{"id":11426,"date":"2025-12-31T17:00:27","date_gmt":"2025-12-31T11:30:27","guid":{"rendered":"https:\/\/namastedev.com\/blog\/?p=11426"},"modified":"2025-12-31T17:03:43","modified_gmt":"2025-12-31T11:33:43","slug":"min-cost-to-connect-all-points","status":"publish","type":"post","link":"https:\/\/namastedev.com\/blog\/min-cost-to-connect-all-points\/","title":{"rendered":"Min Cost to Connect All Points"},"content":{"rendered":"\n<!-- PrismJS for Syntax Highlighting -->\n<link href=\"https:\/\/cdn.jsdelivr.net\/npm\/prismjs@1.29.0\/themes\/prism-tomorrow.min.css\" rel=\"stylesheet\">\n<script src=\"https:\/\/cdn.jsdelivr.net\/npm\/prismjs@1.29.0\/prism.min.js\"><\/script>\n<script src=\"https:\/\/cdn.jsdelivr.net\/npm\/prismjs@1.29.0\/plugins\/autoloader\/prism-autoloader.min.js\"><\/script>\n\n<style>\n.wp_blog_theme {\n  --primary: #E58C32;\n  --secondary: #030302;\n  --light-bg: #fef9f4;\n  --text-dark: #2d2d2d;\n  --tab-radius: 12px;\n  --shadow: 0 4px 12px rgba(0, 0, 0, 0.08);\n  --code-bg: #001f3f;\n  --code-text: #d4f1ff;\n}\n\n.wp_blog_container {\n  font-family: 'Segoe UI', sans-serif;\n  background: var(--light-bg);\n  margin: 0;\n  padding: 0;\n  color: var(--text-dark);\n}\n\n\/* Heading *\/\n.wp_blog_main-heading {\n  text-align: center;\n  font-size: 2.4rem;\n  color: var(--primary);\n  margin-top: 2.5rem;\n  font-weight: bold;\n}\n\n\/* Explanation Card *\/\n.wp_blog_explanation,\n.wp_blog_code-tabs-container {\n  max-width: 940px;\n  margin: 2rem auto;\n  padding: 2rem;\n  background: white;\n  border-radius: var(--tab-radius);\n  box-shadow: var(--shadow);\n}\n\n\/* Text and Visuals *\/\n.wp_blog_explanation h2 {\n  font-size: 1.4rem;\n  color: var(--primary);\n  margin-bottom: 0.5rem;\n}\n\n.wp_blog_explanation p,\n.wp_blog_explanation li {\n  font-size: 1.05rem;\n  line-height: 1.7;\n  margin: 0.5rem 0;\n}\n\n.wp_blog_explanation code {\n  background: #fef9f4; 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\/* makes it the reference for absolute children *\/\n}\n\n.wp_blog_toggle-btn {\n  position: absolute;\n  top: 1rem;\n  right: 1rem;\n  z-index: 9999;\n  padding: 0.5rem 0.8rem;\n  border-radius: 10%;\n  background: var(--primary);\n  color: white;\n  font-weight: bold;\n  cursor: pointer;\n  border: none;\n  box-shadow: var(--shadow);\n  transition: background 0.3s, transform 0.2s;\n}\n\n.wp_blog_toggle-btn:hover {\n  background: #cc772e;\n}\n\n.wp_blog_theme.dark-mode .wp_blog_code-tabs-container {\n  background: #1e1e1e;\n}\n<\/style>\n\n<div class=\"wp_blog_container wp_blog_theme\">\n  <button id=\"blogNotesThemeToggle\" class=\"wp_blog_toggle-btn\">\ud83c\udf19<\/button>\n  <h1 class=\"wp_blog_main-heading\"><\/h1>\n\n  <div class=\"wp_blog_explanation\">\n    <h2>Problem Statement<\/h2>\n    <p>\n        You are given an array points representing integer coordinates of some points on a 2D-plane, where <code>points[i] = [x<sub>i<\/sub>, y<sub>i<\/sub>]<\/code>.\n    <\/p>\n\n    <p>\n        The cost of connecting two points <code>[x<sub>i<\/sub>, y<sub>i<\/sub>]<\/code> and <code>[x<sub>j<\/sub>, y<sub>j<\/sub>]<\/code> is the manhattan distance between them: <code>|x<sub>i<\/sub> - x<sub>j<\/sub>| + |y<sub>i<\/sub> - y<sub>j<\/sub>|<\/code>, where <code>|val|<\/code> denotes the absolute value of val.\n    <\/p>\n\n    <p>\n        Return <i>the minimum cost to make all points connected<\/i>. All points are connected if there is <strong>exactly one<\/strong> simple path between any two points.