Merge pull request #1503 from Hitesh4278/min-cost-to-connect-all-points

Added Solution of Min cost to connect all points -  Issue No - 1440

Author: ajay-dhangar

dsa-problems/leetcode-problems/1500-1599.md

@@ -518,7 +518,7 @@ export const problems = [
     "problemName": "1584. Min Cost to Connect All Points",
     "difficulty": "Medium",
     "leetCodeLink": "https://leetcode.com/problems/min-cost-to-connect-all-points",
-    "solutionLink": "#"
+    "solutionLink": "/dsa-solutions/lc-solutions/1500-1599/min-cost-to-connect-all-points"
   },
   {
     "problemName": "1585. Check If String Is Transformable With Substring Sort Operations",

dsa-solutions/lc-solutions/1500-1599/1584-min-cost-to-connect-all-points.md

@@ -0,0 +1,376 @@
+---
+id: min-cost-to-connect-all-points
+title: Min Cost to Connect All Points
+sidebar_label: 1584 - Min Cost to Connect All Points
+tags:
+- Array
+- Union Find
+- Graph
+- Minimum Spanning Tree
+description: "This is a solution to the Min Cost to Connect All Points problem on LeetCode."
+---
+
+## Problem Description
+You are given an array points representing integer coordinates of some points on a 2D-plane, where `points[i] = [xi, yi].`
+
+The cost of connecting two points [xi, yi] and [xj, yj] is the manhattan distance between them: `|xi - xj| + |yi - yj|`, where |val| denotes the absolute value of val.
+
+Return the minimum cost to make all points connected. All points are connected if there is exactly one simple path between any two points.
+
+### Examples
+**Example 1:**
+![image](https://assets.leetcode.com/uploads/2020/08/26/d.png)
+```
+Input: points = [[0,0],[2,2],[3,10],[5,2],[7,0]]
+Output: 20
+Explanation:
+![image](https://assets.leetcode.com/uploads/2020/08/26/c.png)
+We can connect the points as shown above to get the minimum cost of 20.
+Notice that there is a unique path between every pair of points.
+
+```
+
+### Constraints
+- `1 <= points.length <= 1000`
+- `-10^6 <= xi, yi <= 10^6`
+- `All pairs (xi, yi) are distinct.`
+
+## Solution for Min Cost to Connect All Points
+### Approach 
+#### Prim's algorithm:
+
+- Prim's algorithm is an algorithm for solving the optimization problem of finding the minimum spanning tree in a weighted connected graph within graph theory. A minimum spanning tree is a subset of the edges of the graph that forms a tree containing all vertices while minimizing the total weight of those edges.
+
+#### Overview of the Algorithm:
+
+- Calculate the distances between each pair of points and use Prim's algorithm to form the minimum spanning tree.
+- Start from an initial point, mark it as visited, and select the point with the smallest distance among the unvisited points.
+- Calculate the distances from the selected point to the unvisited points and store them in a cache.
+- Add the minimum cost edge to the priority queue using the distances from the cache.
+- Repeat the above steps until all points are visited, and calculate the minimum cost.
+#### Specific Steps:
+
+#### Initial State:
+
+- n: Number of points
+- min_cost: Minimum cost (initially 0) and return value
+- visited: A list to indicate if each point is visited (initially all False)
+- pq: Priority queue (initially (0, 0) indicating starting from point 0 with cost 0)
+- cache: Dictionary for caching distances (initially empty)
+#### Each Step:
+
+- Pop cost and point from pq (start from the initial point).
+- If the point is already visited, skip this point.
+- Otherwise, mark this point as visited and add the current cost to the minimum cost.
+- Calculate distances from this point to all unvisited points and store them in the cache. Update the cache if the new distance is smaller.
+- Add the point with the smallest distance among the unvisited points to the priority queue using distances from the cache.
+- Repeat steps 3 to 5 until all points are visited.
+- Return the final minimum cost.
+
+
+<Tabs>
+  <TabItem value="Solution" label="Solution">
+
+    #### Implementation
+    ```jsx live
+    function Solution(arr) {
+    var minCostConnectPoints = function(points) {
+        let cost = 0;
+        const n = points.length;
+        const dist = Array(n).fill(Infinity);
