leetcode-problemset/leetcode-cn/originData/maximum-balanced-subsequence-sum.json

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            "title": "Maximum Balanced Subsequence Sum",
            "titleSlug": "maximum-balanced-subsequence-sum",
            "content": "<p>You are given a <strong>0-indexed</strong> integer array <code>nums</code>.</p>\n\n<p>A <strong>subsequence</strong> of <code>nums</code> having length <code>k</code> and consisting of <strong>indices</strong> <code>i<sub>0</sub>&nbsp;&lt;&nbsp;i<sub>1</sub> &lt;&nbsp;... &lt; i<sub>k-1</sub></code> is <strong>balanced</strong> if the following holds:</p>\n\n<ul>\n\t<li><code>nums[i<sub>j</sub>] - nums[i<sub>j-1</sub>] &gt;= i<sub>j</sub> - i<sub>j-1</sub></code>, for every <code>j</code> in the range <code>[1, k - 1]</code>.</li>\n</ul>\n\n<p>A <strong>subsequence</strong> of <code>nums</code> having length <code>1</code> is considered balanced.</p>\n\n<p>Return <em>an integer denoting the <strong>maximum</strong> possible <strong>sum of elements</strong> in a <strong>balanced</strong> subsequence of </em><code>nums</code>.</p>\n\n<p>A <strong>subsequence</strong> of an array is a new <strong>non-empty</strong> array that is formed from the original array by deleting some (<strong>possibly none</strong>) of the elements without disturbing the relative positions of the remaining elements.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [3,3,5,6]\n<strong>Output:</strong> 14\n<strong>Explanation:</strong> In this example, the subsequence [3,5,6] consisting of indices 0, 2, and 3 can be selected.\nnums[2] - nums[0] &gt;= 2 - 0.\nnums[3] - nums[2] &gt;= 3 - 2.\nHence, it is a balanced subsequence, and its sum is the maximum among the balanced subsequences of nums.\nThe subsequence consisting of indices 1, 2, and 3 is also valid.\nIt can be shown that it is not possible to get a balanced subsequence with a sum greater than 14.</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [5,-1,-3,8]\n<strong>Output:</strong> 13\n<strong>Explanation:</strong> In this example, the subsequence [5,8] consisting of indices 0 and 3 can be selected.\nnums[3] - nums[0] &gt;= 3 - 0.\nHence, it is a balanced subsequence, and its sum is the maximum among the balanced subsequences of nums.\nIt can be shown that it is not possible to get a balanced subsequence with a sum greater than 13.\n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n\n<pre>\n<strong>Input:</strong> nums = [-2,-1]\n<strong>Output:</strong> -1\n<strong>Explanation:</strong> In this example, the subsequence [-1] can be selected.\nIt is a balanced subsequence, and its sum is the maximum among the balanced subsequences of nums.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>\n\t<li><code>-10<sup>9</sup> &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>\n</ul>\n",
            "translatedTitle": "平衡子序列的最大和",
            "translatedContent": "<p>给你一个下标从 <strong>0</strong>&nbsp;开始的整数数组&nbsp;<code>nums</code>&nbsp;。</p>\n\n<p><code>nums</code>&nbsp;一个长度为 <code>k</code>&nbsp;的 <strong>子序列</strong>&nbsp;指的是选出 <code>k</code>&nbsp;个 <strong>下标</strong>&nbsp;<code>i<sub>0</sub>&nbsp;&lt;&nbsp;i<sub>1</sub> &lt;&nbsp;... &lt; i<sub>k-1</sub></code>&nbsp;，如果这个子序列满足以下条件，我们说它是 <strong>平衡的</strong>&nbsp;：</p>\n\n<ul>\n\t<li>对于范围&nbsp;<code>[1, k - 1]</code>&nbsp;内的所有&nbsp;<code>j</code>&nbsp;，<code>nums[i<sub>j</sub>] - nums[i<sub>j-1</sub>] &gt;= i<sub>j</sub> - i<sub>j-1</sub></code>&nbsp;都成立。</li>\n</ul>\n\n<p><code>nums</code>&nbsp;长度为 <code>1</code>&nbsp;的 <strong>子序列</strong>&nbsp;是平衡的。</p>\n\n<p>请你返回一个整数，表示 <code>nums</code>&nbsp;<strong>平衡</strong>&nbsp;子序列里面的 <strong>最大元素和</strong>&nbsp;。