{"id":10218,"date":"2025-10-03T13:44:36","date_gmt":"2025-10-03T08:14:36","guid":{"rendered":"https:\/\/namastedev.com\/blog\/?p=10218"},"modified":"2025-10-03T13:48:27","modified_gmt":"2025-10-03T08:18:27","slug":"longest-increasing-subsequence","status":"publish","type":"post","link":"https:\/\/namastedev.com\/blog\/longest-increasing-subsequence\/","title":{"rendered":"Longest Increasing Subsequence"},"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        Given an integer array nums, return<i> the length of the longest <strong>strictly increasing<\/strong> subsequence<\/i>.\n   <\/p>\n    <h3>Example 1:<\/h2>\n    <p><strong>Input:<\/strong> nums = [10,9,2,5,3,7,101,18]<\/p>\n    <p><strong>Output:<\/strong> 4<\/p>\n    <p><strong>Explanation:<\/strong> The longest increasing subsequence is [2,3,7,101], therefore the length is 4.<\/p>\n\n    <h3>Example 2:<\/h2>\n    <p><strong>Input:<\/strong> nums = [0,1,0,3,2,3]<\/p>\n    <p><strong>Output:<\/strong> 4<\/p>\n\n    <h3>Example 3:<\/h2>\n    <p><strong>Input:<\/strong> nums = [7,7,7,7,7,7,7]<\/p>\n    <p><strong>Output:<\/strong> 1<\/p>\n\n    <h3>Constraints<\/h3>\n    <ul>\n       <li><code>1 <= nums.length <= 2500<\/code><\/li>\n        <li><code>-10<sup>4<\/sup> <= nums[i] <= 10<sup>4<\/sup><\/code><\/li>\n    <\/ul>\n    \n<h2>Approach:<\/h2>\n<ul>\n  <li>Use <strong>Dynamic Programming (DP)<\/strong> to compute the <strong>Longest Increasing Subsequence<\/strong> (LIS).<\/li>\n  <li>Create a <code>dp<\/code> array where <code>dp[i]<\/code> stores the length of the LIS <strong>ending at index<\/strong> <code>i<\/code>.\n    <ul>\n      <li>Initialize all <code>dp[i] = 1<\/code> (each element alone is an LIS).<\/li>\n    <\/ul>\n  <\/li>\n  <li>For each element <code>arr[i]<\/code>, check all previous elements <code>arr[j] (0 \u2264 j < i)<\/code>:\n  <ul>\n    <li>If <code>arr[j] < arr[i]<\/code>, update <code>dp[i] = max(dp[i], dp[j] + 1)<\/code>.<\/li>\n  <\/ul>\n  <\/li>\n\n  <li>Keep track of the maximum LIS length while filling <code>dp<\/code>.<\/li>\n  <li>Return the <code>maximum LIS length<\/code>.<\/li>\n<\/ul>\n\n    <h2>Time & Space Complexity:<\/h2>\n    <p><strong>Time Complexity:<\/strong> <code>O(n<sup>2<\/sup>)<\/code><\/p>\n    <p><strong>Space Complexity:<\/strong> <code>O(n)<\/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  <p><strong>Input:<\/strong> <code>arr = [10, 9, 2, 5, 3, 7, 101, 18]<\/code><\/p>\n  <pre style=\"white-space: pre-wrap; background: var(--code-bg); padding: 1rem; border-radius: 8px; overflow-x: auto; color: var(--code-text);\">\nInitial:\nn = 8\ndp = [1, 1, 1, 1, 1, 1, 1, 1]   (each element LIS = 1 initially)\nlisLength = 1\n\ni = 1 (arr[1] = 9):\n  j = 0 \u2192 arr[0]=10 < 9? No\ndp = [1, 1, 1, 1, 1, 1, 1, 1], lisLength = 1\n\ni = 2 (arr[2] = 2):\n  j = 0 \u2192 10 < 2? No\n  j = 1 \u2192 9 < 2? No\ndp = [1, 1, 1, 1, 1, 1, 1, 1], lisLength = 1\n\ni = 3 (arr[3] = 5):\n  j = 0 \u2192 10 < 5? No\n  j = 1 \u2192 9 < 5? No\n  j = 2 \u2192 2 < 5? Yes \u2192 dp[3] = max(1, dp[2]+1=2) = 2\n  lisLength = 2\ndp = [1, 1, 1, 2, 1, 1, 1, 1]\n\ni = 4 (arr[4] = 3):\n  j = 0 \u2192 10 < 3? No\n  j = 1 \u2192 9 < 3? No\n  j = 2 \u2192 2 < 3? Yes \u2192 dp[4] = max(1, dp[2]+1=2) = 2\n  j = 3 \u2192 5 < 3? No\n  lisLength = 2\ndp = [1, 1, 1, 2, 2, 1, 1, 1]\n\ni = 5 (arr[5] = 7):\n  j = 0 \u2192 10 < 7? No\n  j = 1 \u2192 9 < 7? No\n  j = 2 \u2192 2 < 7? Yes \u2192 dp[5] = max(1, 1+1=2) = 2\n  j = 3 \u2192 5 < 7? Yes \u2192 dp[5] = max(2, 2+1=3) = 3\n  j = 4 \u2192 3 < 7? Yes \u2192 dp[5] = max(3, 2+1=3) = 3\n  lisLength = 3\ndp = [1, 1, 1, 2, 2, 3, 1, 1]\n\ni = 6 (arr[6] = 101):\n  j = 0 \u2192 10 < 101? Yes \u2192 dp[6] = max(1, 1+1=2) = 2\n  j = 1 \u2192 9 < 101? Yes \u2192 dp[6] = max(2, 1+1=2) = 2\n  j = 2 \u2192 2 < 101? Yes \u2192 dp[6] = max(2, 1+1=2) = 2\n  j = 3 \u2192 5 < 101? Yes \u2192 dp[6] = max(2, 2+1=3) = 3\n  j = 4 \u2192 3 < 101? Yes \u2192 dp[6] = max(3, 2+1=3) = 3\n  j = 5 \u2192 7 < 101? Yes \u2192 dp[6] = max(3, 3+1=4) = 4\n  lisLength = 4\ndp = [1, 1, 1, 2, 2, 3, 4, 1]\n\ni = 7 (arr[7] = 18):\n  j = 0 \u2192 10 < 18? Yes \u2192 dp[7] = max(1, 1+1=2) = 2\n  j = 1 \u2192 9 < 18? Yes \u2192 dp[7] = max(2, 1+1=2) = 2\n  j = 2 \u2192 2 < 18? Yes \u2192 dp[7] = max(2, 1+1=2) = 2\n  j = 3 \u2192 5 < 18? Yes \u2192 dp[7] = max(2, 2+1=3) = 3\n  j = 4 \u2192 3 < 18? Yes \u2192 dp[7] = max(3, 2+1=3) = 3\n  j = 5 \u2192 7 < 18? Yes \u2192 dp[7] = max(3, 3+1=4) = 4\n  j = 6 \u2192 101 < 18? No\n  lisLength = 4\ndp = [1, 1, 1, 2, 2, 3, 4, 4]\n\nFinal LIS length = 4\n  <\/pre>\n  <p><strong>Output:<\/strong> <code>4<\/code> (The LIS is [2, 3, 7, 101] or [2, 3, 7, 18])<\/p>\n<\/div>\n\n        <!-- <h2>Visualisation:<\/h2>\n        <img decoding=\"async\" src=\"https:\/\/namastedev.com\/blog\/wp-content\/uploads\/2025\/09\/Screenshot-2025-09-30-at-3.00.03\u202fPM.png\" alt=\"Stocks\" \/>\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\">\nvar lengthOfLIS = function(arr) {\n    let n = arr.length;\n    let dp = Array(n).fill(1);\n    dp[0] = 1;\n    let lisLength = 1;\n    for (let i = 1; i < n; i++) {\n        for (let j = 0; j < i; j++) {\n            if (arr[j] < arr[i]) {\n                dp[i] = Math.max(dp[i], dp[j] + 1);\n                lisLength = Math.max(lisLength, dp[i]);\n            }\n        }\n    }\n    return lisLength;\n};\n<\/code><\/pre>\n    <\/div>\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"py\">\n      <pre><code class=\"language-python\">\ndef lengthOfLIS(arr):\n    n = len(arr)\n    dp = [1] * n\n    dp[0] = 1\n    lisLength = 1\n    for i in range(1, n):\n        for j in range(0, i):\n            if arr[j] < arr[i]:\n                dp[i] = max(dp[i], dp[j] + 1)\n                lisLength = max(lisLength, dp[i])\n    return lisLength\n      <\/code><\/pre>\n    <\/div>\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"java\">\n      <pre><code class=\"language-java\">\npublic class Solution {\n    public static int lengthOfLIS(int[] arr) {\n        int n = arr.length;\n        int[] dp = new int[n];\n        java.util.Arrays.fill(dp, 1);\n        dp[0] = 1;\n        int lisLength = 1;\n        for (int i = 1; i < n; i++) {\n            for (int j = 0; j < i; j++) {\n                if (arr[j] < arr[i]) {\n                    