\n    <\/p>\n    <h3>Example 1:<\/h2>\n    <img decoding=\"async\" src=\"https:\/\/namastedev.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-31-at-4.45.08-PM.png\" alt=\"\">\n    <p><strong>Input:<\/strong> points = [[0,0],[2,2],[3,10],[5,2],[7,0]]<\/p>\n    <p><strong>Output:<\/strong> 20<\/p>\n    <p><strong>Explanation:<\/strong><\/p>\n\n    <img decoding=\"async\" src=\"https:\/\/namastedev.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-31-at-4.45.19-PM.png\" alt=\"\">\n    <p>We can connect the points as shown above to get the minimum cost of 20.\nNotice that there is a unique path between every pair of points.<\/p>\n\n    <h3>Example 2:<\/h2>\n    <p><strong>Input:<\/strong> points = [[3,12],[-2,5],[-4,1]]<\/p>\n    <p><strong>Output:<\/strong> 18<\/p>\n\n    <h3>Constraints<\/h3>\n    <ul>\n        <li><code>1 <= points.length <= 1000<\/code><\/li>\n        <li><code>-10<sup>6<\/sup> <= x<sub>i<\/sub>, y<sub>i<\/sub> <= 10<sup>6<\/sup><\/code><\/li>\n        <li>All pairs <code>(x<sub>i<\/sub>, y<sub>i<\/sub>)<\/code> are distinct.<\/li>\n    <\/ul>\n    \n<h2>Approach<\/h2>\n<ul>\n    <li>We want to <code>connect all points with the minimum total cost<\/code>. The cost between two points is their Manhattan distance. This is essentially a <strong>Minimum Spanning Tree (MST) problem.<\/strong><\/li>\n    <li>Start from any point (here, point 0).<\/li>\n    <li>Maintain a <code>priority queue<\/code> of <strong>edges<\/strong> [distance, node] to pick the smallest edge each time.<\/li>\n    <li>Keep track of <strong>visited points<\/strong> to avoid cycles.<\/li>\n    <li>Each time we pick the smallest edge to an unvisited point, add its distance to <code>minCost<\/code> and mark the point visited.<\/li>\n    <li>Push all edges from the newly visited point to <code>unvisited points<\/code> into the priority queue.<\/li>\n    <li><code>Repeat until all points are visited.<\/code><\/li>\n    <li>The sum of all chosen <code>edge distances (minCost)<\/code>.<\/li>\n<\/ul>\n\n    <h2>Time & Space Complexity<\/h2>\n    <p><strong>Time Complexity:<\/strong> <code>O(n<sup>2<\/sup>log n)<\/code><\/p>\n    <p><strong>Space Complexity:<\/strong> <code>O(n<sup>2<\/sup>)<\/code><\/p>\n\n<h2>Dry Run<\/h2> \n<div style=\"background: var(--light-bg); border-left: 4px solid var(--primary); padding: 1rem; border-radius: var(--tab-radius); margin: 1rem 0; color: var(--text-dark);\"> \n    \n    <p><strong>Input:<\/strong> \n    <code>points = [[0,0],[2,2],[3,10],[5,2]]<\/code> \n    <\/p> <pre style=\"white-space: pre-wrap; background: var(--code-bg); padding: 1rem; border-radius: 8px; overflow-x: auto; color: var(--code-text);\"> \n    Step 0: Start Function: minCostConnectPoints(points) \n    \n    \n    Step 1: Initialize Variables n = 4 visited = [false, false, false, false] minCost = 0 edgesUsed = 0 pq = [[0, 0]] \/\/ [distance, node] \n    \n\n    Step 2: Start while loop (edgesUsed < n) Pop pq \u2192 [0, 0] node = 0, nodeDist = 0 visited[0] = true minCost = 0 + 0 = 0 \n    edgesUsed = 0 + 1 = 1 \n    Explore neighbors of node 0: - nextNode = 1 \u2192 distance = |0-2| + |0-2| = 4 \u2192 push [4,1] - nextNode = 2 \u2192 distance = |0-3| + |0-10| = 13 \u2192 push [13,2] - nextNode = 3 \u2192 distance = |0-5| + |0-2| = 7 \u2192 push [7,3] pq = [[4,1],[13,2],[7,3]] \n    \n\n\n    Step 3: Next iteration Pop pq \u2192 [4, 1] node = 1, \n    nodeDist = 4 visited[1] = true minCost = 0 + 4 = 4 \n    edgesUsed = 1 + 1 = 2 \n    Explore neighbors of node 