{"id":10113,"date":"2025-09-26T14:27:53","date_gmt":"2025-09-26T08:57:53","guid":{"rendered":"https:\/\/namastedev.com\/blog\/?p=10113"},"modified":"2025-09-26T14:49:52","modified_gmt":"2025-09-26T09:19:52","slug":"fibonacci-numbers-using-dp","status":"publish","type":"post","link":"https:\/\/namastedev.com\/blog\/fibonacci-numbers-using-dp\/","title":{"rendered":"Fibonacci Numbers using DP"},"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>Fibonacci Numbers using DP<\/h2>\n    <p><strong>Prerequisite: Recursion<\/strong><\/p>\n    <h2>N = 4<\/h2>\n    <img decoding=\"async\" src=\"https:\/\/namastedev.com\/blog\/wp-content\/uploads\/2025\/09\/Screenshot-2025-09-25-at-4.25.19\u202fPM.png\" alt=\"\">\n\n    <h2>This recursion has exponential time complexity, as the recursion tree grows significantly when n increases.<\/h2>\n    <p><strong>For example, if n = 6.<\/strong><\/p>\n    <img decoding=\"async\" src=\"https:\/\/namastedev.com\/blog\/wp-content\/uploads\/2025\/09\/Screenshot-2025-09-25-at-4.40.48\u202fPM.png\" alt=\"\">\n\n    <h2>Overlapping Subproblems:<\/h2>\n    <p>\n        As you can see, some <code>subproblems repeat<\/code>. If you <strong>encounter the same subproblems<\/strong> again and again, this is known as <strong>overlapping subproblems<\/strong>.\n    <\/p>\n\n    <h2>Optimal Substructure:<\/h2>\n    <p>\n        When <code>solving smaller subproblems<\/code> helps in solving the <strong>bigger problem<\/strong>, the problem is said to have an <code>optimal substructure<\/code>.\n    <\/p>\n\n    <h2>Note: <\/h2>\n    <p>\n        <strong>If any problem satisfies these <code>two properties<\/code> <strong>(overlapping subproblems and optimal substructure)<\/strong>, it is an ideal candidate for <strong>Dynamic Programming (DP)<\/strong>. <\/strong>\n    <\/p>\n\n    <h2>Time Complexity:<\/h2>\n        <ul>\n            <li>\n                The <strong>time complexity<\/strong> of this problem <code>using plain (naive)<\/code> recursion is <code>O(2<sup>n<\/sup>).<\/code>\n            <\/li>\n\n            <li>\n                Now, applying DP to this problem: DP works on the <strong>DRY (Don\u2019t Repeat Yourself) principle<\/strong>. Instead of recalculating overlapping subproblems, <strong>we save their results and reuse them<\/strong>.\n            <\/li>\n        <\/ul>\n        \n    <img decoding=\"async\" src=\"https:\/\/namastedev.com\/blog\/wp-content\/uploads\/2025\/09\/Screenshot-2025-09-25-at-4.44.26\u202fPM.png\" alt=\"\">\n    <p>\n        <strong>In this above image (Focus on the left side -\u2192 it symbolizes an arrow), <code>we maintain a storage<\/code> (memoization table or DP array) to keep previously computed results.<\/strong>\n    <\/p>\n    <ul>\n        <li>\n            By doing this, we can <code>fetch already computed results<\/code> from the <strong>storage<\/strong>. This way, we don\u2019t need to calculate the same <code>subproblems again and again<\/code>.\n        <\/li>\n        <li>\n            <code>By storing these values<\/code>, the <strong>time complexity<\/strong> reduces from <code>O(2<sup>n<\/sup>) to O(n).<\/code>.\n        <\/li>\n    <\/ul>\n\n    <h2>Fib(100):<\/h2>\n    <p>This would <code>take longer than universe\u2019s age<\/code> with naive <code>recursion<\/code>, but with DP you can solve it <strong>instantelously<\/strong>.<\/p>\n\n    <h2>Fib(n):<\/h2> <strong><p>where <code>n<\/code> is for bigger values.<\/p><\/strong>\n    <p>Exponential growth leads to <code>Astronomically large number of operations<\/code>, making it computationally infeasible within a <strong>reasonable<\/strong> timeframe.