{"id":878,"date":"2025-08-11T17:02:19","date_gmt":"2025-08-11T11:32:19","guid":{"rendered":"https:\/\/codeanddebug.in\/blog\/?p=878"},"modified":"2025-08-11T17:02:23","modified_gmt":"2025-08-11T11:32:23","slug":"subset-sum-problem","status":"publish","type":"post","link":"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/","title":{"rendered":"Subset Sum Problem | Solved using Dynamic Programming"},"content":{"rendered":"\n<p>Given an array of positive integers&nbsp;<strong>arr[]<\/strong>&nbsp;and a value&nbsp;<strong>sum<\/strong>, determine if there is a subset of&nbsp;<strong>arr[]<\/strong>&nbsp;with sum equal to given&nbsp;<strong>sum<\/strong>.&nbsp;<\/p>\n\n\n\n<p>Here&#8217;s the [<strong><a href=\"https:\/\/www.geeksforgeeks.org\/problems\/subset-sum-problem-1611555638\/1\" target=\"_blank\" rel=\"noreferrer noopener\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><span style=\"text-decoration: underline;\">Problem Link<\/span><\/mark><\/a><\/strong>] to begin with.<\/p>\n\n\n\n<p><strong>Examples:<\/strong><\/p>\n\n\n\n<pre class=\"wp-block-preformatted\"><strong>Input<\/strong>: arr[] = [3, 34, 4, 12, 5, 2], sum = 9<br><strong>Output:<\/strong> true\u00a0<br><strong>Explanation<\/strong>: Here there exists a subset with target sum = 9, 4+3+2 = 9.<\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\"><strong>Input<\/strong>: arr[] = [3, 34, 4, 12, 5, 2], sum = 30\n<strong>Output:<\/strong> false\n<strong>Explanation<\/strong>: There is no subset with target sum 30.<\/pre>\n\n\n\n<pre class=\"wp-block-preformatted\"><strong>Input<\/strong>: arr[] = [1, 2, 3], sum = 6\n<strong>Output:<\/strong> true<br><strong>Explanation<\/strong>: The entire array can be taken as a subset, giving 1 + 2 + 3 = 6.<\/pre>\n\n\n\n<p><strong>Constraints:<\/strong><br>1 &lt;= arr.size() &lt;= 200<br>1&lt;= arr[i] &lt;= 200<br>1&lt;= sum &lt;= 10<sup>4<\/sup><\/p>\n\n\n<div class=\"wp-block-ub-table-of-contents-block ub_table-of-contents\" id=\"ub_table-of-contents-094396f3-1fdf-4c43-b0b8-7e1baf32d20c\" data-linktodivider=\"false\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" data-enablesmoothscroll=\"false\" data-initiallyhideonmobile=\"false\" data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-header-container\" style=\"\">\n\t\t\t<div class=\"ub_table-of-contents-header\" style=\"text-align: left; \">\n\t\t\t\t<div class=\"ub_table-of-contents-title\">Contents:<\/div>\n\t\t\t\t\n\t\t\t<\/div>\n\t\t<\/div><div class=\"ub_table-of-contents-extra-container\" style=\"\">\n\t\t\t<div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \">\n\t\t\t\t<ul style=\"\"><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#0-solution-1-recursion-brute-force\" style=\"\">Solution 1: Recursion (Brute Force)<\/a><ul><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#1-intuition-and-approach\" style=\"\">Intuition and Approach<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#2-code-implementation\" style=\"\">Code Implementation<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#3-code-explanation\" style=\"\">Code Explanation<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#4-time-and-space-complexity\" style=\"\">Time and Space Complexity<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#5-conclusion\" style=\"\">Conclusion<\/a><\/li><\/ul><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#6-solution-2-memoization-top-down-dp\" style=\"\">Solution 2: Memoization (Top-Down DP)<\/a><ul><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#7-intuition-and-approach\" style=\"\">Intuition and Approach<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#8-code-implementation\" style=\"\">Code Implementation<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#9-code-explanation\" style=\"\">Code Explanation<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#10-time-and-space-complexity\" style=\"\">Time and Space Complexity<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#11-conclusion\" style=\"\">Conclusion<\/a><\/li><\/ul><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#12-solution-3-tabulation-bottom-up-dp\" style=\"\">Solution 3: Tabulation (Bottom-Up DP)<\/a><ul><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#13-intuition-and-approach\" style=\"\">Intuition and Approach<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#14-code-implementation\" style=\"\">Code Implementation<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#15-code-explanation\" style=\"\">Code Explanation<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#16-time-and-space-complexity\" style=\"\">Time and Space Complexity<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#17-conclusion\" style=\"\">Conclusion<\/a><\/li><\/ul><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#18-solution-4-tabulation-space-optimization\" style=\"\">Solution 4: Tabulation (Space Optimization)<\/a><ul><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#19-intuition-and-approach\" style=\"\">Intuition and Approach<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#20-code-implementation\" style=\"\">Code Implementation<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#21-code-explanation\" style=\"\">Code Explanation<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#22-time-and-space-complexity\" style=\"\">Time and Space Complexity<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#23-conclusion\" style=\"\">Conclusion<\/a><\/li><\/ul><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/subset-sum-problem\/#24-final-notes\" style=\"\">Final Notes<\/a><\/li><\/ul>\n\t\t\t<\/div>\n\t\t<\/div><\/div>\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"0-solution-1-recursion-brute-force\">Solution 1: Recursion (Brute Force)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"1-intuition-and-approach\">Intuition and Approach<\/h3>\n\n\n\n<p>At each index, you decide to include the current element (if it doesn\u2019t exceed target) or exclude it. This forms a binary decision tree. Base cases:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If target == 0 \u2192 True (empty subset achieves 0).