1: - nextNode = 2 \u2192 distance = |2-3| + |2-10| = 9 \u2192 push [9,2] - nextNode = 3 \u2192 distance = |2-5| + |2-2| = 3 \u2192 push [3,3] pq = [[3,3],[7,3],[13,2],[9,2]] \n    \n\n\n    Step 4: Next iteration Pop pq \u2192 [3,3] node = 3, \n    nodeDist = 3 visited[3] = true minCost = 4 + 3 = 7 \n    edgesUsed = 2 + 1 = 3 \n    Explore neighbors of node 3: - nextNode = 2 \u2192 distance = |5-3| + |2-10| = 10 \u2192 push [10,2] pq = [[7,3],[9,2],[13,2],[10,2]] \n    \n\n\n    Step 5: Next iteration Pop pq \u2192 [7,3] \n    node 3 already visited \u2192 skip Pop pq \u2192 [9,2] node = 2, \n    nodeDist = 9 visited[2] = true minCost = 7 + 9 = 16 \n    edgesUsed = 3 + 1 = 4 \n    All nodes visited \u2192 exit while loop\n    \n    \n    Step 6: Return Result minCost = 16 <\/pre> \n    \n    <p><strong>Output:<\/strong> <code>16<\/code><\/p> <\/div>\n\n\n        <!-- <h2>Visualisation:<\/h2>\n        <img decoding=\"async\" src=\"https:\/\/namastedev.com\/blog\/wp-content\/uploads\/2025\/09\/4.png\" alt=\"\"> -->\n   \n<\/div>\n\n\n  <div class=\"wp_blog_code-tabs-container\">\n    <div class=\"wp_blog_code-tabs-header\">\n      <button class=\"wp_blog_code-tab-button active\" data-lang=\"js\">JavaScript<\/button>\n      <button class=\"wp_blog_code-tab-button\" data-lang=\"py\">Python<\/button>\n      <button class=\"wp_blog_code-tab-button\" data-lang=\"java\">Java<\/button>\n      <button class=\"wp_blog_code-tab-button\" data-lang=\"cpp\">C++<\/button>\n      <button class=\"wp_blog_code-tab-button\" data-lang=\"c\">C<\/button>\n      <button class=\"wp_blog_code-tab-button\" data-lang=\"cs\">C#<\/button>\n    <\/div>\n\n    <div class=\"wp_blog_code-tab-content active\" data-lang=\"js\">\n      <pre><code class=\"language-javascript\">\n\nvar minCostConnectPoints = function(points) {\n    let n = points.length;\n    let pq = new MinPriorityQueue(x => x[0]);\n\n    let minCost = 0;\n    let visited = new Array(n).fill(false);\n    pq.push([0, 0]);\n    let edgesUsed = 0;\n\n    while(edgesUsed < n){\n        let [nodeDist, node] = pq.pop();\n        if(visited[node]) continue;\n        visited[node] = true;\n        minCost = minCost + nodeDist;\n        edgesUsed++;\n    \n    for(let nextNode = 0; nextNode < n; ++nextNode) {\n        if(!visited[nextNode]) {\n            let nextDist =\n  Math.abs(points[node][0] - points[nextNode][0]) +\n  Math.abs(points[node][1] - points[nextNode][1]);\n\n           pq.push([nextDist, nextNode]);\n         }\n       }\n    }\n    return minCost;\n};\n<\/code><\/pre>\n    <\/div>\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"py\">\n      <pre><code class=\"language-python\">\nimport heapq\n\ndef minCostConnectPoints(points):\n    n = len(points)\n    minCost = 0\n    visited = [False] * n\n    edgesUsed = 0\n\n    pq = [(0, 0)]  # (distance, node)\n\n    while edgesUsed < n:\n        nodeDist, node = heapq.heappop(pq)\n        if visited[node]:\n            continue\n        visited[node] = True\n        minCost += nodeDist\n        edgesUsed += 1\n\n        for nextNode in range(n):\n            if not visited[nextNode]:\n                nextDist = abs(points[node][0] - points[nextNode][0]) + \\\n                           abs(points[node][1] - points[nextNode][1])\n                heapq.heappush(pq, (nextDist, nextNode))\n    return minCost\n      <\/code><\/pre>\n    <\/div>\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"java\">\n      <pre><code class=\"language-java\">\nimport java.util.