Min Cost to Connect All Points

class Solution {
public:
    int connectPointsWithMinimumCost(vector<vector<int>>& coords) {
        int pointCount = coords.size();
        int totalCost = 0;
        int connections = 0;
        
        vector<bool> isVisited(pointCount);
        
        vector<int> leastDist(pointCount, INT_MAX);
        leastDist[0] = 0;
        
        while (connections < pointCount) {
            int minEdge = INT_MAX;
            int chosenPoint = -1;
            
            // Select the minimum edge node not yet added.
            for (int i = 0; i < pointCount; ++i) {
                if (!isVisited[i] && minEdge > leastDist[i]) {
                    minEdge = leastDist[i];
                    chosenPoint = i;
                }
            }
            
            totalCost += minEdge;
            connections++;
            isVisited[chosenPoint] = true;
            
            // Update the distance for adjacent points.
            for (int adj = 0; adj < pointCount; ++adj) {
                int dist = abs(coords[chosenPoint][0] - coords[adj][0]) + 
                           abs(coords[chosenPoint][1] - coords[adj][1]);
                
                if (!isVisited[adj] && leastDist[adj] > dist) {
                    leastDist[adj] = dist;
                }
            }
        }
        
        return totalCost;
    }
};

This C++ solution to the problem involves finding the minimum cost to connect all points in a given set of 2D coordinates. The approach used here is similar to Prim's algorithm for minimum spanning tree (MST). Here's how the solution operates:

• Begin by initializing some key variables: totalCost to track the total cost of connecting all points, connections to count the number of connections made, and isVisited a boolean vector to keep track of visited nodes.
• leastDist vector is initialized with maximal integers (INT_MAX) except for the starting point, which is set to 0 to signify initiation from point 0.
• Use a while loop to continue creating edges until all points are connected. In each iteration:
  • Locate the unvisited point with the smallest edge distance using a loop, marking this point as the chosenPoint.
  • Add the smallest edge distance minEdge to the totalCost and mark the chosenPoint as visited.
  • Iterate over all other points to update the minimum distances to each if the direct edge from chosenPoint is smaller than the previously recorded distances in leastDist.
• Return the totalCost which now represents the minimum cost required to connect all the given points using the smallest possible sum of distances.

Each step ensures that the next point connected has the smallest edge distance from any of the already connected points, efficiently forming an MST and thus solving the problem optimally.

class MinimumSpanningTree {
    public int calculateMinimumCost(int[][] coordinates) {
        int totalPoints = coordinates.length;
        int totalCost = 0;
        int edgesIncluded = 0;

        boolean[] visited = new boolean[totalPoints];

        int[] distance = new int[totalPoints];
        distance[0] = 0;

        for (int i = 1; i < totalPoints; ++i) {
            distance[i] = Integer.MAX_VALUE;
        }

        while (edgesIncluded < totalPoints) {
            int minEdge = Integer.MAX_VALUE;
            int nearestNode = -1;

            for (int node = 0; node < totalPoints; ++node) {
                if (!visited[node] && minEdge > distance[node]) {
                    minEdge = distance[node];
                    nearestNode = node;
                }
            }

            totalCost += minEdge;
            edgesIncluded++;
            visited[nearestNode] = true;

            for (int adjacent = 0; adjacent < totalPoints; ++adjacent) {
                int dist = Math.abs(coordinates[nearestNode][0] - coordinates[adjacent][0]) +
                           Math.abs(coordinates[nearestNode][1] - coordinates[adjacent][1]);

                if (!visited[adjacent] && distance[adjacent] > dist) {
                    distance[adjacent] = dist;
                }
            }
        }

        return totalCost;
    }
}

The given Java code implements a solution to the problem of finding the minimum cost to connect all points in a plane using Prim's algorithm, a classical algorithm to find the minimum spanning tree (MST) in a weighted undirected graph.

• Start by initializing structures to track visited nodes, the minimum distance of each point from the set of included points, and setting the distance to the first point as 0 (start point).
• Loop until all nodes are included in the MST. During each iteration:
  • Identify the unvisited node that is nearest to the set of included points by finding the minimum edge connecting it.
  • Add the nearest node to the set of included points and update the total cost with the distance value of this nearest node.
  • Update the minimum distances for all adjacent points that are not yet included in the MST. The distance between any two points is calculated using the Manhattan distance formula.

Ensure the total cost is accumulated with each added edge and finally return the total cost, which represents the minimum cost to connect all given points.