dp[i] = Math.max(dp[i], dp[j] + 1);\n                    lisLength = Math.max(lisLength, dp[i]);\n                }\n            }\n        }\n        return lisLength;\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;vector.h&gt;     \n#include &lt;algorithm.h&gt;  \n\nint lengthOfLIS(const std::vector<int>& arr) {\n    int n = arr.size();\n    std::vector<int> dp(n, 1);\n    dp[0] = 1;\n    int lisLength = 1;\n    for (int i = 1; i < n; ++i) {\n        for (int j = 0; j < i; ++j) {\n            if (arr[j] < arr[i]) {\n                dp[i] = std::max(dp[i], dp[j] + 1);\n                lisLength = std::max(lisLength, dp[i]);\n            }\n        }\n    }\n    return lisLength;\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;stdlib.h&gt;\nint lengthOfLIS(int *arr, int n) {\n    int *dp = (int*) malloc(n * sizeof(int));\n    for (int i = 0; i < n; i++) dp[i] = 1;\n    dp[0] = 1;\n    int lisLength = 1;\n    for (int i = 1; i < n; i++) {\n        for (int j = 0; j < i; j++) {\n            if (arr[j] < arr[i]) {\n                int candidate = dp[j] + 1;\n                if (candidate > dp[i]) dp[i] = candidate;\n                if (dp[i] > lisLength) lisLength = dp[i];\n            }\n        }\n    }\n    free(dp);\n    return lisLength;\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;\n\npublic class Solution {\n    public static int LengthOfLIS(int[] arr) {\n        int n = arr.Length;\n        int[] dp = new int[n];\n        for (int i = 0; i < n; i++) dp[i] = 1;\n        dp[0] = 1;\n        int lisLength = 1;\n        for (int i = 1; i < n; i++) {\n            for (int j = 0; j < i; j++) {\n                if (arr[j] < arr[i]) {\n                    dp[i] = Math.Max(dp[i], dp[j] + 1);\n                    lisLength = Math.Max(lisLength, dp[i]);\n                }\n            }\n        }\n        return lisLength;\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: Given an integer array nums, return the length of the longest strictly increasing subsequence. Example 1: Input: nums = [10,9,2,5,3,7,101,18] Output: 4 Explanation: The longest increasing subsequence is [2,3,7,101], therefore the length is 4. Example 2: Input: nums = [0,1,0,3,2,3] Output: 4 Example 3: Input: nums = [7,7,7,7,7,7,7] Output: 1 Constraints 1<\/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":[210,260,176,175,211,811,810,174,172,173],"tags":[],"class_list":["post-10218","post","type-post","status-publish","format-standard","category-algorithms","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\/10218","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=10218"}],"version-history":[{"count":1,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10218\/revisions"}],"predecessor-version":[{"id":10219,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10218\/revisions\/10219"}],"wp:attachment":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/media?parent=10218"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/categories?post=10218"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/tags?post=10218"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}