*;\n\nclass Solution {\n    public int minCostConnectPoints(int[][] points) {\n        int n = points.length;\n        boolean[] visited = new boolean[n];\n        int minCost = 0, edgesUsed = 0;\n\n        PriorityQueue<int[]> pq = new PriorityQueue<>(Comparator.comparingInt(a -> a[0]));\n        pq.offer(new int[]{0, 0}); \/\/ {distance, node}\n\n        while (edgesUsed < n) {\n            int[] curr = pq.poll();\n            int nodeDist = curr[0];\n            int node = curr[1];\n\n            if (visited[node]) continue;\n\n            visited[node] = true;\n            minCost += nodeDist;\n            edgesUsed++;\n\n            for (int nextNode = 0; nextNode < n; nextNode++) {\n                if (!visited[nextNode]) {\n                    int nextDist = Math.abs(points[node][0] - points[nextNode][0]) +\n                                   Math.abs(points[node][1] - points[nextNode][1]);\n                    pq.offer(new int[]{nextDist, nextNode});\n                }\n            }\n        }\n        return minCost;\n    }\n}\n    <\/code><\/pre>\n    <\/div>\n\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"cpp\">\n      <pre><code class=\"language-cpp\">\n#include &lt;bits\/stdc++.h&gt;\nusing namespace std;\n\nint minCostConnectPoints(vector<vector<int>>& points) {\n    int n = points.size();\n    vector<bool> visited(n, false);\n    int minCost = 0, edgesUsed = 0;\n\n    priority_queue<pair<int, int>, vector<pair<int,int>>, greater<pair<int,int>>> pq;\n    pq.push({0, 0}); \/\/ {distance, node}\n\n    while (edgesUsed < n) {\n        auto [nodeDist, node] = pq.top(); pq.pop();\n        if (visited[node]) continue;\n\n        visited[node] = true;\n        minCost += nodeDist;\n        edgesUsed++;\n\n        for (int nextNode = 0; nextNode < n; ++nextNode) {\n            if (!visited[nextNode]) {\n                int nextDist = abs(points[node][0] - points[nextNode][0]) +\n                               abs(points[node][1] - points[nextNode][1]);\n                pq.push({nextDist, nextNode});\n            }\n        }\n    }\n    return minCost;\n}\n<\/code><\/pre>\n    <\/div>\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"c\">\n      <pre><code class=\"language-c\">\n#include &lt;stdio.h&gt;\n#include &lt;stdlib.h&gt;\n#include &lt;math.h&gt;\n#include &lt;limits.h&gt;\n\nint minCostConnectPoints(int** points, int n) {\n    int* visited = (int*)calloc(n, sizeof(int));\n    int minCost = 0, edgesUsed = 0;\n    int* minDist = (int*)malloc(n * sizeof(int));\n\n    for (int i = 0; i < n; i++) minDist[i] = INT_MAX;\n    minDist[0] = 0;\n\n    while (edgesUsed < n) {\n        int u = -1;\n        for (int i = 0; i < n; i++) {\n            if (!visited[i] &#038;&#038; (u == -1 || minDist[i] < minDist[u]))\n                u = i;\n        }\n\n        visited[u] = 1;\n        minCost += minDist[u];\n        edgesUsed++;\n\n        for (int v = 0; v < n; v++) {\n            if (!visited[v]) {\n                int dist = abs(points[u][0] - points[v][0]) + abs(points[u][1] - points[v][1]);\n                if (dist < minDist[v]) minDist[v] = dist;\n            }\n        }\n    }\n    free(visited);\n    free(minDist);\n    return minCost;\n}\n <\/code><\/pre>\n    <\/div>\n\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"cs\">\n      <pre><code class=\"language-csharp\">\nusing System;\nusing System.Collections.Generic;\n\npublic class Solution {\n    public int MinCostConnectPoints(int[][] points) {\n        