<\/li>\n\n\n\n<li>If index == 0 \u2192 True only if arr equals target.<br>This explores all subsets but with exponential time due to overlapping subproblems.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"2-code-implementation\">Code Implementation<\/h3>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#D4D4D4;--cbp-line-number-width:calc(2 * 0.6 * .875rem);line-height:1.5rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"class Solution:\n    def solve(self, index, target, arr):\n        # Base: target achieved\n        if target == 0:\n            return True\n        # Base: only first element available\n        if index == 0:\n            if arr == target:\n                return True\n            return False\n        # Choice: pick only if feasible\n        if arr[index] &gt; target:\n            pick = False\n        else:\n            pick = self.solve(index - 1, target - arr[index], arr)\n        # Choice: not pick current element\n        not_pick = self.solve(index - 1, target, arr)\n        # Return if any choice leads to True\n        return pick or not_pick\n\n    def isSubsetSum(self, arr, sum):\n        n = len(arr)\n        # Start from the last index aiming for target sum\n        return self.solve(n - 1, sum, arr)\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #569CD6\">class<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #4EC9B0\">Solution<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #569CD6\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">solve<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">index<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">target<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">arr<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Base: target achieved<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> target == <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">True<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Base: only first element available<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> index == <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> arr == target:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">True<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">False<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Choice: pick only if feasible<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> arr[index] &gt; target:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            pick = <\/span><span style=\"color: #569CD6\">False<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">else<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            pick = <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, target - arr[index], arr)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Choice: not pick current element<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        not_pick = <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, target, arr)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Return if any choice leads to True<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> pick <\/span><span style=\"color: #569CD6\">or<\/span><span style=\"color: #D4D4D4\"> not_pick<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #569CD6\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">isSubsetSum<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">arr<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">sum<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        n = <\/span><span style=\"color: #DCDCAA\">len<\/span><span style=\"color: #D4D4D4\">(arr)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Start from the last index aiming for target sum<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(n - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">, arr)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"3-code-explanation\">Code Explanation<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The function solve(index, target) tries both decisions: include arr[index] (if allowed) or exclude it.<\/li>\n\n\n\n<li>If target hits 0 at any depth, it signals a valid subset.<\/li>\n\n\n\n<li>The first-element base case avoids negative indices and settles the smallest subproblem.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"4-time-and-space-complexity\">Time and Space Complexity<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Precise: Time O(2^n), as each element can be picked or not; Space O(n) recursion depth.<\/li>\n\n\n\n<li>Simplified: Exponential time, linear stack space.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"5-conclusion\">Conclusion<\/h3>\n\n\n\n<p>Good for intuition and very small inputs, but inefficient due to repeated subproblems.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-solution-2-memoization-top-down-dp\">Solution 2: Memoization (Top-Down DP)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"7-intuition-and-approach\">Intuition and Approach<\/h3>\n\n\n\n<p>The same (index, target) states recur many times. Cache results in a 2D table dp[index][target]:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Before computing, return cached dp if already set.<\/li>\n\n\n\n<li>Otherwise, compute pick\/not-pick as recursion and store True\/False.