{"id":10225,"date":"2025-10-03T14:46:22","date_gmt":"2025-10-03T09:16:22","guid":{"rendered":"https:\/\/namastedev.com\/blog\/?p=10225"},"modified":"2025-10-03T14:49:47","modified_gmt":"2025-10-03T09:19:47","slug":"unique-paths","status":"publish","type":"post","link":"https:\/\/namastedev.com\/blog\/unique-paths\/","title":{"rendered":"Unique Paths"},"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;   \/* light bg instead of dark blue *\/\n  color: #E58C32;        \/* brand orange *\/\n  padding: 3px 6px;\n  border-radius: 4px;\n  font-family: 'Courier New', monospace;\n  font-weight: 600;      \/* optional, makes it pop *\/\n}\n\n.wp_blog_explanation img {\n  max-width: 100%;\n  border-radius: var(--tab-radius);\n  margin-top: 1rem;\n  box-shadow: 0 2px 12px rgba(0, 0, 0, 0.06);\n}\n\n\/* Tab Buttons *\/\n.wp_blog_code-tabs-header {\n  display: flex;\n  flex-wrap: wrap;\n  gap: 0.5rem;\n  margin-bottom: 1rem;\n}\n\n.wp_blog_code-tab-button {\n  padding: 0.6rem 1.2rem;\n  border: 1px solid var(--primary);\n  background: white;\n  color: var(--primary);\n  border-radius: 50px;\n  font-weight: 600;\n  cursor: pointer;\n  transition: all 0.3s ease;\n}\n\n.wp_blog_code-tab-button:hover {\n  background: var(--secondary);\n}\n\n.wp_blog_code-tab-button.active {\n  background: var(--primary);\n  color: white;\n}\n\n\/* Code Content *\/\n.wp_blog_code-tab-content {\n  display: none;\n  background: var(--code-bg);\n  border-radius: var(--tab-radius);\n}\n\n.wp_blog_code-tab-content.active {\n  display: block;\n}\n\n.wp_blog_code-tab-content pre {\n  margin: 0;\n  padding: 1.5rem;\n  font-size: 1rem;\n  overflow-x: auto;\n  background: var(--code-bg);\n  border-radius: var(--tab-radius);\n  color: var(--code-text);\n}\n\n\/* Dark mode variables *\/\n.wp_blog_theme.dark-mode {\n  --light-bg: #121212;\n  --text-dark: #f5f5f5;\n  --shadow: 0 4px 12px rgba(255, 255, 255, 0.08);\n  --code-bg: #1e1e1e;\n  --code-text: #c5f0ff;\n}\n\n.wp_blog_theme.dark-mode .wp_blog_explanation {\n  background: #1e1e1e;\n}\n\n\/* Dark mode code highlight *\/\n.wp_blog_theme.dark-mode .wp_blog_explanation code {\n  background: #333;\n  color: #ffd27f;\n}\n\n.wp_blog_theme {\n  position: relative; \/* 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        There is a robot on an <code>m x n<\/code> grid. The robot is initially located at the <strong>top-left corner<\/strong> (i.e., <code>grid[0][0]<\/code>). The robot tries to move to the <strong>bottom-right corner<\/strong> (i.e., <code>grid[m - 1][n - 1]<\/code>). The robot can only move either down or right at any point in time.\n    <\/p>\n\n    <p>\n        Given the two integers <code>m<\/code> and <code>n<\/code>, return <i>the number of possible unique paths that the robot can take to reach the bottom-right corner<\/i>.\n    <\/p>\n\n    <p>\n        The test cases are generated so that the answer will be less than or equal to <code>2 * 10<sup>9<\/sup><\/code>.\n    <\/p>\n\n    <p>\n        The answer is <strong>guaranteed<\/strong> to fit into a signed <strong>32-bit<\/strong> integer.