int n = points.Length;\n        bool[] visited = new bool[n];\n        int minCost = 0, edgesUsed = 0;\n\n        var pq = new SortedSet<(int dist, int node)>();\n        pq.Add((0, 0));\n\n        while (edgesUsed < n) {\n            var curr = pq.Min;\n            pq.Remove(curr);\n            int nodeDist = curr.dist;\n            int node = curr.node;\n\n            if (visited[node]) continue;\n\n            visited[node] = true;\n            minCost += nodeDist;\n            edgesUsed++;\n\n            for (int nextNode = 0; nextNode < n; nextNode++) {\n                if (!visited[nextNode]) {\n                    int nextDist = Math.Abs(points[node][0] - points[nextNode][0]) +\n                                   Math.Abs(points[node][1] - points[nextNode][1]);\n                    pq.Add((nextDist, nextNode));\n                }\n            }\n        }\n\n        return minCost;\n    }\n}\n      <\/code><\/pre>\n    <\/div>\n  <\/div>\n<\/div>\n\n<script>\ndocument.addEventListener(\"DOMContentLoaded\", () => {\n  const buttons = document.querySelectorAll(\".wp_blog_code-tab-button\");\n  const contents = document.querySelectorAll(\".wp_blog_code-tab-content\");\n\n  buttons.forEach((button) => {\n    button.addEventListener(\"click\", () => {\n      const lang = button.getAttribute(\"data-lang\");\n\n      buttons.forEach((btn) => btn.classList.remove(\"active\"));\n      contents.forEach((content) => content.classList.remove(\"active\"));\n\n      button.classList.add(\"active\");\n      document\n        .querySelector(`.wp_blog_code-tab-content[data-lang=\"${lang}\"]`)\n        .classList.add(\"active\");\n    });\n  });\n\n  const themeToggle = document.getElementById(\"blogNotesThemeToggle\");\n  const themeContainer = document.querySelector(\".wp_blog_theme\");\n\n  themeToggle.addEventListener(\"click\", () => {\n    themeContainer.classList.toggle(\"dark-mode\");\n    themeToggle.textContent =\n      themeContainer.classList.contains(\"dark-mode\") ? \"\u2600\ufe0f\" : \"\ud83c\udf19\";\n  });\n});\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>\ud83c\udf19 Problem Statement 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 &#8211; xj| + |yi &#8211; yj|, where |val| denotes the absolute value of val.<\/p>\n","protected":false},"author":108,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[210,322,260,176,175,211,811,810,174,172,173],"tags":[],"class_list":["post-11426","post","type-post","status-publish","format-standard","category-algorithms","category-algorithms-and-data-structures","category-c-c-plus-plus","category-csharp","category-cplusplus","category-data-structures","category-data-structures-and-algorithms","category-dsa","category-java","category-javascript","category-python"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/11426","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/users\/108"}],"replies":[{"embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/comments?post=11426"}],"version-history":[{"count":1,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/11426\/revisions"}],"predecessor-version":[{"id":11427,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/11426\/revisions\/11427"}],"wp:attachment":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/media?parent=11426"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/categories?post=11426"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/tags?post=11426"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}