{"id":1005,"date":"2025-08-27T19:25:46","date_gmt":"2025-08-27T13:55:46","guid":{"rendered":"https:\/\/codeanddebug.in\/blog\/?p=1005"},"modified":"2025-08-27T19:25:48","modified_gmt":"2025-08-27T13:55:48","slug":"best-time-to-buy-and-sell-stock-ii","status":"publish","type":"post","link":"https:\/\/codeanddebug.in\/blog\/best-time-to-buy-and-sell-stock-ii\/","title":{"rendered":"Best Time to Buy and Sell Stock II | Leetcode 122 | Recursion to Tabulation"},"content":{"rendered":"\n<p>You are given an array <code>prices<\/code> where <code>prices[i]<\/code> is the price of a stock on day <code>i<\/code>. You may complete as many transactions as you like. You must sell the stock before you buy again. Return the <strong>maximum profit<\/strong> you can achieve.<\/p>\n\n\n\n<p>Here&#8217;s the [<strong><a href=\"https:\/\/leetcode.com\/problems\/best-time-to-buy-and-sell-stock-ii\/description\/\" 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>Example<\/strong><\/p>\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(1 * 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=\"Input:  prices = [7,1,5,3,6,4]\nOutput: 7\nExplanation: Buy on day 2 (price 1), sell on day 3 (price 5), profit 4.\nBuy on day 4 (price 3), sell on day 5 (price 6), profit 3. Total 7.\" 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\">Input<\/span><span style=\"color: #D4D4D4\">:  <\/span><span style=\"color: #CE9178\">prices = [7,1,5,3,6,4]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #569CD6\">Output<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #B5CEA8\">7<\/span><\/span>\n<span class=\"line\"><span style=\"color: #569CD6\">Explanation<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #CE9178\">Buy on day 2 (price 1), sell on day 3 (price 5), profit 4.<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">Buy on day 4 (price 3), sell on day 5 (price 6), profit 3. Total 7.<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>Key idea: whenever there is an ascending step from <code>prices[i]<\/code> to <code>prices[i+1]<\/code>, it is profitable to capture it.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Recursion<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Intuition and Approach<\/h3>\n\n\n\n<p>Model decisions as a state machine on day <code>index<\/code> and a flag <code>buy<\/code>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If <code>buy == 1<\/code>, you can either buy today or skip.<\/li>\n\n\n\n<li>If <code>buy == 0<\/code>, you can either sell today or skip.<\/li>\n<\/ul>\n\n\n\n<p>Recurse to the next day and choose the better of the two options for each state. Base case at the end returns 0 profit.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">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, buy, prices):\n        if index == len(prices):\n            return 0\n        if buy == 1:\n            buy = -prices[index] + self.solve(index + 1, 0, prices)\n            not_buy = 0 + self.solve(index + 1, 1, prices)\n            profit = max(buy, not_buy)\n        else:\n            sell = prices[index] + self.solve(index + 1, 1, prices)\n            not_sell = 0 + self.solve(index + 1, 0, prices)\n            profit = max(sell, not_sell)\n        return profit\n\n    def maxProfit(self, prices: List[int]) -&gt; int:\n        n = len(prices)\n        return self.solve(0, 1, prices)\" 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\">buy<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">prices<\/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\"> index == <\/span><span style=\"color: #DCDCAA\">len<\/span><span style=\"color: #D4D4D4\">(prices):<\/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: #B5CEA8\">0<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> buy == <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            buy = -prices[index] + <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, prices)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            not_buy = <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, prices)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            profit = <\/span><span style=\"color: #DCDCAA\">max<\/span><span style=\"color: #D4D4D4\">(buy, not_buy)<\/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\">            sell = prices[index] + <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, prices)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            not_sell = <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, prices)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            profit = <\/span><span style=\"color: #DCDCAA\">max<\/span><span style=\"color: #D4D4D4\">(sell, not_sell)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> profit<\/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\">maxProfit<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">prices<\/span><span style=\"color: #D4D4D4\">: List[<\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">]) -&gt; <\/span><span style=\"color: #4EC9B0\">int<\/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\">(prices)<\/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(<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, prices)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Code explanation<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>At each day, decide to act or skip based on <code>buy<\/code> state.<\/li>\n\n\n\n<li>Buying subtracts price, selling adds price, skipping adds zero.<\/li>\n\n\n\n<li>Maximum profit of subproblems composes the global maximum.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Time and Space Complexity<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Precise: each state branches twice without caching, so exponential time in <code>n<\/code>. Recursion depth up to <code>n<\/code>, so space <code>O(n)<\/code>.<\/li>\n\n\n\n<li>Simplified: exponential time, linear space.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>Correct but impractical for larger inputs 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\">Memoization<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Intuition and Approach<\/h3>\n\n\n\n<p>Cache the results for each <code>(index, buy)<\/code> to avoid recomputation. There are <code>n * 2<\/code> states, each solved once.