{"id":10134,"date":"2025-09-26T16:14:34","date_gmt":"2025-09-26T10:44:34","guid":{"rendered":"https:\/\/namastedev.com\/blog\/?p=10134"},"modified":"2025-09-26T16:16:25","modified_gmt":"2025-09-26T10:46:25","slug":"top-down-and-bottom-up-dp-approach","status":"publish","type":"post","link":"https:\/\/namastedev.com\/blog\/top-down-and-bottom-up-dp-approach\/","title":{"rendered":"Top Down and Bottom Up DP approach"},"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>Two ways to implement Dynamic Programming:<\/h2>\n    <p>1. <code>Bottom-Up (Tabulation):<\/code> This approach <strong>uses iteration<\/strong> to solve the problem.<\/p>\n    <p>2. <code>Top-Down (Memoization):<\/code> This approach <strong>uses recursion<\/strong> along with <strong>memoization<\/strong>.><\/p>\n\n    <h2>Top-Down Approach<\/h2>\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        The image with <code>fib(6)<\/code> <strong>demonstrates the Top-Down approach<\/strong> using <code>recursion<\/code> and <code>memoization<\/code>.\n    <\/p>\n\n    <p><strong>Where should we store the values?<\/strong><\/p>\n    <p>We can store them in an <strong>array, map, object, or other data structures<\/strong>. Fetching data from a <code>map or object<\/code> is generally very fast.<\/p>\n\n    <p><strong>Now, let\u2019s solve the Fibonacci number problem using the Top-Down approach.<\/strong><\/h2>\n    <h2>Problem Statement<\/h2>\n    <p>\n        The <code>Fibonacci numbers<\/code>, commonly denoted <code>F(n)<\/code> form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from <code>0<\/code> and <code>1<\/code>. That is,\n    <\/p>\n    <p>\n        <code>\n            F(0) = 0, F(1) = 1\n            F(n) = F(n - 1) + F(n - 2), for n > 1.\n        <\/code>\n    <\/p>\n\n    <p><strong>Given n, calculate F(n)<\/strong>.<\/p>\n\n    <h2>Code: <\/h2>\n    <pre>\n        <code>\n          <strong>\n            let store = {}\n            var fib = function(n) {\n                if(n <= 1) {\n                    return n;\n                }\n                if(!store[n]){\n                    store[n] = fib(n - 1) + fib(n - 2);\n                }\n                return store[n];\n            };\n          <\/strong>\n        <\/code>\n    <\/pre>\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>n = 5<\/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 fib(5)\nstore = {}\n\nfib(5):\n- n = 5 (not <= 1)\n- store[5] not present\n- Compute fib(4) + fib(3)\n\nfib(4):\n- n = 4\n- store[4] not present\n- Compute fib(3) + fib(2)\n\n  fib(3):\n  - n = 3\n  - store[3] not present\n  - Compute fib(2) + fib(1)\n\n    fib(2):\n    - n = 2\n    - store[2] not present\n    - Compute fib(1) + fib(0)\n\n      fib(1) = 1 (base case)\n      fib(0) = 0 (base case)\n      \u2192 store[2] = 1 + 0 = 1\n\n    fib(1) = 1 (base case)\n\n    \u2192 store[3] = fib(2) + fib(1) = 1 + 1 = 2\n\n  fib(2): Already in store \u2192 return 1\n\n  \u2192 store[4] = fib(3) + fib(2) = 2 + 1 = 3\n\nfib(3):\n- Already in store \u2192 2\n\n\u2192 store[5] = fib(4) + fib(3) = 3 + 2 = 5\n\nStep 3: End\nReturn store[5] = 5\n  <\/pre> \n  \n  <p><strong>Output:<\/strong> <code>5<\/code><\/p> \n<\/div>\n\n<h2>Bottom Up Approach<\/h2>\n<p><strong>This approach uses iteration.<\/strong><\/p>\n<p>E.g: <strong>Fibonacci series: 0, 1, 1, 2, 3, 5, 8, 13,. . . <\/strong><\/p>\n\n<ul>\n    <li>When we build the solution in this way, it is called the <code>Bottom-Up approach<\/code>.<\/li>\n    <li>Basically, <strong>we move from 0 to n<\/strong> (instead of n to 0, which we use in the Top-Down recursive approach). <code>For iteration<\/code>, we typically use <strong>loops such as for or while<\/strong>.<\/li>\n<\/ul>\n    <h2>Problem Statement<\/h2>\n    <p>\n        The Fibonacci numbers, commonly denoted F(n) form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,\n    <\/p>\n<pre>\n    <p><code>Fib(i) = fib(i-1) + fib(i-2) <\/code><\/p>\n    <code>\n    var fib = function(n) {\n        let dp = [0, 1];\n        for(let i=2; i<=n; i++){\n            dp[i] = dp[i-1] + dp[i-2];\n        }\n        return dp[n];\n    };\n    <\/code>\n<\/pre>\n<p><strong>This approach is known as Bottom-Up.<\/strong><\/p>\n<\/div>\n\n    <h2>Comparison of the Top-Down and Bottom-Up Approaches<\/h2>\n    <img decoding=\"async\" src=\"https:\/\/namastedev.com\/blog\/wp-content\/uploads\/2025\/09\/Screenshot-2025-09-26-at-4.03.03\u202fPM.png\" alt=\"\">\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\") ? 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Bottom-Up (Tabulation): This approach uses iteration to solve the problem. 2. Top-Down (Memoization): This approach uses recursion along with memoization.> Top-Down Approach The image with fib(6) demonstrates the Top-Down approach using recursion and memoization. Where should we store the values? We can store them in an array,<\/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-10134","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\/10134","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=10134"}],"version-history":[{"count":1,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10134\/revisions"}],"predecessor-version":[{"id":10136,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10134\/revisions\/10136"}],"wp:attachment":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/media?parent=10134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/categories?post=10134"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/tags?post=10134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}