+        dist[0] = 0;
+        let next = 0;
+
+        for (let step = 1; step < n; step++) {
+            let min = Infinity;
+            let point = -1;
+            for (let i = 1; i < n; i++) {
+                if (dist[i] > 0) {
+                    dist[i] = Math.min(dist[i], Math.abs(points[i][0] - points[next][0]) + Math.abs(points[i][1] - points[next][1]));
+                    if (dist[i] < min) {
+                        min = dist[i];
+                        point = i;
+                    }
+                }
+            }
+            cost += min;
+            dist[point] = 0;
+            next = point;
+        }
+        
+        return cost;
+    };
+      const input = [[0,0],[2,2],[3,10],[5,2],[7,0]]
+      const output =minCostConnectPoints(input)
+      return (
+        <div>
+          <p>
+            <b>Input: </b>
+            {JSON.stringify(input)}
+          </p>
+          <p>
+            <b>Output:</b> {output.toString()}
+          </p>
+        </div>
+      );
+    }
+    ```
+
+    #### Complexity Analysis
+
+    - Time Complexity: $O(n^2 * log(n)) $ 
+    - Space Complexity: $ O(n)$
+
+   ## Code in Different Languages
+   <Tabs>
+  <TabItem value="JavaScript" label="JavaScript">
+  <SolutionAuthor name="@hiteshgahanolia"/>
+   ```javascript
+    manhattDist(v1, v2) {
+        return Math.abs(v1[0] - v2[0]) + Math.abs(v1[1] - v2[1]);
+    }
+
+    minCostConnectPoints(points) {
+        const n = points.length;
+        const adj = Array.from({ length: n }, () => []);
+
+        for (let i = 0; i < n; i++) {
+            for (let j = i + 1; j < n; j++) {
+                const dist = this.manhattDist(points[i], points[j]);
+                adj[i].push([j, dist]);
+                adj[j].push([i, dist]);
+            }
+        }
+
+        const pq = new MinPriorityQueue({ priority: x => x[0] });
+        const vis = Array(n).fill(false);
+        pq.enqueue([0, 0]);
+        let cost = 0;
+
+        while (!pq.isEmpty()) {
+            const [topEdgwWt, currNode] = pq.dequeue().element;
+
+            if (vis[currNode]) continue;
+            vis[currNode] = true;
+            cost += topEdgwWt;
+
+            for (const [adjPoint, edWt] of adj[currNode]) {
+                if (!vis[adjPoint]) {
+                    pq.enqueue([edWt, adjPoint]);
+                }
+            }
+        }
+
+        return cost;
+    }
+    ```
+
+  </TabItem>
+  <TabItem value="TypeScript" label="TypeScript">
+  <SolutionAuthor name="@hiteshgahanolia"/> 
+   ```typescript
+   class Solution {
+    manhattDist(v1: number[], v2: number[]): number {
+        return Math.abs(v1[0] - v2[0]) + Math.abs(v1[1] - v2[1]);
+    }
+
+    minCostConnectPoints(points: number[][]): number {
+        const n = points.length;
+        const adj: [number, number][][] = Array.from({ length: n }, () => []);
+
+        for (let i = 0; i < n; i++) {
+            for (let j = i + 1; j < n; j++) {
+                const dist = this.manhattDist(points[i], points[j]);
+                adj[i].push([j, dist]);
+                adj[j].push([i, dist]);
+            }
+        }
+
+        const pq = new MinPriorityQueue({ priority: (x: [number, number]) => x[0] });
+        const vis = Array(n).fill(false);
+        pq.enqueue([0, 0]);
+        let cost = 0;
+
+        while (!pq.isEmpty()) {
+            const [topEdgwWt, currNode] = pq.dequeue().element;
+
+            if (vis[currNode]) continue;
+            vis[currNode] = true;
+            cost += topEdgwWt;
+
+            for (const [adjPoint, edWt] of adj[currNode]) {
+                if (!vis[adjPoint]) {
+                    pq.enqueue([edWt, adjPoint]);
+                }
+            }
+        }
+
+        return cost;
+    }
+}
+
+    ```
+  </TabItem>
+  <TabItem value="Python" label="Python">
+  <SolutionAuthor name="@hiteshgahanolia"/>
+   ```python
+   import heapq
+
+class Solution:
+    def manhattDist(self, v1, v2):
+        return abs(v1[0] - v2[0]) + abs(v1[1] - v2[1])
+
+    def minCostConnectPoints(self, points):
+        n = len(points)
+        adj = [[] for _ in range(n)]
+
+        for i in range(n):
+            for j in range(i + 1, n):
