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leetcode-problemset/leetcode-cn/problem (English)/位计数深度为 K 的整数数目 II(English) [number-of-integers-with-popcount-depth-equal-to-k-ii].html
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<p>You are given an integer array <code>nums</code>.</p>
<p>For any positive integer <code>x</code>, define the following sequence:</p>
<ul>
<li><code>p<sub>0</sub> = x</code></li>
<li><code>p<sub>i+1</sub> = popcount(p<sub>i</sub>)</code> for all <code>i &gt;= 0</code>, where <code>popcount(y)</code> is the number of set bits (1&#39;s) in the binary representation of <code>y</code>.</li>
</ul>
<p>This sequence will eventually reach the value 1.</p>
<p>The <strong>popcount-depth</strong> of <code>x</code> is defined as the <strong>smallest</strong> integer <code>d &gt;= 0</code> such that <code>p<sub>d</sub> = 1</code>.</p>
<p>For example, if <code>x = 7</code> (binary representation <code>&quot;111&quot;</code>). Then, the sequence is: <code>7 &rarr; 3 &rarr; 2 &rarr; 1</code>, so the popcount-depth of 7 is 3.</p>
<p>You are also given a 2D integer array <code>queries</code>, where each <code>queries[i]</code> is either:</p>
<ul>
<li><code>[1, l, r, k]</code> - <strong>Determine</strong> the number of indices <code>j</code> such that <code>l &lt;= j &lt;= r</code> and the <strong>popcount-depth</strong> of <code>nums[j]</code> is equal to <code>k</code>.</li>
<li><code>[2, idx, val]</code> - <strong>Update</strong> <code>nums[idx]</code> to <code>val</code>.</li>
</ul>
<p>Return an integer array <code>answer</code>, where <code>answer[i]</code> is the number of indices for the <code>i<sup>th</sup></code> query of type <code>[1, l, r, k]</code>.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [2,4], queries = [[1,0,1,1],[2,1,1],[1,0,1,0]]</span></p>
<p><strong>Output:</strong> <span class="example-io">[2,1]</span></p>
<p><strong>Explanation:</strong></p>
<table style="border: 1px solid black;">
<thead>
<tr>
<th style="border: 1px solid black;"><code>i</code></th>
<th style="border: 1px solid black;"><code>queries[i]</code></th>
<th style="border: 1px solid black;"><code>nums</code></th>
<th style="border: 1px solid black;">binary(<code>nums</code>)</th>
<th style="border: 1px solid black;">popcount-<br />
depth</th>
<th style="border: 1px solid black;"><code>[l, r]</code></th>
<th style="border: 1px solid black;"><code>k</code></th>
<th style="border: 1px solid black;">Valid<br />
<code>nums[j]</code></th>
<th style="border: 1px solid black;">updated<br />
<code>nums</code></th>
<th style="border: 1px solid black;">Answer</th>
</tr>
</thead>
<tbody>
<tr>
<td style="border: 1px solid black;">0</td>
<td style="border: 1px solid black;">[1,0,1,1]</td>
<td style="border: 1px solid black;">[2,4]</td>
<td style="border: 1px solid black;">[10, 100]</td>
<td style="border: 1px solid black;">[1, 1]</td>
<td style="border: 1px solid black;">[0, 1]</td>
<td style="border: 1px solid black;">1</td>
<td style="border: 1px solid black;">[0, 1]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">2</td>
</tr>
<tr>
<td style="border: 1px solid black;">1</td>
<td style="border: 1px solid black;">[2,1,1]</td>
<td style="border: 1px solid black;">[2,4]</td>
<td style="border: 1px solid black;">[10, 100]</td>
<td style="border: 1px solid black;">[1, 1]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">[2,1]</td>
<td style="border: 1px solid black;">&mdash;</td>
