (x_m, y_m).

3. Calculate total distance: Compute the Manhattan distance from each house to the meeting point and sum up these distances. This can be done efficiently by using a helper function f(arr, x) that sums the absolute differences between each element v in arr and the median x. The total distance is f(rows, x_m) + f(cols, y_m).

Time Complexity Analysis:

• Collecting coordinates: O(m*n), where 'm' and 'n' are the dimensions of the grid.
• Sorting columns: O(k log k), where 'k' is the number of houses (k <= m*n).
• Calculating distances: O(k)
• Overall: O(m*n) as sorting complexity is dominated by the coordinate collection.

Space Complexity Analysis:

• Space used to store rows and cols lists: O(k), where k is the number of houses.
• Space for sorting (in-place sorting can reduce auxiliary space): O(log k) or O(k) depending on the sorting algorithm.
• Overall: O(k)

Code Examples (Python, Java, C++, Go, Rust):

The code provided in the problem statement implements this algorithm effectively in various programming languages. Each implementation follows these steps and achieves the same time and space complexity. Key differences among the implementations are mostly syntactic sugar, data structure choices (e.g. ArrayList vs vector), and the handling of median calculation. The Rust code uses a helper function to calculate the Manhattan distance more clearly, and all implementations efficiently compute the median using integer division.

Example in Python (as provided, slightly more detailed comments):

class Solution:
    def minTotalDistance(self, grid: List[List[int]]) -> int:
        # Helper function to calculate sum of absolute differences from median
        def f(arr, x):
            return sum(abs(v - x) for v in arr)
 
        rows, cols = [], []  # Store house coordinates
        for i, row in enumerate(grid):
            for j, v in enumerate(row):
                if v:  #If it's a house
                    rows.append(i)
                    cols.append(j)
 
        #Find medians
        cols.sort() #Sort cols to find median easily
        x_median = rows[len(rows) // 2] #Median of row coordinates
        y_median = cols[len(cols) // 2] #Median of col coordinates
 
 
        #Calculate and return total distance
        return f(rows, x_median) + f(cols, y_median)

The other language examples (Java, C++, Go, and Rust) follow a very similar structure, adapting the syntax and data structures appropriate for each language. They all aim for the same efficient algorithm with O(m*n) time complexity and O(k) space complexity.