\n    <\/p>\n\n    <h3>Example 1:<\/h2>\n        <img decoding=\"async\" src=\"https:\/\/namastedev.com\/blog\/wp-content\/uploads\/2025\/10\/Screenshot-2025-10-03-at-2.24.39\u202fPM.png\" alt=\"\">\n    <p><strong>Input:<\/strong> m = 3, n = 7<\/p>\n    <p><strong>Output:<\/strong> 28<\/p>\n\n    <h3>Example 2:<\/h2>\n    <p><strong>Input:<\/strong> m = 3, n = 2<\/p>\n    <p><strong>Output:<\/strong> 3<\/p>\n    <p><strong>Explanation:<\/strong>From the top-left corner, there are a total of 3 ways to reach the bottom-right corner:\n    <ol>\n        <li>Right -> Down -> Down<\/li>\n        <li>Down -> Down -> Right<\/li>\n        <li>Down -> Right -> Down<\/li>\n    <\/ol>\n    <\/p>\n\n    <h3>Constraints<\/h3>\n    <ul>\n       <li>1 <= m, n <= 100<\/li>\n    <\/ul>\n    \n<h2>Approach:<\/h2>\n<ul>\n    <li><strong>DP Definition: <\/strong>Let <code>dp[i][j]<\/code> represent the number of unique paths to reach cell <code>(i,j)<\/code><\/li>\n    <li><strong>Bases cases are like this: <\/strong>\n    <ul>\n        <li>First row \u2192 Only one way (all moves right).<\/li>\n        <li>First column \u2192 Only one way (all moves down).<\/li>\n    <\/ul>\n    <\/li>\n\n    <li><strong>Transition: <\/strong>To reach (i,j), you can come from:\n    <ul>\n        <li><strong>Above<\/strong>: <code>(i-1, j)<\/code><\/li>\n        <li><strong>Left<\/strong>: <code>(i, j-1)<\/code><\/li>\n    <\/ul>\n    <\/li>\n\n    <li><strong>Final result<\/strong> is at <code>bottom-right cell \u2192 dp[m-1][n-1]<\/code>.<\/li>\n<\/ul>\n\n    <h2>Time &#038; Space Complexity:<\/h2>\n    <p><strong>Time Complexity:<\/strong> <code>O(m * n)<\/code><\/p>\n    <p><strong>Space Complexity:<\/strong> <code>O(m * 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\n  <p><strong>Input:<\/strong> <code>m = 3, n = 3<\/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);\">\nInitial dp matrix (all -1):\n[[-1, -1, -1],\n [-1, -1, -1],\n [-1, -1, -1]]\n\nStep 1: Fill first column with 1\n[[1, -1, -1],\n [1, -1, -1],\n [1, -1, -1]]\n\nStep 2: Fill first row with 1\n[[1, 1, 1],\n [1, -1, -1],\n [1, -1, -1]]\n\nStep 3: Fill remaining cells using dp[i][j] = dp[i-1][j] + dp[i][j-1]\n\n i = 1, j = 1:\n   dp[1][1] = dp[0][1] + dp[1][0] = 1 + 1 = 2\n   [[1, 1, 1],\n    [1, 2, -1],\n    [1, -1, -1]]\n\n i = 1, j = 2:\n   dp[1][2] = dp[0][2] + dp[1][1] = 1 + 2 = 3\n   [[1, 1, 1],\n    [1, 2, 3],\n    [1, -1, -1]]\n\n i = 2, j = 1:\n   dp[2][1] = dp[1][1] + dp[2][0] = 2 + 1 = 3\n   [[1, 1, 1],\n    [1, 2, 3],\n    [1, 3, -1]]\n\n i = 2, j = 2:\n   dp[2][2] = dp[1][2] + dp[2][1] = 3 + 3 = 6\n   [[1, 1, 1],\n    [1, 2, 3],\n    [1, 3, 6]]\n\nFinal dp matrix:\n[[1, 1, 1],\n [1, 2, 3],\n [1, 3, 6]]\n\nAnswer = dp[m-1][n-1] = dp[2][2] = 6\n  <\/pre>\n\n  <p><strong>Output:<\/strong> <code>6<\/code><\/p>\n<\/div>\n\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 uniquePaths = function(m, n) {\n    let dp = Array.from({ length: m }, () => Array(n).fill(0));\n\n    for (let i = 0; i < m; i++) dp[i][0] = 1;\n    for (let j = 0; j < n; j++) dp[0][j] = 1;\n\n    for (let i = 1; i < m; i++) {\n        for (let j = 1; j < n; j++) {\n            dp[i][j] = dp[i-1][j] + dp[i][j-1];\n        }\n    }\n    return dp[m-1][n-1];\n};\n<\/code><\/pre>\n    <\/div>\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"py\">\n      <pre><code class=\"language-python\">\ndef uniquePaths(m, n):\n    dp = [[0] * n for _ in range(m)]\n\n    for i in range(m):\n        