</p>\n\n<p>一个数组的 <strong>子序列</strong>&nbsp;指的是从原数组中删除一些元素（<strong>也可能一个元素也不删除</strong>）后，剩余元素保持相对顺序得到的 <strong>非空</strong>&nbsp;新数组。</p>\n\n<p>&nbsp;</p>\n\n<p><strong class=\"example\">示例 1：</strong></p>\n\n<pre>\n<b>输入：</b>nums = [3,3,5,6]\n<b>输出：</b>14\n<b>解释：</b>这个例子中，选择子序列 [3,5,6] ，下标为 0 ，2 和 3 的元素被选中。\nnums[2] - nums[0] &gt;= 2 - 0 。\nnums[3] - nums[2] &gt;= 3 - 2 。\n所以，这是一个平衡子序列，且它的和是所有平衡子序列里最大的。\n包含下标 1 ，2 和 3 的子序列也是一个平衡的子序列。\n最大平衡子序列和为 14 。</pre>\n\n<p><strong class=\"example\">示例 2：</strong></p>\n\n<pre>\n<b>输入：</b>nums = [5,-1,-3,8]\n<b>输出：</b>13\n<b>解释：</b>这个例子中，选择子序列 [5,8] ，下标为 0 和 3 的元素被选中。\nnums[3] - nums[0] &gt;= 3 - 0 。\n所以，这是一个平衡子序列，且它的和是所有平衡子序列里最大的。\n最大平衡子序列和为 13 。\n</pre>\n\n<p><strong class=\"example\">示例 3：</strong></p>\n\n<pre>\n<b>输入：</b>nums = [-2,-1]\n<b>输出：</b>-1\n<b>解释：</b>这个例子中，选择子序列 [-1] 。\n这是一个平衡子序列，而且它的和是 nums 所有平衡子序列里最大的。\n</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;= 10<sup>5</sup></code></li>\n\t<li><code>-10<sup>9</sup> &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>\n</ul>\n",
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                "Let <code>dp[x]</code> represent the maximum sum of a balanced subsequence ending at <code>x</code>.",
                "Rewriting the formula <code>nums[i<sub>j</sub>] - nums[i<sub>j-1</sub>] >= i<sub>j</sub> - i<sub>j-1</sub></code> gives <code>nums[i<sub>j</sub>] - i<sub>j</sub> >= nums[i<sub>j-1</sub>] - i<sub>j-1</sub></code>.",
                "So, for some index <code>x</code>, we need to find an index <code>y</code>, <code>y < x</code>, such that <code>dp[x] = nums[x] + dp[y]</code> is maximized, and <code>nums[x] - x >= nums[y] - y</code>.",
                "There are many ways to achieve this. One method involves sorting the values of <code>nums[x] - x</code> for all indices <code>x</code> and using a segment/Fenwick tree with coordinate compression.",
                "Hence, using a dictionary or map, let's call it <code>dict</code>, where <code>dict[nums[x] - x]</code> represents the position of the value, <code>nums[x] - x</code>, in the segment tree.",
                "The tree is initialized with zeros initially.",
                "For indices <code>x</code> in order from <code>[0, n - 1]</code>, <code>dp[x] = max(nums[x]</code>, <code>nums[x]</code> + the maximum query from the tree in the range <code>[0, dict[nums[x] - x]])</code>, and if <code>dp[x]</code> is greater than the value in the tree at position <code>dict[nums[x] - x]</code>, we update the value in the tree.",
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href=\\\"https:\\/\\/github.com\\/emirpasic\\/gods\\/tree\\/v1.18.1\\\" target=\\\"_blank\\\">https:\\/\\/godoc.org\\/github.com\\/emirpasic\\/gods@v1.18.1<\\/a> \\u7b2c\\u4e09\\u65b9\\u5e93\\u3002<\\/p>\"],\"python3\":[\"Python3\",\"<p>\\u7248\\u672c\\uff1a<code>Python 3.10<\\/code><\\/p>\\r\\n\\r\\n<p>\\u4e3a\\u4e86\\u65b9\\u4fbf\\u8d77\\u89c1\\uff0c\\u5927\\u90e8\\u5206\\u5e38\\u7528\\u5e93\\u5df2\\u7ecf\\u88ab\\u81ea\\u52a8 \\u5bfc\\u5165\\uff0c\\u5982<a href=\\\"https:\\/\\/docs.python.org\\/3\\/library\\/array.html\\\" target=\\\"_blank\\\">array<\\/a>, <a href=\\\"https:\\/\\/docs.python.org\\/3\\/library\\/bisect.html\\\" target=\\\"_blank\\\">bisect<\\/a>, <a href=\\\"https:\\/\\/docs.python.org\\/3\\/library\\/collections.html\\\" target=\\\"_blank\\\">collections<\\/a>\\u3002 \\u5982\\u679c\\u60a8\\u9700\\u8981\\u4f7f\\u7528\\u5176\\u4ed6\\u5e93\\u51fd\\u6570\\uff0c\\u8bf7\\u81ea\\u884c\\u5bfc\\u5165\\u3002<\\/p>\\r\\n\\r\\n<p>\\u5982\\u9700\\u4f7f\\u7528 Map\\/TreeMap 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