</tr>
<tr>
<td style="border: 1px solid black;">2</td>
<td style="border: 1px solid black;">[1,0,1,0]</td>
<td style="border: 1px solid black;">[2,1]</td>
<td style="border: 1px solid black;">[10, 1]</td>
<td style="border: 1px solid black;">[1, 0]</td>
<td style="border: 1px solid black;">[0, 1]</td>
<td style="border: 1px solid black;">0</td>
<td style="border: 1px solid black;">[1]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">1</td>
</tr>
</tbody>
</table>
<p>Thus, the final <code>answer</code> is <code>[2, 1]</code>.</p>
</div>
<p><strong class="example">Example 2:</strong></p>
<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [3,5,6], queries = [[1,0,2,2],[2,1,4],[1,1,2,1],[1,0,1,0]]</span></p>
<p><strong>Output:</strong> <span class="example-io">[3,1,0]</span></p>
<p><strong>Explanation:</strong></p>
<table style="border: 1px solid black;">
<thead>
<tr>
<th style="border: 1px solid black;"><code>i</code></th>
<th style="border: 1px solid black;"><code>queries[i]</code></th>
<th style="border: 1px solid black;"><code>nums</code></th>
<th style="border: 1px solid black;">binary(<code>nums</code>)</th>
<th style="border: 1px solid black;">popcount-<br />
depth</th>
<th style="border: 1px solid black;"><code>[l, r]</code></th>
<th style="border: 1px solid black;"><code>k</code></th>
<th style="border: 1px solid black;">Valid<br />
<code>nums[j]</code></th>
<th style="border: 1px solid black;">updated<br />
<code>nums</code></th>
<th style="border: 1px solid black;">Answer</th>
</tr>
</thead>
<tbody>
<tr>
<td style="border: 1px solid black;">0</td>
<td style="border: 1px solid black;">[1,0,2,2]</td>
<td style="border: 1px solid black;">[3, 5, 6]</td>
<td style="border: 1px solid black;">[11, 101, 110]</td>
<td style="border: 1px solid black;">[2, 2, 2]</td>
<td style="border: 1px solid black;">[0, 2]</td>
<td style="border: 1px solid black;">2</td>
<td style="border: 1px solid black;">[0, 1, 2]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">3</td>
</tr>
<tr>
<td style="border: 1px solid black;">1</td>
<td style="border: 1px solid black;">[2,1,4]</td>
<td style="border: 1px solid black;">[3, 5, 6]</td>
<td style="border: 1px solid black;">[11, 101, 110]</td>
<td style="border: 1px solid black;">[2, 2, 2]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">[3, 4, 6]</td>
<td style="border: 1px solid black;">&mdash;</td>
</tr>
<tr>
<td style="border: 1px solid black;">2</td>
<td style="border: 1px solid black;">[1,1,2,1]</td>
<td style="border: 1px solid black;">[3, 4, 6]</td>
<td style="border: 1px solid black;">[11, 100, 110]</td>
<td style="border: 1px solid black;">[2, 1, 2]</td>
<td style="border: 1px solid black;">[1, 2]</td>
<td style="border: 1px solid black;">1</td>
<td style="border: 1px solid black;">[1]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">1</td>
</tr>
<tr>
<td style="border: 1px solid black;">3</td>
<td style="border: 1px solid black;">[1,0,1,0]</td>
<td style="border: 1px solid black;">[3, 4, 6]</td>
<td style="border: 1px solid black;">[11, 100, 110]</td>
<td style="border: 1px solid black;">[2, 1, 2]</td>
<td style="border: 1px solid black;">[0, 1]</td>
<td style="border: 1px solid black;">0</td>
<td style="border: 1px solid black;">[]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">0</td>
</tr>
</tbody>
</table>
<p>Thus, the final <code>answer</code> is <code>[3, 1, 0]</code>.</p>
</div>
<p><strong class="example">Example 3:</strong></p>
