leetcode-problemset/leetcode-cn/originData/recover-the-original-array.json

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        "question": {
            "questionId": "2241",
            "questionFrontendId": "2122",
            "categoryTitle": "Algorithms",
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            "title": "Recover the Original Array",
            "titleSlug": "recover-the-original-array",
            "content": "<p>Alice had a <strong>0-indexed</strong> array <code>arr</code> consisting of <code>n</code> <strong>positive</strong> integers. She chose an arbitrary <strong>positive integer</strong> <code>k</code> and created two new <strong>0-indexed</strong> integer arrays <code>lower</code> and <code>higher</code> in the following manner:</p>\n\n<ol>\n\t<li><code>lower[i] = arr[i] - k</code>, for every index <code>i</code> where <code>0 &lt;= i &lt; n</code></li>\n\t<li><code>higher[i] = arr[i] + k</code>, for every index <code>i</code> where <code>0 &lt;= i &lt; n</code></li>\n</ol>\n\n<p>Unfortunately, Alice lost all three arrays. However, she remembers the integers that were present in the arrays <code>lower</code> and <code>higher</code>, but not the array each integer belonged to. Help Alice and recover the original array.</p>\n\n<p>Given an array <code>nums</code> consisting of <code>2n</code> integers, where <strong>exactly</strong> <code>n</code> of the integers were present in <code>lower</code> and the remaining in <code>higher</code>, return <em>the <strong>original</strong> array</em> <code>arr</code>. In case the answer is not unique, return <em><strong>any</strong> valid array</em>.</p>\n\n<p><strong>Note:</strong> The test cases are generated such that there exists <strong>at least one</strong> valid array <code>arr</code>.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [2,10,6,4,8,12]\n<strong>Output:</strong> [3,7,11]\n<strong>Explanation:</strong>\nIf arr = [3,7,11] and k = 1, we get lower = [2,6,10] and higher = [4,8,12].\nCombining lower and higher gives us [2,6,10,4,8,12], which is a permutation of nums.\nAnother valid possibility is that arr = [5,7,9] and k = 3. In that case, lower = [2,4,6] and higher = [8,10,12]. \n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [1,1,3,3]\n<strong>Output:</strong> [2,2]\n<strong>Explanation:</strong>\nIf arr = [2,2] and k = 1, we get lower = [1,1] and higher = [3,3].\nCombining lower and higher gives us [1,1,3,3], which is equal to nums.\nNote that arr cannot be [1,3] because in that case, the only possible way to obtain [1,1,3,3] is with k = 0.\nThis is invalid since k must be positive.\n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [5,435]\n<strong>Output:</strong> [220]\n<strong>Explanation:</strong>\nThe only possible combination is arr = [220] and k = 215. Using them, we get lower = [5] and higher = [435].\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2 * n == nums.length</code></li>\n\t<li><code>1 &lt;= n &lt;= 1000</code></li>\n\t<li><code>1 &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>\n\t<li>The test cases are generated such that there exists <strong>at least one</strong> valid array <code>arr</code>.</li>\n</ul>\n",
            "translatedTitle": "还原原数组",
            "translatedContent": "<p>Alice 有一个下标从 <strong>0</strong> 开始的数组 <code>arr</code> ，由 <code>n</code> 个正整数组成。她会选择一个任意的 <strong>正整数 </strong><code>k</code> 并按下述方式创建两个下标从 <strong>0</strong> 开始的新整数数组 <code>lower</code> 和 <code>higher</code> ：</p>\n\n<ol>\n\t<li>对每个满足 <code>0 &lt;= i &lt; n</code> 的下标 <code>i</code> ，<code>lower[i] = arr[i] - k</code></li>\n\t<li>对每个满足 <code>0 &lt;= i &lt; n</code> 的下标 <code>i</code> ，<code>higher[i] = arr[i] + k</code></li>\n</ol>\n\n<p>不幸地是，Alice 丢失了全部三个数组。但是，她记住了在数组 <code>lower</code> 和 <code>higher</code> 中出现的整数，但不知道每个整数属于哪个数组。请你帮助 Alice 还原原数组。</p>\n\n<p>给你一个由 2n 个整数组成的整数数组 <code>nums</code> ，其中 <strong>恰好</strong> <code>n</code> 个整数出现在 <code>lower</code> ，剩下的出现在 <code>higher</code> ，还原并返回 <strong>原数组</strong> <code>arr</code> 。如果出现答案不唯一的情况，返回 <strong>任一</strong> 有效数组。</p>\n\n<p><strong>注意：</strong>生成的测试用例保证存在 <strong>至少一个</strong> 有效数组 <code>arr</code> 。</p>\n\n<p>&nbsp;</p>\n\n<p><strong>示例 1：</strong></p>\n\n<pre><strong>输入：</strong>nums = [2,10,6,4,8,12]\n<strong>输出：</strong>[3,7,11]\n<strong>解释：</strong>\n如果 arr = [3,7,11] 且 k = 1 ，那么 lower = [2,6,10] 且 higher = [4,8,12] 。\n组合 lower 和 higher 得到 [2,6,10,4,8,12] ，这是 nums 的一个排列。\n另一个有效的数组是 arr = [5,7,9] 且 k = 3 。在这种情况下，lower = [2,4,6] 且 higher = [8,10,12] 。\n</pre>\n\n<p><strong>示例 2：</strong></p>\n\n<pre><strong>输入：</strong>nums = [1,1,3,3]\n<strong>输出：</strong>[2,2]\n<strong>解释：</strong>\n如果 arr = [2,2] 且 k = 1 ，那么 lower = [1,1] 且 higher = [3,3] 。\n组合 lower 和 higher 得到 [1,1,3,3] ，这是 nums 的一个排列。\n注意，数组不能是 [1,3] ，因为在这种情况下，获得 [1,1,3,3] 唯一可行的方案是 k = 0 。\n这种方案是无效的，k 必须是一个正整数。\n</pre>\n\n<p><strong>示例 3：</strong></p>\n\n<pre><strong>输入：</strong>nums = [5,435]\n<strong>输出：</strong>[220]\n<strong>解释：</strong>\n唯一可行的组合是 arr = [220] 且 k = 215 。在这种情况下，lower = [5] 且 higher = [435] 。</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>2 * n == nums.length</code></li>\n\t<li><code>1 &lt;= n &lt;= 1000</code></li>\n\t<li><code>1 &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>\n\t<li>生成的测试用例保证存在 <strong>至少一个</strong> 有效数组 <code>arr</code></li>\n</ul>\n",
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                "If we fix the value of k, how can we check if an original array exists for the fixed k?",
                "The smallest value of nums is obtained by subtracting k from the smallest value of the original array. How can we use this to reduce the search space for finding a valid k?",
                "You can compute every possible k by using the smallest value of nums (as lower[i]) against every other value in nums (as the corresponding higher[i]).",
                "For every computed k, greedily pair up the values in nums. This can be done sorting nums, then using a map to store previous values and searching that map for a corresponding lower[i] for the current nums[j] (as higher[i])."
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