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leetcode-problemset/leetcode-cn/originData/find-the-maximum-number-of-elements-in-subset.json
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"questionFrontendId": "3020",
"categoryTitle": "Algorithms",
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"title": "Find the Maximum Number of Elements in Subset",
"titleSlug": "find-the-maximum-number-of-elements-in-subset",
"content": "<p>You are given an array of <strong>positive</strong> integers <code>nums</code>.</p>\n\n<p>You need to select a <span data-keyword=\"subset\">subset</span> of <code>nums</code> which satisfies the following condition:</p>\n\n<ul>\n\t<li>You can place the selected elements in a <strong>0-indexed</strong> array such that it follows the pattern: <code>[x, x<sup>2</sup>, x<sup>4</sup>, ..., x<sup>k/2</sup>, x<sup>k</sup>, x<sup>k/2</sup>, ..., x<sup>4</sup>, x<sup>2</sup>, x]</code> (<strong>Note</strong> that <code>k</code> can be be any <strong>non-negative</strong> power of <code>2</code>). For example, <code>[2, 4, 16, 4, 2]</code> and <code>[3, 9, 3]</code> follow the pattern while <code>[2, 4, 8, 4, 2]</code> does not.</li>\n</ul>\n\n<p>Return <em>the <strong>maximum</strong> number of elements in a subset that satisfies these conditions.</em></p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [5,4,1,2,2]\n<strong>Output:</strong> 3\n<strong>Explanation:</strong> We can select the subset {4,2,2}, which can be placed in the array as [2,4,2] which follows the pattern and 2<sup>2</sup> == 4. Hence the answer is 3.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [1,3,2,4]\n<strong>Output:</strong> 1\n<strong>Explanation:</strong> We can select the subset {1}, which can be placed in the array as [1] which follows the pattern. Hence the answer is 1. Note that we could have also selected the subsets {2}, {4}, or {3}, there may be multiple subsets which provide the same answer. \n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>\n\t<li><code>1 &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>\n</ul>\n",
"translatedTitle": "子集中元素的最大数量",
"translatedContent": "<p>给你一个<strong> 正整数 </strong>数组 <code>nums</code> 。</p>\n\n<p>你需要从数组中选出一个满足下述条件的<span data-keyword=\"subset\">子集</span></p>\n\n<ul>\n\t<li>你可以将选中的元素放置在一个下标从 <strong>0</strong> 开始的数组中,并使其遵循以下模式:<code>[x, x<sup>2</sup>, x<sup>4</sup>, ..., x<sup>k/2</sup>, x<sup>k</sup>, x<sup>k/2</sup>, ..., x<sup>4</sup>, x<sup>2</sup>, x]</code><strong>注意</strong><code>k</code> 可以是任何 <strong>非负</strong> 的 2 的幂)。例如,<code>[2, 4, 16, 4, 2]</code> 和 <code>[3, 9, 3]</code> 都符合这一模式,而 <code>[2, 4, 8, 4, 2]</code> 则不符合。</li>\n</ul>\n\n<p>返回满足这些条件的子集中,元素数量的 <strong>最大值 </strong><em>。</em></p>\n\n<p>&nbsp;</p>\n\n<p><strong class=\"example\">示例 1</strong></p>\n\n<pre>\n<strong>输入:</strong>nums = [5,4,1,2,2]\n<strong>输出:</strong>3\n<strong>解释:</strong>选择子集 {4,2,2} ,将其放在数组 [2,4,2] 中,它遵循该模式,且 2<sup>2</sup> == 4 。因此答案是 3 。\n</pre>\n\n<p><strong class=\"example\">示例 2</strong></p>\n\n<pre>\n<strong>输入:</strong>nums = [1,3,2,4]\n<strong>输出:</strong>1\n<strong>解释:</strong>选择子集 {1},将其放在数组 [1] 中,它遵循该模式。因此答案是 1 。注意我们也可以选择子集 {2} 、{4} 或 {3} ,可能存在多个子集都能得到相同的答案。\n</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示:</strong></p>\n\n<ul>\n\t<li><code>2 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>\n\t<li><code>1 &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>\n</ul>\n",
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"We can select an odd number of <code>1</code>s.",
"Put all the values into a HashSet. We can start from each <code>x > 1</code> as the smallest chosen value and we can find the longest subset by checking the new values (which are the square of the previous value) in the set by brute force.",
"Note when <code>x > 1</code>, <code>x<sup>2</sup></code>, <code>x<sup>4</sup></code>, <code>x<sup>8</sup></code>, … increases very fast, the longest subset with smallest value x cannot be very long. (The length is <code>O(log(log(10<sup>9</sup>)))</code>.",
"Hence we can directly check all lengths less than <code>10</code> for all values of <code>x</code>."
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