<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [1,2], queries = [[1,0,1,1],[2,0,3],[1,0,0,1],[1,0,0,2]]</span></p>
<p><strong>Output:</strong> <span class="example-io">[1,0,1]</span></p>
<p><strong>Explanation:</strong></p>
<table style="border: 1px solid black;">
<thead>
<tr>
<th style="border: 1px solid black;"><code>i</code></th>
<th style="border: 1px solid black;"><code>queries[i]</code></th>
<th style="border: 1px solid black;"><code>nums</code></th>
<th style="border: 1px solid black;">binary(<code>nums</code>)</th>
<th style="border: 1px solid black;">popcount-<br />
depth</th>
<th style="border: 1px solid black;"><code>[l, r]</code></th>
<th style="border: 1px solid black;"><code>k</code></th>
<th style="border: 1px solid black;">Valid<br />
<code>nums[j]</code></th>
<th style="border: 1px solid black;">updated<br />
<code>nums</code></th>
<th style="border: 1px solid black;">Answer</th>
</tr>
</thead>
<tbody>
<tr>
<td style="border: 1px solid black;">0</td>
<td style="border: 1px solid black;">[1,0,1,1]</td>
<td style="border: 1px solid black;">[1, 2]</td>
<td style="border: 1px solid black;">[1, 10]</td>
<td style="border: 1px solid black;">[0, 1]</td>
<td style="border: 1px solid black;">[0, 1]</td>
<td style="border: 1px solid black;">1</td>
<td style="border: 1px solid black;">[1]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">1</td>
</tr>
<tr>
<td style="border: 1px solid black;">1</td>
<td style="border: 1px solid black;">[2,0,3]</td>
<td style="border: 1px solid black;">[1, 2]</td>
<td style="border: 1px solid black;">[1, 10]</td>
<td style="border: 1px solid black;">[0, 1]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">[3, 2]</td>
<td style="border: 1px solid black;">&nbsp;</td>
</tr>
<tr>
<td style="border: 1px solid black;">2</td>
<td style="border: 1px solid black;">[1,0,0,1]</td>
<td style="border: 1px solid black;">[3, 2]</td>
<td style="border: 1px solid black;">[11, 10]</td>
<td style="border: 1px solid black;">[2, 1]</td>
<td style="border: 1px solid black;">[0, 0]</td>
<td style="border: 1px solid black;">1</td>
<td style="border: 1px solid black;">[]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">0</td>
</tr>
<tr>
<td style="border: 1px solid black;">3</td>
<td style="border: 1px solid black;">[1,0,0,2]</td>
<td style="border: 1px solid black;">[3, 2]</td>
<td style="border: 1px solid black;">[11, 10]</td>
<td style="border: 1px solid black;">[2, 1]</td>
<td style="border: 1px solid black;">[0, 0]</td>
<td style="border: 1px solid black;">2</td>
<td style="border: 1px solid black;">[0]</td>
<td style="border: 1px solid black;">&mdash;</td>
<td style="border: 1px solid black;">1</td>
</tr>
</tbody>
</table>
<p>Thus, the final <code>answer</code> is <code>[1, 0, 1]</code>.</p>
</div>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= n == nums.length &lt;= 10<sup>5</sup></code></li>
<li><code>1 &lt;= nums[i] &lt;= 10<sup>15</sup></code></li>
<li><code>1 &lt;= queries.length &lt;= 10<sup>5</sup></code></li>
<li><code>queries[i].length == 3</code> or <code>4</code>
<ul>
<li><code>queries[i] == [1, l, r, k]</code> or,</li>
<li><code>queries[i] == [2, idx, val]</code></li>
<li><code>0 &lt;= l &lt;= r &lt;= n - 1</code></li>
<li><code>0 &lt;= k &lt;= 5</code></li>
<li><code>0 &lt;= idx &lt;= n - 1</code></li>
<li><code>1 &lt;= val &lt;= 10<sup>15</sup></code></li>
</ul>
</li>
</ul>
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