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"8-code-implementation\">Code Implementation<\/h3>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#D4D4D4;--cbp-line-number-width:calc(2 * 0.6 * .875rem);line-height:1.5rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"class Solution:\n    def solve(self, index, target, arr, dp):\n        # If target achieved\n        if target == 0:\n            return True\n        # If only first element can be used\n        if index == 0:\n            if arr[0] == target:\n                return True\n            return False\n        # Use memoized result if available\n        if dp[index][target] != -1:\n            return dp[index][target]\n        # Try picking current element if feasible\n        if arr[index] &gt; target:\n            pick = False\n        else:\n            pick = self.solve(index - 1, target - arr[index], arr, dp)\n        # Or skip current element\n        not_pick = self.solve(index - 1, target, arr, dp)\n        # Store boolean result in dp\n        return pick or not_pick\n\n    def isSubsetSum(self, arr, sum):\n        n = len(arr)\n        # dp[index][target] = -1 (unknown), else True\/False\n        dp = [[-1 for _ in range(sum + 1)] for _ in range(n)]\n        return self.solve(n - 1, sum, arr, dp)\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #569CD6\">class<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #4EC9B0\">Solution<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #569CD6\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">solve<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">index<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">target<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">arr<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">dp<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># If target achieved<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> target == <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">True<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># If only first element can be used<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> index == <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> arr[<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">] == target:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">True<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">False<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Use memoized result if available<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> dp[index][target] != -<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> dp[index][target]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Try picking current element if feasible<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> arr[index] &gt; target:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            pick = <\/span><span style=\"color: #569CD6\">False<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">else<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            pick = <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, target - arr[index], arr, dp)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Or skip current element<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        not_pick = <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, target, arr, dp)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Store boolean result in dp<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> pick <\/span><span style=\"color: #569CD6\">or<\/span><span style=\"color: #D4D4D4\"> not_pick<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #569CD6\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">isSubsetSum<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">arr<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">sum<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        n = <\/span><span style=\"color: #DCDCAA\">len<\/span><span style=\"color: #D4D4D4\">(arr)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># dp[index][target] = -1 (unknown), else True\/False<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        dp = [[-<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> _ <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">)] <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> _ <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(n)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(n - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">, arr, dp)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"9-code-explanation\">Code Explanation<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>dp[index][target] caches whether a subset of arr[0..index] can form target.<\/li>\n\n\n\n<li>Converts exponential recursion into polynomial by avoiding recomputation of identical states.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"10-time-and-space-complexity\">Time and Space Complexity<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Precise<\/strong>: Time O(n * sum), Space O(n * sum) for dp + O(n) stack.<\/li>\n\n\n\n<li><strong>Simplified<\/strong>: Pseudo-polynomial in target, linear stack.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"11-conclusion\">Conclusion<\/h3>\n\n\n\n<p>Efficient and interview-ready; ideal when sum is not too large.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"12-solution-3-tabulation-bottom-up-dp\">Solution 3: Tabulation (Bottom-Up DP)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"13-intuition-and-approach\">Intuition and Approach<\/h3>\n\n\n\n<p>Build a boolean table dp[n][sum+1] where dp[i][t] indicates if you can make sum t using arr[0..i].<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Initialize dp[i] = True for all i (target 0 is always achievable).<\/li>\n\n\n\n<li>Initialize dp[arr] = True if it doesn\u2019t exceed sum.<\/li>\n\n\n\n<li>Transition:\n<ul class=\"wp-block-list\">\n<li>If arr[i] > t \u2192 pick = False else pick = dp[i-1][t &#8211; arr[i]]<\/li>\n\n\n\n<li>not_pick = dp[i-1][t]<\/li>\n\n\n\n<li>dp[i][t] = pick or not_pick<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"14-code-implementation\">Code Implementation<\/h3>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#D4D4D4;--cbp-line-number-width:calc(2 * 0.6 * .875rem);line-height:1.5rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"class Solution:\n    def isSubsetSum(self, arr, sum):\n        n = len(arr)\n        # dp[i][t] -&gt; can we make sum t using first i elements (0..i)?\n        dp = [[False for _ in range(sum + 1)] for _ in range(n)]\n        # Sum 0 achievable by picking empty subset\n        for index in range(n):\n            dp[index] = True\n        # Handle first element\n        if arr &lt;= sum:\n            dp[arr] = True\n        # Fill table for remaining elements\n        for index in range(1, n):\n            for target in range(0, sum + 1):\n                if arr[index] &gt; target:\n                    pick = False\n                else:\n                    pick = dp[index - 1][target - arr[index]]\n                not_pick = dp[index - 1][target]\n                dp[index][target] = pick or not_pick\n        # Final answer: can we form 'sum' using all elements?\n        return dp[n - 1][sum]\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #569CD6\">class<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #4EC9B0\">Solution<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #569CD6\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">isSubsetSum<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">arr<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">sum<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        n = <\/span><span style=\"color: #DCDCAA\">len<\/span><span style=\"color: #D4D4D4\">(arr)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># dp[i][t] -&gt; can we make sum t using first i elements (0..i)?<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        dp = [[<\/span><span style=\"color: #569CD6\">False<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> _ <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">)] <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> _ <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(n)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Sum 0 achievable by picking empty subset<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> index <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(n):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            dp[index] = <\/span><span style=\"color: #569CD6\">True<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Handle first element<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> arr &lt;= <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            dp[arr] = <\/span><span style=\"color: #569CD6\">True<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Fill table for remaining elements<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> index <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, n):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> target <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> arr[index] &gt; target:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    pick = <\/span><span style=\"color: #569CD6\">False<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">else<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    pick = dp[index - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">][target - arr[index]]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                not_pick = dp[index - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">][target]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                dp[index][target] = pick <\/span><span style=\"color: #569CD6\">or<\/span><span style=\"color: #D4D4D4\"> not_pick<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Final answer: can we form &#39;sum&#39; using all elements?<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> dp[n - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">][<\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"15-code-explanation\">Code Explanation<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Iteratively fills achievable sums per prefix of the array.<\/li>\n\n\n\n<li>Each state depends only on the previous row, enabling further space optimization.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"16-time-and-space-complexity\">Time and Space Complexity<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Precise<\/strong>: Time O(n * sum), Space O(n * sum).<\/li>\n\n\n\n<li><strong>Simplified<\/strong>: Pseudo-polynomial time\/space.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"17-conclusion\">Conclusion<\/h3>\n\n\n\n<p>Deterministic and easy to debug; preferred when memory is acceptable.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"18-solution-4-tabulation-space-optimization\">Solution 4: Tabulation (Space Optimization)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"19-intuition-and-approach\">Intuition and Approach<\/h3>\n\n\n\n<p>Current dp row depends only on the previous row, so we can roll arrays:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Maintain prev (for i-1) and curr (for i).<\/li>\n\n\n\n<li>After computing curr for all targets, assign prev = curr.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"20-code-implementation\">Code Implementation<\/h3>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#D4D4D4;--cbp-line-number-width:calc(2 * 0.6 * .875rem);line-height:1.5rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"class Solution:\n    def isSubsetSum(self, arr, sum):\n        n = len(arr)\n        # prev[t] -&gt; achievable with elements up to previous index\n        prev = [False for _ in range(sum + 1)]\n        prev = True  # sum 0 is always achievable\n        if arr &lt;= sum:\n            prev[arr] = True\n        # Iterate elements\n        for index in range(1, n):\n            curr = [False for _ in range(sum + 1)]\n            for target in range(0, sum + 1):\n                if arr[index] &gt; target:\n                    pick = False\n                else:\n                    pick = prev[target - arr[index]]\n                not_pick = prev[target]\n                curr[target] = pick or not_pick\n            # Move window\n            prev = curr\n        return prev[sum]\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #569CD6\">class<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #4EC9B0\">Solution<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #569CD6\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">isSubsetSum<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">arr<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">sum<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        n = <\/span><span style=\"color: #DCDCAA\">len<\/span><span style=\"color: #D4D4D4\">(arr)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># prev[t] -&gt; achievable with elements up to previous index<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        prev = [<\/span><span style=\"color: #569CD6\">False<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> _ <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        prev = <\/span><span style=\"color: #569CD6\">True<\/span><span style=\"color: #D4D4D4\">  <\/span><span style=\"color: #6A9955\"># sum 0 is always achievable<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> arr &lt;= <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            prev[arr] = <\/span><span style=\"color: #569CD6\">True<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Iterate elements<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> index <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, n):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            curr = [<\/span><span style=\"color: #569CD6\">False<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> _ <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> target <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> arr[index] &gt; target:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    pick = <\/span><span style=\"color: #569CD6\">False<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">else<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    pick = prev[target - arr[index]]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                not_pick = prev[target]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                curr[target] = pick <\/span><span style=\"color: #569CD6\">or<\/span><span style=\"color: #D4D4D4\"> not_pick<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #6A9955\"># Move window<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            prev = curr<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> prev[<\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"21-code-explanation\">Code Explanation<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Uses two 1D arrays to track reachable sums, preserving correctness with O(sum) space.<\/li>\n\n\n\n<li>Same transitions as full table but memory efficient.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"22-time-and-space-complexity\">Time and Space Complexity<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Precise<\/strong>: Time O(n * sum), Space O(sum).<\/li>\n\n\n\n<li><strong>Simplified<\/strong>: Pseudo-polynomial time, linear space in target.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"23-conclusion\">Conclusion<\/h3>\n\n\n\n<p>Best practical version for large n with moderate target sum; commonly expected in interviews.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"24-final-notes\">Final Notes<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>All approaches rely on the fundamental pick\/not-pick decision per element.<\/li>\n\n\n\n<li>Base cases ensure correctness: reaching target 0 is always True; the single-item check at index 0 anchors recursion\/DP.<\/li>\n\n\n\n<li>Memoization and tabulation both achieve O(n * sum) time; the space-optimized tabulation reduces memory to O(sum).<\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-container-core-buttons-is-layout-16018d1d wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link wp-element-button\" href=\"https:\/\/codeanddebug.in\/course\/zero-to-hero-python-dsa\" target=\"_blank\" rel=\"noreferrer noopener\">Join our Advance DSA COURSE<\/a><\/div>\n<\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><em>For any changes to the article, kindly email at <a href=\"mailto:code@codeanddebug.in\" target=\"_blank\" rel=\"noreferrer noopener\">code@codeanddebug.in<\/a> or contact us at <a href=\"tel:+91-9712928220\" target=\"_blank\" rel=\"noreferrer noopener\">+91-9712928220<\/a>.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given an array of positive integers&nbsp;arr[]&nbsp;and a value&nbsp;sum, determine if there is a subset of&nbsp;arr[]&nbsp;with sum equal to given&nbsp;sum.&nbsp; Here&#8217;s the [Problem Link] to begin with. Examples: Input: arr[] = [3, 34, 4, 12, 5, 2], sum = 9Output: true\u00a0Explanation: Here there exists a subset with target sum = 9, 4+3+2 = 9. 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