<\/p>\n   \n   <h2>DP Vs Greedy: (When to use:)<\/h2>\n   <ul>\n    <li>\n        <strong>Greedy<\/strong>: Make the <code>best local choice<\/code> and hope to reach the <code>Global Optimal Solution<\/code>.\n    <\/li>\n    <li>\n        <strong>DP<\/strong>: <code>Explore all possibilities store result<\/code> of <strong>subproblem<\/strong> and reach the result.\n    <\/li>\n   <\/ul>\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 = [1, 0, 2]<\/code><\/p> \n  \n  <pre style=\"white-space: pre-wrap; background: var(--code-bg); padding: 1rem; border-radius: 8px; overflow-x: auto; color: var(--code-text);\">\n\nStep 0: Start Function: candy(arr)\narr = [1, 0, 2]\nn = arr.length = 3\nInitialize ltr = Array(n).fill(1) \u2192 [1, 1, 1]\n\nStep 1: Left-to-right pass (populate ltr)\nIteration (i = 1): arr[1] = 0, arr[0] = 1\n- Condition: 0 > 1 \u2192 false\n- ltr stays [1, 1, 1]\n\nIteration (i = 2): arr[2] = 2, arr[1] = 0\n- Condition: 2 > 0 \u2192 true\n- ltr[2] = ltr[1] + 1 = 1 + 1 = 2\n- ltr becomes [1, 1, 2]\n\nStep 2: Right-to-left pass (populate rtl)\nInitialize rtl = Array(n).fill(1) \u2192 [1, 1, 1]\n\nIteration (i = 1): arr[1] = 0, arr[2] = 2\n- Condition: 0 > 2 \u2192 false\n- rtl stays [1, 1, 1]\n\nIteration (i = 0): arr[0] = 1, arr[1] = 0\n- Condition: 1 > 0 \u2192 true\n- rtl[0] = rtl[1] + 1 = 1 + 1 = 2\n- rtl becomes [2, 1, 1]\n\nStep 3: Sum up using max(ltr[i], rtl[i])\nans = 0\n\ni = 0: max(2, 1) = 2 \u2192 ans = 2  \ni = 1: max(1, 1) = 1 \u2192 ans = 3  \ni = 2: max(1, 2) = 2 \u2192 ans = 5  \n\nStep 4: End\nReturn ans = 5\n  <\/pre> \n  \n  <p><strong>Output:<\/strong> <code>5<\/code><\/p> \n<\/div> -->\n\n<\/div>\n\n\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 candy = function(arr) {\n    let n = arr.length;\n    let ltr = Array(n).fill(1);\n\n    for(let i=1; i<n; i++){\n        if(arr[i] > arr[i-1]){\n            ltr[i] = ltr[i-1] + 1;\n        }\n    }\n\n    let rtl = Array(n).fill(1);\n    for(let i=n-2; i>=0; i--){\n        if(arr[i] > arr[i+1]) {\n            rtl[i] = rtl[i+1] + 1;\n        }\n    }\n    let ans = 0; \n    for(let i=0; i < n; i++){\n        ans = ans + Math.max(rtl[i], ltr[i]);\n    }\n    return ans;\n};\n<\/code><\/pre>\n    <\/div> -->\n    <!-- <div class=\"wp_blog_code-tab-content\" data-lang=\"py\">\n      <pre><code class=\"language-python\">\ndef candy(arr):\n    n = len(arr)\n    if n == 0:\n        return 0\n    ltr = [1] * n\n    rtl = [1] * n\n    for i in range(1, n):\n        if arr[i] > arr[i-1]:\n            ltr[i] = ltr[i-1] + 1\n    for i in range(n-2, -1, -1):\n        if arr[i] > arr[i+1]:\n            rtl[i] = rtl[i+1] + 1\n    ans = 0\n    for i in range(n):\n        ans += max(ltr[i], rtl[i])\n    return ans\nif __name__ == \"__main__\":\n    print(candy([1, 0, 2]))\n      <\/code><\/pre>\n    <\/div> -->\n    <!-- <div class=\"wp_blog_code-tab-content\" data-lang=\"java\">\n      <pre><code class=\"language-java\">\npublic class CandyProblem {\n    public static int candy(int[] arr) {\n        int n = arr.length;\n        if (n == 0) return 0;\n\n        int[] ltr = new int[n];\n        int[] rtl = new int[n];\n        for (int i = 0; i < n; i++) { ltr[i] = 1; rtl[i] = 1; }\n\n        for (int i = 1; i < n; i++) {\n            if (arr[i] > arr[i - 1]) ltr[i] = ltr[i - 1] + 1;\n        }\n\n        for (int i = n - 2; i >= 0; i--) {\n            if (arr[i] > arr[i + 1]) rtl[i] = rtl[i + 1] + 1;\n        }\n\n        int ans = 0;\n        for (int i = 0; i < n; i++) ans += Math.max(ltr[i], rtl[i]);\n        return ans;\n    }\n    public static void main(String[] args) {\n        int[] ratings = {1, 0, 2};\n        