dp[i][0] = 1\n    for j in range(n):\n        dp[0][j] = 1\n\n    for i in range(1, m):\n        for j in range(1, n):\n            dp[i][j] = dp[i-1][j] + dp[i][j-1]\n\n    return dp[m-1][n-1]\n      <\/code><\/pre>\n    <\/div>\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"java\">\n      <pre><code class=\"language-java\">\nclass Solution {\n    public int uniquePaths(int m, int n) {\n        int[][] dp = new int[m][n];\n\n        for (int i = 0; i < m; i++) dp[i][0] = 1;\n        for (int j = 0; j < n; j++) dp[0][j] = 1;\n\n        for (int i = 1; i < m; i++) {\n            for (int j = 1; j < n; j++) {\n                dp[i][j] = dp[i-1][j] + dp[i][j-1];\n            }\n        }\n\n        return dp[m-1][n-1];\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;     \nusing namespace std;\n\nclass Solution {\npublic:\n    int uniquePaths(int m, int n) {\n        vector<vector<int>> dp(m, vector<int>(n, 0));\n\n        for (int i = 0; i < m; i++) dp[i][0] = 1;\n        for (int j = 0; j < n; j++) dp[0][j] = 1;\n\n        for (int i = 1; i < m; i++) {\n            for (int j = 1; j < n; j++) {\n                dp[i][j] = dp[i-1][j] + dp[i][j-1];\n            }\n        }\n\n        return dp[m-1][n-1];\n    }\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\nint uniquePaths(int m, int n) {\n    int dp[m][n];\n\n    for (int i = 0; i < m; i++) dp[i][0] = 1;\n    for (int j = 0; j < n; j++) dp[0][j] = 1;\n\n    for (int i = 1; i < m; i++) {\n        for (int j = 1; j < n; j++) {\n            dp[i][j] = dp[i-1][j] + dp[i][j-1];\n        }\n    }\n\n    return dp[m-1][n-1];\n}\n <\/code><\/pre>\n    <\/div>\n\n    <div class=\"wp_blog_code-tab-content\" data-lang=\"cs\">\n      <pre><code class=\"language-csharp\">\npublic class Solution {\n    public int UniquePaths(int m, int n) {\n        int[,] dp = new int[m, n];\n\n        for (int i = 0; i < m; i++) dp[i, 0] = 1;\n        for (int j = 0; j < n; j++) dp[0, j] = 1;\n\n        for (int i = 1; i < m; i++) {\n            for (int j = 1; j < n; j++) {\n                dp[i, j] = dp[i - 1, j] + dp[i, j - 1];\n            }\n        }\n\n        return dp[m - 1, n - 1];\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: There is a robot on an m x n grid. The robot is initially located at the top-left corner (i.e., grid[0][0]). The robot tries to move to the bottom-right corner (i.e., grid[m &#8211; 1][n &#8211; 1]). The robot can only move either down or right at any point in time. Given the<\/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],"tags":[],"class_list":["post-10225","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"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10225","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=10225"}],"version-history":[{"count":1,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10225\/revisions"}],"predecessor-version":[{"id":10226,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/10225\/revisions\/10226"}],"wp:attachment":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/media?parent=10225"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/categories?post=10225"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/tags?post=10225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}