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">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, buy, prices, dp):\n        if index == len(prices):\n            return 0\n        if dp[index][buy] != -1:\n            return dp[index][buy]\n        if buy == 1:\n            buy_p = -prices[index] + self.solve(index + 1, 0, prices, dp)\n            not_buy = 0 + self.solve(index + 1, 1, prices, dp)\n            profit = max(buy_p, not_buy)\n        else:\n            sell = prices[index] + self.solve(index + 1, 1, prices, dp)\n            not_sell = 0 + self.solve(index + 1, 0, prices, dp)\n            profit = max(sell, not_sell)\n        dp[index][buy] = profit\n        return dp[index][buy]\n\n    def maxProfit(self, prices: List[int]) -&gt; int:\n        n = len(prices)\n        dp = [[-1, -1] for _ in range(n)]\n        return self.solve(0, 1, prices, 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\">buy<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">prices<\/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: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> index == <\/span><span style=\"color: #DCDCAA\">len<\/span><span style=\"color: #D4D4D4\">(prices):<\/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: #B5CEA8\">0<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> dp[index][buy] != -<\/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][buy]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> buy == <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            buy_p = -prices[index] + <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, prices, dp)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            not_buy = <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, prices, dp)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            profit = <\/span><span style=\"color: #DCDCAA\">max<\/span><span style=\"color: #D4D4D4\">(buy_p, not_buy)<\/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\">            sell = prices[index] + <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, prices, dp)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            not_sell = <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #569CD6\">self<\/span><span style=\"color: #D4D4D4\">.solve(index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, prices, dp)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            profit = <\/span><span style=\"color: #DCDCAA\">max<\/span><span style=\"color: #D4D4D4\">(sell, not_sell)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        dp[index][buy] = profit<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> dp[index][buy]<\/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\">maxProfit<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">prices<\/span><span style=\"color: #D4D4D4\">: List[<\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">]) -&gt; <\/span><span style=\"color: #4EC9B0\">int<\/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\">(prices)<\/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: #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(<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, prices, dp)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Code explanation<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>dp[index][buy]<\/code> stores the best profit from day <code>index<\/code> in the given state.<\/li>\n\n\n\n<li>Lookups prune the exponential tree to linear number of states.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Time and Space Complexity<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Precise: states <code>2n<\/code>, each with O(1) transitions, so <code>O(n)<\/code> time. DP table <code>O(n)<\/code>. Recursion stack <code>O(n)<\/code>.<\/li>\n\n\n\n<li>Simplified: <code>O(n)<\/code> time, <code>O(n)<\/code> space.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>A fast top down solution that passes constraints comfortably.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Tabulation<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Intuition and Approach<\/h3>\n\n\n\n<p>Bottom up DP over days from right to left. <code>dp[index][buy]<\/code> holds the best profit from day <code>index<\/code> with state <code>buy<\/code>. Initialize base row at <code>index = n<\/code> to 0.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">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 maxProfit(self, prices: List[int]) -&gt; int:\n        n = len(prices)\n        dp = [[-1, -1] for _ in range(n + 1)]\n        dp[n][0] = 0\n        dp[n][1] = 0\n        for index in range(n - 1, -1, -1):\n            for buy in range(0, 2):\n                if buy == 1:\n                    buy_p = -prices[index] + dp[index + 1][0]\n                    not_buy = 0 + dp[index + 1][1]\n                    profit = max(buy_p, not_buy)\n                else:\n                    sell = prices[index] + dp[index + 1][1]\n                    not_sell = 0 + dp[index + 1][0]\n                    profit = max(sell, not_sell)\n                dp[index][buy] = profit\n        return dp[0][1]\" 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\">maxProfit<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">prices<\/span><span style=\"color: #D4D4D4\">: List[<\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">]) -&gt; <\/span><span style=\"color: #4EC9B0\">int<\/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\">(prices)<\/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: #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 style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        dp[n][<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">] = <\/span><span style=\"color: #B5CEA8\">0<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        dp[n][<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">] = <\/span><span style=\"color: #B5CEA8\">0<\/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 style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, -<\/span><span style=\"color: #B5CEA8\">1<\/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\"> buy <\/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: #B5CEA8\">2<\/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\"> buy == <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    buy_p = -prices[index] + dp[index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">][<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    not_buy = <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> + dp[index + <\/span><span style=\"color: #B5CEA8\">1<\/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\">                    profit = <\/span><span style=\"color: #DCDCAA\">max<\/span><span style=\"color: #D4D4D4\">(buy_p, not_buy)<\/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\">                    sell = prices[index] + dp[index + <\/span><span style=\"color: #B5CEA8\">1<\/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\">                    not_sell = <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> + dp[index + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">][<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    profit = <\/span><span style=\"color: #DCDCAA\">max<\/span><span style=\"color: #D4D4D4\">(sell, not_sell)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                dp[index][buy] = profit<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> dp[<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">][<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Code explanation<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Fill the table from the end to the start using the same transition as memoization.