System.out.println(candy(ratings)); \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\nusing namespace std;\nint candy(const vector<int>& arr) {\n    int n = arr.size();\n    if (n == 0) return 0;\n    vector<int> ltr(n, 1), rtl(n, 1);\n    for (int i = 1; i < n; ++i) {\n        if (arr[i] > arr[i-1]) ltr[i] = ltr[i-1] + 1;\n    }\n    for (int i = n - 2; i >= 0; --i) {\n        if (arr[i] > arr[i+1]) rtl[i] = rtl[i+1] + 1;\n    }\n    int ans = 0;\n    for (int i = 0; i < n; ++i) ans += max(ltr[i], rtl[i]);\n    return ans;\n}\nint main() {\n    vector<int> ratings = {1, 0, 2};\n    cout << candy(ratings) << '\\n';\n    return 0;\n}\n<\/code><\/pre>\n    <\/div> \n\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\nint candy(int *arr, int n) {\n    if (n == 0) return 0;\n    int *ltr = (int*)malloc(sizeof(int) * n);\n    int *rtl = (int*)malloc(sizeof(int) * n);\n    for (int i = 0; i < n; i++) { ltr[i] = 1; rtl[i] = 1; }\n\n    for (int i = 1; i < n; i++) {\n        if (arr[i] > arr[i-1]) ltr[i] = ltr[i-1] + 1;\n    }\n    for (int i = n - 2; i >= 0; i--) {\n        if (arr[i] > arr[i+1]) rtl[i] = rtl[i+1] + 1;\n    }\n    int ans = 0;\n    for (int i = 0; i < n; i++) ans += (ltr[i] > rtl[i] ? ltr[i] : rtl[i]);\n\n    free(ltr);\n    free(rtl);\n    return ans;\n}\nint main() {\n    int ratings[] = {1, 0, 2};\n    int n = sizeof(ratings) \/ sizeof(ratings[0]);\n    printf(\"%d\\n\", candy(ratings, n)); \n    return 0;\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;\nclass Program {\n    static int Candy(int[] arr) {\n        int n = arr.Length;\n        if (n == 0) return 0;\n        int[] ltr = new int[n];\n        int[] rtl = new int[n];\n        for (int i = 0; i < n; i++) { ltr[i] = 1; rtl[i] = 1; }\n        for (int i = 1; i < n; i++) {\n            if (arr[i] > arr[i - 1]) ltr[i] = ltr[i - 1] + 1;\n        }\n        for (int i = n - 2; i >= 0; i--) {\n            if (arr[i] > arr[i + 1]) rtl[i] = rtl[i + 1] + 1;\n        }\n        int ans = 0;\n        for (int i = 0; i < n; i++) ans += Math.Max(ltr[i], rtl[i]);\n        return ans;\n    }\n    static void Main() {\n        int[] ratings = {1, 0, 2};\n        Console.WriteLine(Candy(ratings)); \n    }\n}\n      <\/code><\/pre>\n    <\/div> -->\n\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 Fibonacci Numbers using DP Prerequisite: Recursion N = 4 This recursion has exponential time complexity, as the recursion tree grows significantly when n increases. For example, if n = 6. Overlapping Subproblems: As you can see, some subproblems repeat. If you encounter the same subproblems again and again, this is known as overlapping subproblems.<\/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":[210,322,176,175,211,174,172,173],"tags":[],"class_list":["post-10113","post","type-post","status-publish","format-standard","category-algorithms","category-algorithms-and-data-structures","category-csharp","category-cplusplus","category-data-structures","category-java","category-javascript","category-python"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10113","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=10113"}],"version-history":[{"count":1,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10113\/revisions"}],"predecessor-version":[{"id":10125,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10113\/revisions\/10125"}],"wp:attachment":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/media?parent=10113"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/categories?post=10113"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/tags?post=10113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}