"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> < i<sub>1</sub> < ... < 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>] >= 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> </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] >= 2 - 0.\nnums[3] - nums[2] >= 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] >= 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> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 <= nums.length <= 10<sup>5</sup></code></li>\n\t<li><code>-10<sup>9</sup> <= nums[i] <= 10<sup>9</sup></code></li>\n</ul>\n",
"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.",
"The answer to the problem is the maximum value in <code>dp</code>."
"envInfo":"{\"cpp\": [\"C++\", \"<p>Compiled with <code> clang 11 </code> using the latest C++ 20 standard.</p>\\r\\n\\r\\n<p>Your code is compiled with level two optimization (<code>-O2</code>). <a href=\\\"https://github.com/google/sanitizers/wiki/AddressSanitizer\\\" target=\\\"_blank\\\">AddressSanitizer</a> is also enabled to help detect out-of-bounds and use-after-free bugs.</p>\\r\\n\\r\\n<p>Most standard library headers are already included automatically for your convenience.</p>\"], \"java\": [\"Java\", \"<p><code>OpenJDK 17</code>. Java 8 features such as lambda expressions and stream API can be used. </p>\\r\\n\\r\\n<p>Most standard library headers are already included automatically for your convenience.</p>\\r\\n<p>Includes <code>Pair</code> class from https://docs.oracle.com/javase/8/javafx/api/javafx/util/Pair.html.</p>\"], \"python\": [\"Python\", \"<p><code>Python 2.7.12</code>.</p>\\r\\n\\r\\n<p>Most libraries are already imported automatically for your convenience, such as <a href=\\\"https://docs.python.org/2/library/array.html\\\" target=\\\"_blank\\\">array</a>, <a href=\\\"https://docs.python.org/2/library/bisect.html\\\" target=\\\"_blank\\\">bisect</a>, <a href=\\\"https://docs.python.org/2/library/collections.html\\\" target=\\\"_blank\\\">collections</a>. If you need more libraries, you can import it yourself.</p>\\r\\n\\r\\n<p>For Map/TreeMap data structure, you may use <a href=\\\"http://www.grantjenks.com/docs/sortedcontainers/\\\" target=\\\"_blank\\\">sortedcontainers</a> library.</p>\\r\\n\\r\\n<p>Note that Python 2.7 <a href=\\\"https://www.python.org/dev/peps/pep-0373/\\\" target=\\\"_blank\\\">will not be maintained past 2020</a>. For the latest Python, please choose Python3 instead.</p>\"], \"c\": [\"C\", \"<p>Compiled with <code>gcc 8.2</code> using the gnu11 standard.</p>\\r\\n\\r\\n<p>Your code is compiled with level one optimization (<code>-O1</code>). <a href=\\\"https://github.com/google/sanitizers/wiki/AddressSanitizer\\\" target=\\\"_blank\\\">AddressSanitizer</a> is also enabled to help detect out-of-bounds and use-after-free bugs.</p>\\r\\n\\r\\n<p>Most standard library headers are already included automatically for your convenience.</p>\\r\\n\\r\\n<p>For hash table operations, you may use <a href=\\\"https://troydhanson.github.io/uthash/\\\" target=\\\"_blank\\\">uthash</a>. \\\"uthash.h\\\" is included by default. Below are some examples:</p>\\r\\n\\r\\n<p><b>1. Adding an item to a hash.</b>\\r\\n<pre>\\r\\nstruct hash_entry {\\r\\n int id; /* we'll use this field as the key */\\r\\n char name[10];\\r\\n UT_hash_handle hh; /* makes this structure hashable */\\r\\n};\\r\\n\\r\\nstruct hash_entry *users = NULL;\\r\\n\\r\\nvoid add_user(struct hash_entry *s) {\\r\\n HASH_ADD_INT(users, id, s);\\r\\n}\\r\\n</pre>\\r\\n</p>\\r\\n\\r\\n<p><b>2. Looking up an item in a hash:</b>\\r\\n<pre>\\r\\nstruct hash_entry *find_user(int user_id) {\\r\\n struct hash_entry *s;\\r\\n HASH_FIND_INT(users, &user_id, s);\\r\\n return s;\\r\\n}\\r\\n</pre>\\r\\n</p>\\r\\n\\r\\n<p><b>3. Deleting an item in a hash:</b>\\r\\n<pre>\\r\\nvoid delete_user(struct hash_entry *user) {\\r\\n HASH_DEL(users, user); \\r\\n}\\r\\n</pre>\\r\\n</p>\"], \"csharp\": [\"C#\", \"<p><a href=\\\"https://learn.microsoft.com/en-us/dotnet/csharp/whats-new/csharp-10\\\" target=\\\"_blank\\\">C# 10 with .NET 6 runtime</a></p>\"], \"javascript\": [\"JavaScript\", \"<p><code>Node.js 16.13.2</code>.</p>\\r\\n\\r\\n<p>Your code is run with <code>--harmony</code> flag, enabling <a href=\\\"http://node.green/\\\" target=\\\"_blank\\\">new ES6 features</a>.</p>\\r\\n\\r\\n<p><a href=\\\"https://lodash.com\\\" target=\\\"_blank\\\">lodash.js</a> library is included by default.</p>\\r\\n\\r\\n<p>For Priority Queue / Queue data structures, you may use 5.3.0 version of <a href=\\\"https://github.com/datastructures-js/priority-queue/tree/fb4fdb984834421279aeb081df7af624d17c2a03\\\" target=\\\"_blank\\\">datastructures-js/priority-queue</a> and 4.2.1 version of <a href=\\\"https://githu
