leetcode-problemset/leetcode-cn/originData/sum-of-imbalance-numbers-of-all-subarrays.json

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            "categoryTitle": "Algorithms",
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            "title": "Sum of Imbalance Numbers of All Subarrays",
            "titleSlug": "sum-of-imbalance-numbers-of-all-subarrays",
            "content": "<p>The <strong>imbalance number</strong> of a <strong>0-indexed</strong> integer array <code>arr</code> of length <code>n</code> is defined as the number of indices in <code>sarr = sorted(arr)</code> such that:</p>\n\n<ul>\n\t<li><code>0 &lt;= i &lt; n - 1</code>, and</li>\n\t<li><code>sarr[i+1] - sarr[i] &gt; 1</code></li>\n</ul>\n\n<p>Here, <code>sorted(arr)</code> is the function that returns the sorted version of <code>arr</code>.</p>\n\n<p>Given a <strong>0-indexed</strong> integer array <code>nums</code>, return <em>the <strong>sum of imbalance numbers</strong> of all its <strong>subarrays</strong></em>.</p>\n\n<p>A <strong>subarray</strong> is a contiguous <strong>non-empty</strong> sequence of elements within an array.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [2,3,1,4]\n<strong>Output:</strong> 3\n<strong>Explanation:</strong> There are 3 subarrays with non-zero<strong> </strong>imbalance numbers:\n- Subarray [3, 1] with an imbalance number of 1.\n- Subarray [3, 1, 4] with an imbalance number of 1.\n- Subarray [1, 4] with an imbalance number of 1.\nThe imbalance number of all other subarrays is 0. Hence, the sum of imbalance numbers of all the subarrays of nums is 3. \n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [1,3,3,3,5]\n<strong>Output:</strong> 8\n<strong>Explanation:</strong> There are 7 subarrays with non-zero imbalance numbers:\n- Subarray [1, 3] with an imbalance number of 1.\n- Subarray [1, 3, 3] with an imbalance number of 1.\n- Subarray [1, 3, 3, 3] with an imbalance number of 1.\n- Subarray [1, 3, 3, 3, 5] with an imbalance number of 2. \n- Subarray [3, 3, 3, 5] with an imbalance number of 1. \n- Subarray [3, 3, 5] with an imbalance number of 1.\n- Subarray [3, 5] with an imbalance number of 1.\nThe imbalance number of all other subarrays is 0. Hence, the sum of imbalance numbers of all the subarrays of nums is 8. </pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= nums.length &lt;= 1000</code></li>\n\t<li><code>1 &lt;= nums[i] &lt;= nums.length</code></li>\n</ul>\n",
            "translatedTitle": "所有子数组中不平衡数字之和",
            "translatedContent": "<p>一个长度为 <code>n</code>&nbsp;下标从 <strong>0</strong>&nbsp;开始的整数数组 <code>arr</code>&nbsp;的 <strong>不平衡数字</strong>&nbsp;定义为，在&nbsp;<code>sarr = sorted(arr)</code>&nbsp;数组中，满足以下条件的下标数目：</p>\n\n<ul>\n\t<li><code>0 &lt;= i &lt; n - 1</code>&nbsp;，和</li>\n\t<li><code>sarr[i+1] - sarr[i] &gt; 1</code></li>\n</ul>\n\n<p>这里，<code>sorted(arr)</code>&nbsp;表示将数组 <code>arr</code>&nbsp;排序后得到的数组。</p>\n\n<p>给你一个下标从 <strong>0</strong>&nbsp;开始的整数数组&nbsp;<code>nums</code>&nbsp;，请你返回它所有&nbsp;<strong>子数组</strong>&nbsp;的&nbsp;<strong>不平衡数字</strong>&nbsp;之和。</p>\n\n<p>子数组指的是一个数组中连续一段 <strong>非空</strong>&nbsp;的元素序列。</p>\n\n<p>&nbsp;</p>\n\n<p><strong>示例 1：</strong></p>\n\n<pre><strong>输入：</strong>nums = [2,3,1,4]\n<b>输出：</b>3\n<b>解释：</b>总共有 3 个子数组有非 0 不平衡数字：\n- 子数组 [3, 1] ，不平衡数字为 1 。\n- 子数组 [3, 1, 4] ，不平衡数字为 1 。\n- 子数组 [1, 4] ，不平衡数字为 1 。\n其他所有子数组的不平衡数字都是 0 ，所以所有子数组的不平衡数字之和为 3 。\n</pre>\n\n<p><strong>示例 2：</strong></p>\n\n<pre><b>输入：</b>nums = [1,3,3,3,5]\n<b>输出：</b>8\n<b>解释：</b>总共有 7 个子数组有非 0 不平衡数字：\n- 子数组 [1, 3] ，不平衡数字为 1 。\n- 子数组 [1, 3, 3] ，不平衡数字为 1 。\n- 子数组 [1, 3, 3, 3] ，不平衡数字为 1 。\n- 子数组 [1, 3, 3, 3, 5] ，不平衡数字为 2 。\n- 子数组 [3, 3, 3, 5] ，不平衡数字为 1 。\n- 子数组 [3, 3, 5] ，不平衡数字为 1 。\n- 子数组 [3, 5] ，不平衡数字为 1 。\n其他所有子数组的不平衡数字都是 0 ，所以所有子数组的不平衡数字之和为 8 。</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= nums.length &lt;= 1000</code></li>\n\t<li><code>1 &lt;= nums[i] &lt;= nums.length</code></li>\n</ul>\n",
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                "Iterate over all subarrays in a nested fashion. Namely, for each left endpoint, start from nums[left] and add elements nums[left + 1], nums[left + 2], etc.",
                "To keep track of the imbalance value, maintain a set of added elements.",
                "Increment the imbalance value whenever a new number is not adjacent (+/- 1) to other old numbers. For example, when you add 3 to [1, 5], or when you add 5 to [1, 3]. For a formal proof, consider three cases: new value is (i) largest, (ii) smallest, (iii) between two old numbers.",
                "Decrement the imbalance value whenever a new number is adjacent (+/- 1) to two old numbers. For example, when you add 3 to [2, 4]. The imbalance value does not change in the case of one adjacent old number."
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