This implementation is efficient for solving the problem given its direct use of greedy techniques facilitated by Prim's algorithm. Adjustments can be made for optimization, but primarily, the approach will handle a variety of inputs owing to its robust handling of graph properties through adjacency measurements and systematic node inclusion.

let calculateMinimumCost = function(coordinates) {
    let totalPoints = coordinates.length;
    let totalCost = 0;
    let edgesCount = 0;

    let visited = Array(totalPoints).fill(false);

    let distance = Array(totalPoints).fill(Number.MAX_SAFE_INTEGER);
    distance[0] = 0;

    while (edgesCount < totalPoints) {
        let smallestEdge = Number.MAX_SAFE_INTEGER;
        let currentNode = -1;

        for (let i = 0; i < totalPoints; ++i) {
            if (!visited[i] && smallestEdge > distance[i]) {
                smallestEdge = distance[i];
                currentNode = i;
            }
        }

        totalCost += smallestEdge;
        edgesCount++;
        visited[currentNode] = true;

        for (let next = 0; next < totalPoints; ++next) {
            let cost = Math.abs(coordinates[currentNode][0] - coordinates[next][0]) + 
                       Math.abs(coordinates[currentNode][1] - coordinates[next][1]);

            if (!visited[next] && distance[next] > cost) {
                distance[next] = cost;
            }
        }
    }

    return totalCost;
};

This JavaScript implementation provides a solution to the problem of determining the minimum cost to connect all points where the input is a list of coordinates. The procedure uses Prim’s Algorithm, a greedy algorithm that is effective for finding the minimum spanning tree (MST) in a weighted and undirected graph. Here’s how the code works:

• Coordinate Array Initialization: An array coordinates stores each point as a pair [x, y].

• Distance Initialization: An array distance keeps track of minimum distances. All distances are initialized to Number.MAX_SAFE_INTEGER, except for the first point, which is set to zero to start the algorithm.

• Visited Nodes Tracking: An array visited tracks whether a point has been included in the MST. By default, all entries are false.

• Variables Setup: totalPoints defines the number of points, totalCost tracks the current total minimum cost, and edgesCount keeps the count of edges included in the MST.

• Prim's Algorithm Execution:

  • Iterate while the count of edges is less than the total points.
  • Find the unvisited node with the smallest tentative distance—this becomes the current node.
  • Update the minimum total cost by adding the smallest distance found.
  • Mark this node as visited.
  • Update the distances to the neighboring nodes. If the computed distance (Manhattan distance between the current node and any other unvisited node) is less than the stored distance in the distance array, it is updated.

• Cost Calculation: The loop continues until all points are connected, and totalCost will hold the minimum cost to connect all given points.

This implementation is efficient for the problem at hand and correctly leverages Prim's algorithm to ensure that the resulting graph has the minimum weight and includes all vertices.

class Solution:
    def calculateMinimumCost(self, points: List[List[int]]) -> int:
        num_points = len(points)
        total_cost = 0
        edges_included = 0
        
        # Array to keep track of visited points.
        visited_points = [False] * num_points
        
        edge_costs = [math.inf] * num_points
        edge_costs[0] = 0
        
        while edges_included < num_points:
            smallest_edge = math.inf
            selected_point = -1
            
            # Identifying the point with smallest edge cost not already added.
            for index in range(num_points):
                if not visited_points[index] and smallest_edge > edge_costs[index]:
                    smallest_edge = edge_costs[index]
                    selected_point = index
            
            total_cost += smallest_edge
            edges_included += 1
            visited_points[selected_point] = True
            
            # Updating the minimum edge costs for neighboring points.
            for neighbor in range(num_points):
                dist = abs(points[selected_point][0] - points[neighbor][0]) +\
                       abs(points[selected_point][1] - points[neighbor][1])
                
                if not visited_points[neighbor] and edge_costs[neighbor] > dist:
                    edge_costs[neighbor] = dist
        
        return total_cost

The provided Python solution addresses the problem of minimizing the cost to connect all points. The algorithm runs in a Prim's-like manner, optimized for connecting points by selecting the smallest incremental edge at each step until all points have been included.

Here's how the implementation works:

• Initialize variables for the number of points (num_points), the total cost (total_cost), and the count of edges included (edges_included).
• Create a boolean list visited_points to track if a point has already been included in the cost calculation.
• Initialize edge_costs with infinity and set the cost of the starting point (index 0) to 0.
• Utilize a while loop to add edges until all points are connected (edges_included < num_points):
  1. Identify the non-visited point with the smallest connection cost.
  2. Add this smallest connection cost to total_cost and mark the point as visited.
  3. Update the edge costs for all neighboring points.
• The edge cost update is based on the Manhattan distance between points.

This method ensures that the total cost covers all points with the least possible addition of edge costs, effectively solving the problem with efficiency in both time and space.