+                dist = self.manhattDist(points[i], points[j])
+                adj[i].append((j, dist))
+                adj[j].append((i, dist))
+
+        pq = [(0, 0)]  # (distance, point)
+        vis = [False] * n
+        cost = 0
+
+        while pq:
+            topEdgwWt, currNode = heapq.heappop(pq)
+
+            if vis[currNode]:
+                continue
+            vis[currNode] = True
+            cost += topEdgwWt
+
+            for adjPoint, edWt in adj[currNode]:
+                if not vis[adjPoint]:
+                    heapq.heappush(pq, (edWt, adjPoint))
+
+        return cost
+
+    ```
+
+  </TabItem>
+  <TabItem value="Java" label="Java">
+  <SolutionAuthor name="@hiteshgahanolia"/>
+   ```java
+   import java.util.*;
+
+class Solution {
+    private int manhattDist(int[] v1, int[] v2) {
+        return Math.abs(v1[0] - v2[0]) + Math.abs(v1[1] - v2[1]);
+    }
+
+    public int minCostConnectPoints(int[][] points) {
+        int n = points.length;
+        List<List<int[]>> adj = new ArrayList<>();
+
+        for (int i = 0; i < n; i++) {
+            adj.add(new ArrayList<>());
+        }
+
+        for (int i = 0; i < n; i++) {
+            for (int j = i + 1; j < n; j++) {
+                int dist = manhattDist(points[i], points[j]);
+                adj.get(i).add(new int[]{j, dist});
+                adj.get(j).add(new int[]{i, dist});
+            }
+        }
+
+        PriorityQueue<int[]> pq = new PriorityQueue<>(Comparator.comparingInt(a -> a[0]));
+        boolean[] vis = new boolean[n];
+        pq.add(new int[]{0, 0}); // {distance, point}
+        int cost = 0;
+
+        while (!pq.isEmpty()) {
+            int[] curr = pq.poll();
+            int topEdgwWt = curr[0];
+            int currNode = curr[1];
+
+            if (vis[currNode]) continue;
+            vis[currNode] = true;
+            cost += topEdgwWt;
+
+            for (int[] neighbor : adj.get(currNode)) {
+                int adjPoint = neighbor[0];
+                int edWt = neighbor[1];
+                if (!vis[adjPoint]) {
+                    pq.add(new int[]{edWt, adjPoint});
+                }
+            }
+        }
+
+        return cost;
+    }
+}
+
+    ```
+
+
+  </TabItem>
+  <TabItem value="C++" label="C++">
+  <SolutionAuthor name="@hiteshgahanolia"/>
+   ```cpp
+   class Solution {
+public:
+    int manhattDist(vector<int>&v1 , vector<int>&v2)
+    {
+        return abs(abs(v1[0]-v2[0]) + abs(v1[1]-v2[1]));
+    }
+    int minCostConnectPoints(vector<vector<int>>& points) {
+        int n = points.size();
+        vector<pair<int,int>> adj[n]; //{point , cost or Manhattan dist}
+
+        //we have to make a adjacency list fom every point to every other point
+        for(int i=0; i<n ; i++)
+        {
+            for(int j=i+1;j<n ; j++)
+            {
+                adj[i].push_back({j , manhattDist(points[i] , points[j])});
+                adj[j].push_back({i , manhattDist(points[i] , points[j])});
+            }
+        }
+
+        priority_queue<pair<int,int> , vector<pair<int,int>> , greater<pair<int,int>> > pq;
+        // {distance , point}
+        vector<int> vis(n,0);
+        pq.push({0,0}); //starting from 0 as source
+        int Cost=0;
+        while(!pq.empty())
+        {
+            auto it = pq.top(); 
+            pq.pop();
+            int CurrNode = it.second ;
+            int topEdgwWt = it.first;
+            
+            if(vis[CurrNode]==1) continue;
+            
+            vis[CurrNode]=1;
+            
+            Cost+=topEdgwWt;
+            for(auto it: adj[CurrNode])
+            {
+                int adjPoint=it.first;
+                int edWt=it.second;
+                if(!vis[adjPoint])
+                {
+                    pq.push({edWt , adjPoint});
+                }
+            }
+        }
+        return Cost;
+    }
+};
+```
+</TabItem>
+</Tabs>
+
+  </TabItem>
+</Tabs>
+
+## References
+
+- **LeetCode Problem**: [ Min Cost to Connect All Points](https://leetcode.com/problems/min-cost-to-connect-all-points/description/)
+
+- **Solution Link**: [LeetCode Solution](https://leetcode.com/problems/min-cost-to-connect-all-points/description/)
+