"content":"<p>You are given a <strong>0-indexed</strong> array <code>arr</code> consisting of <code>n</code> positive integers, and a positive integer <code>k</code>.</p>\n\n<p>The array <code>arr</code> is called <strong>K-increasing</strong> if <code>arr[i-k] <= arr[i]</code> holds for every index <code>i</code>, where <code>k <= i <= n-1</code>.</p>\n\n<ul>\n\t<li>For example, <code>arr = [4, 1, 5, 2, 6, 2]</code> is K-increasing for <code>k = 2</code> because:\n\n\t<ul>\n\t\t<li><code>arr[0] <= arr[2] (4 <= 5)</code></li>\n\t\t<li><code>arr[1] <= arr[3] (1 <= 2)</code></li>\n\t\t<li><code>arr[2] <= arr[4] (5 <= 6)</code></li>\n\t\t<li><code>arr[3] <= arr[5] (2 <= 2)</code></li>\n\t</ul>\n\t</li>\n\t<li>However, the same <code>arr</code> is not K-increasing for <code>k = 1</code> (because <code>arr[0] > arr[1]</code>) or <code>k = 3</code> (because <code>arr[0] > arr[3]</code>).</li>\n</ul>\n\n<p>In one <strong>operation</strong>, you can choose an index <code>i</code> and <strong>change</strong> <code>arr[i]</code> into <strong>any</strong> positive integer.</p>\n\n<p>Return <em>the <strong>minimum number of operations</strong> required to make the array K-increasing for the given </em><code>k</code>.</p>\n\n<p> </p>\n<p><strong>Example 1:</strong></p>\n\n<pre>\n<strong>Input:</strong> arr = [5,4,3,2,1], k = 1\n<strong>Output:</strong> 4\n<strong>Explanation:\n</strong>For k = 1, the resultant array has to be non-decreasing.\nSome of the K-increasing arrays that can be formed are [5,<u><strong>6</strong></u>,<u><strong>7</strong></u>,<u><strong>8</strong></u>,<u><strong>9</strong></u>], [<u><strong>1</strong></u>,<u><strong>1</strong></u>,<u><strong>1</strong></u>,<u><strong>1</strong></u>,1], [<u><strong>2</strong></u>,<u><strong>2</strong></u>,3,<u><strong>4</strong></u>,<u><strong>4</strong></u>]. All of them require 4 operations.\nIt is suboptimal to change the array to, for example, [<u><strong>6</strong></u>,<u><strong>7</strong></u>,<u><strong>8</strong></u>,<u><strong>9</strong></u>,<u><strong>10</strong></u>] because it would take 5 operations.\nIt can be shown that we cannot make the array K-increasing in less than 4 operations.\n</pre>\n\n<p><strong>Example 2:</strong></p>\n\n<pre>\n<strong>Input:</strong> arr = [4,1,5,2,6,2], k = 2\n<strong>Output:</strong> 0\n<strong>Explanation:</strong>\nThis is the same example as the one in the problem description.\nHere, for every index i where 2 <= i <= 5, arr[i-2] <=<b> </b>arr[i].\nSince the given array is already K-increasing, we do not need to perform any operations.</pre>\n\n<p><strong>Example 3:</strong></p>\n\n<pre>\n<strong>Input:</strong> arr = [4,1,5,2,6,2], k = 3\n<strong>Output:</strong> 2\n<strong>Explanation:</strong>\nIndices 3 and 5 are the only ones not satisfying arr[i-3] <= arr[i] for 3 <= i <= 5.\nOne of the ways we can make the array K-increasing is by changing arr[3] to 4 and arr[5] to 5.\nThe array will now be [4,1,5,<u><strong>4</strong></u>,6,<u><strong>5</strong></u>].\nNote that there can be other ways to make the array K-increasing, but none of them require less than 2 operations.\n</pre>\n\n<p> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 <= arr.length <= 10<sup>5</sup></code></li>\n\t<li><code>1 <= arr[i], k <= arr.length</code></li>\n</ul>\n",
"Can we divide the array into non-overlapping subsequences and simplify the problem?",
"In the final array, arr[i-k] ≤ arr[i] should hold. We can use this to divide the array into at most k non-overlapping sequences, where arr[i] will belong to the (i%k)th sequence.",
"Now our problem boils down to performing the minimum operations on each sequence such that it becomes non-decreasing. Our answer will be the sum of operations on each sequence.",
"Which indices of a sequence should we not change in order to count the minimum operations? Can finding the longest non-decreasing subsequence of the sequence help?"
"envInfo":"{\"cpp\": [\"C++\", \"<p>Compiled with <code> clang 11 </code> using the latest C++ 17 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 gnu99 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://docs.microsoft.com/en-us/dotnet/csharp/whats-new/csharp-9\\\" target=\\\"_blank\\\">C# 10 with .NET 6 runtime</a></p>\\r\\n\\r\\n<p>Your code is compiled with debug flag enabled (<code>/debug</code>).</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 <a href=\\\"https://github.com/datastructures-js/priority-queue\\\" target=\\\"_blank\\\">datastructures-js/priority-queue</a> and <a href=\\\"https:
