deploy: b85f63b

Author: yanglbme

en/lc/547/index.html

@@ -18875,18 +18875,18 @@
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#solution-1" class="md-nav__link">
+  <a href="#solution-1-dfs" class="md-nav__link">
     <span class="md-ellipsis">
-      Solution 1
+      Solution 1: DFS
     </span>
   </a>
 
 </li>
 
           <li class="md-nav__item">
-  <a href="#solution-2" class="md-nav__link">
+  <a href="#solution-2-union-find" class="md-nav__link">
     <span class="md-ellipsis">
-      Solution 2
+      Solution 2: Union-Find
     </span>
   </a>
 
@@ -82160,7 +82160,11 @@ <h2 id="description">Description</h2>
 <h2 id="solutions">Solutions</h2>
 <!-- solution:start -->
 
-<h3 id="solution-1">Solution 1</h3>
+<h3 id="solution-1-dfs">Solution 1: DFS</h3>
+<p>We create an array $\textit{vis}$ to record whether each city has been visited.</p>
+<p>Next, we traverse each city $i$. If the city has not been visited, we start a depth-first search from that city. Using the matrix $\textit{isConnected}$, we find the cities directly connected to this city. These cities and the current city belong to the same province. We continue the depth-first search for these cities until all cities in the same province have been visited. This counts as one province, so we increment the answer $\textit{ans}$ by $1$. Then, we move to the next unvisited city and repeat the process until all cities have been traversed.</p>
+<p>Finally, return the answer.</p>
+<p>The time complexity is $O(n^2)$, and the space complexity is $O(n)$. Here, $n$ is the number of cities.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:6"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python3</label><label for="__tabbed_1_2">Java</label><label for="__tabbed_1_3">C++</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">TypeScript</label><label for="__tabbed_1_6">Rust</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -82284,7 +82288,7 @@ <h3 id="solution-1">Solution 1</h3>
 <span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">ans</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <span class="w">        </span><span class="kt">bool</span><span class="w"> </span><span class="n">vis</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
 <span class="w">        </span><span class="n">memset</span><span class="p">(</span><span class="n">vis</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="n">vis</span><span class="p">));</span>
-<span class="w">        </span><span class="n">function</span><span class="o">&lt;</span><span class="kt">void</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dfs</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="o">&amp;</span><span class="p">](</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<span class="w">        </span><span class="k">auto</span><span class="w"> </span><span class="n">dfs</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="o">&amp;</span><span class="p">](</span><span class="k">this</span><span class="w"> </span><span class="k">auto</span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">dfs</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">-&gt;</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="p">{</span>
 <span class="w">            </span><span class="n">vis</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
 <span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">vis</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">isConnected</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
@@ -82445,7 +82449,11 @@ <h3 id="solution-1">Solution 1</h3>
 
 <!-- solution:start -->
 
-<h3 id="solution-2">Solution 2</h3>
+<h3 id="solution-2-union-find">Solution 2: Union-Find</h3>
+<p>We can also use the union-find data structure to maintain each connected component. Initially, each city belongs to a different connected component, so the number of provinces is $n$.</p>
+<p>Next, we traverse the matrix $\textit{isConnected}$. If there is a connection between two cities $(i, j)$ and they belong to two different connected components, they will be merged into one connected component, and the number of provinces is decremented by $1$.</p>
+<p>Finally, return the number of provinces.</p>
+<p>The time complexity is $O(n^2 \times \log n)$, and the space complexity is $O(n)$. Here, $n$ is the number of cities, and $\log n$ is the time complexity of path compression in the union-find data structure.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:5"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python3</label><label for="__tabbed_2_2">Java</label><label for="__tabbed_2_3">C++</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">TypeScript</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -82583,7 +82591,7 @@ <h3 id="solution-2">Solution 2</h3>
 <span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">isConnected</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
 <span class="w">        </span><span class="n">iota</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
-<span class="w">        </span><span class="n">function</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="o">&gt;</span><span class="w"> </span><span class="n">find</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="o">&amp;</span><span class="p">](</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">-&gt;</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
+<span class="w">        </span><span class="k">auto</span><span class="w"> </span><span class="n">find</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="o">&amp;</span><span class="p">](</span><span class="k">this</span><span class="w"> </span><span class="k">auto</span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">find</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">-&gt;</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <span class="w">                </span><span class="n">p</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">find</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">x</span><span class="p">]);</span>
 <span class="w">            </span><span class="p">}</span>
@@ -82686,15 +82694,9 @@ <h3 id="solution-2">Solution 2</h3>
 <span class="normal">21</span>
 <span class="normal">22</span>
 <span class="normal">23</span>
-<span class="normal">24</span>
-<span class="normal">25</span>
-<span class="normal">26</span>
-<span class="normal">27</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="kd">function</span><span class="w"> </span><span class="nx">findCircleNum</span><span class="p">(</span><span class="nx">isConnected</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][])</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
+<span class="normal">24</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="kd">function</span><span class="w"> </span><span class="nx">findCircleNum</span><span class="p">(</span><span class="nx">isConnected</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][])</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">isConnected</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">p</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
-<span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<span class="w">        </span><span class="nx">p</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
-<span class="w">    </span><span class="p">}</span>
+<span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">p</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">(</span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">i</span><span class="p">);</span>
 <span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">find</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">x</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">p</span><span class="p">[</span><span class="nx">x</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="nx">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <span class="w">            </span><span class="nx">p</span><span class="p">[</span><span class="nx">x</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">find</span><span class="p">(</span><span class="nx">p</span><span class="p">[</span><span class="nx">x</span><span class="p">]);</span>