<\/li>\n\n\n\n<li>The answer is the profit at day 0 when you are allowed to buy.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Time and Space Complexity<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Precise: <code>O(n * 2)<\/code> time and <code>O(n * 2)<\/code> space for the table.<\/li>\n\n\n\n<li>Simplified: <code>O(n)<\/code> time, <code>O(n)<\/code> space.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>Iterative and stack free. Mirrors the memoized recurrence.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Tabulation (Space Optimization)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Intuition and Approach<\/h3>\n\n\n\n<p>The value at <code>index<\/code> depends only on <code>index + 1<\/code>. Keep two 2-element arrays: <code>ahead<\/code> for day <code>index + 1<\/code>, <code>curr<\/code> for day <code>index<\/code>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">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 maxProfit(self, prices: List[int]) -&gt; int:\n        n = len(prices)\n\n        ahead = [0, 0]\n        # ahead[0]=0\n        # ahead[1]=0\n\n        for index in range(n - 1, -1, -1):\n            curr = [-1, -1]\n            for buy in range(0, 2):\n                if buy == 1:\n                    buy_p = -prices[index] + ahead[0]\n                    not_buy = 0 + ahead[1]\n                    profit = max(buy_p, not_buy)\n                else:\n                    sell = prices[index] + ahead[1]\n                    not_sell = 0 + ahead[0]\n                    profit = max(sell, not_sell)\n                curr[buy] = profit\n            ahead = curr\n        return ahead[1]\" 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\">maxProfit<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">prices<\/span><span style=\"color: #D4D4D4\">: List[<\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">]) -&gt; <\/span><span style=\"color: #4EC9B0\">int<\/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\">(prices)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        ahead = [<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, <\/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: #6A9955\"># ahead[0]=0<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># ahead[1]=0<\/span><\/span>\n<span class=\"line\"><\/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 style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, -<\/span><span style=\"color: #B5CEA8\">1<\/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\">            curr = [-<\/span><span style=\"color: #B5CEA8\">1<\/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\"> buy <\/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: #B5CEA8\">2<\/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\"> buy == <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    buy_p = -prices[index] + ahead[<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    not_buy = <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> + ahead[<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    profit = <\/span><span style=\"color: #DCDCAA\">max<\/span><span style=\"color: #D4D4D4\">(buy_p, not_buy)<\/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\">                    sell = prices[index] + ahead[<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    not_sell = <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> + ahead[<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    profit = <\/span><span style=\"color: #DCDCAA\">max<\/span><span style=\"color: #D4D4D4\">(sell, not_sell)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                curr[buy] = profit<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            ahead = curr<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> ahead[<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Code explanation<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>ahead[0]<\/code> and <code>ahead[1]<\/code> represent best profits for the next day in states sellable and buyable respectively.<\/li>\n\n\n\n<li>Build <code>curr<\/code> from <code>ahead<\/code>, then roll forward.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Time and Space Complexity<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Precise: single pass over days with constant state work gives <code>O(n)<\/code> time and <code>O(1)<\/code> extra space.<\/li>\n\n\n\n<li>Simplified: <code>O(n)<\/code> time, <code>O(1)<\/code> space.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>The most memory efficient DP version. Recommended for production.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Final takeaway<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The problem reduces to choosing buy or sell actions at each day to maximize profit.<\/li>\n\n\n\n<li>Recursion describes the choice structure but is exponential.<\/li>\n\n\n\n<li>Memoization and tabulation bring it to linear time.<\/li>\n\n\n\n<li>Space optimization keeps linear time with constant extra memory.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\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<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\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>You are given an array prices where prices[i] is the price of a stock on day i. You may complete as many transactions as you like. You must sell the stock before you buy again. Return the maximum profit you can achieve. Here&#8217;s the [Problem Link] to begin with. Example Key idea: whenever there is<\/p>\n","protected":false},"author":1,"featured_media":1008,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[3,6],"tags":[43,19],"class_list":{"0":"post-1005","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-structures-and-algorithm","8":"category-intermediate","9":"tag-dynamic-programming-on-stocks","10":"tag-medium"},"featured_image_src":"https:\/\/codeanddebug.in\/blog\/wp-content\/uploads\/2025\/08\/best-time-to-buy-and-sell-stockII-featured-image.png","author_info":{"display_name":"codeanddebug","author_link":"https:\/\/codeanddebug.in\/blog\/author\/codeanddebug\/"},"_links":{"self":[{"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts\/1005","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/comments?post=1005"}],"version-history":[{"count":1,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts\/1005\/revisions"}],"predecessor-version":[{"id":1009,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts\/1005\/revisions\/1009"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/media\/1008"}],"wp:attachment":[{"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/media?parent=1005"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/categories?post=1005"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/tags?post=1005"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}