deploy: f795901

Author: yanglbme

en/lc/781/index.html

@@ -24839,18 +24839,9 @@
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#solution-1" class="md-nav__link">
+  <a href="#solution-1-dynamic-programming" class="md-nav__link">
     <span class="md-ellipsis">
-      Solution 1
-    </span>
-  </a>
-  
-</li>
-        
-          <li class="md-nav__item">
-  <a href="#solution-2" class="md-nav__link">
-    <span class="md-ellipsis">
-      Solution 2
+      Solution 1: Dynamic Programming
     </span>
   </a>
 
@@ -86519,7 +86510,25 @@ <h2 id="description">Description</h2>
 <h2 id="solutions">Solutions</h2>
 <!-- solution:start -->
 
-<h3 id="solution-1">Solution 1</h3>
+<h3 id="solution-1-dynamic-programming">Solution 1: Dynamic Programming</h3>
+<p>First, we need to understand the problem. The problem is essentially asking us to find the number of ways to tile a <span class="arithmatex">\(2 \times n\)</span> board, where each square on the board can only be covered by one tile.</p>
+<p>There are two types of tiles: <code>2 x 1</code> and <code>L</code> shapes, and both types of tiles can be rotated. We denote the rotated tiles as <code>1 x 2</code> and <code>L'</code> shapes.</p>
+<p>We define <span class="arithmatex">\(f[i][j]\)</span> to represent the number of ways to tile the first <span class="arithmatex">\(2 \times i\)</span> board, where <span class="arithmatex">\(j\)</span> represents the state of the last column. The last column has 4 states:</p>
+<ul>
+<li>The last column is fully covered, denoted as <span class="arithmatex">\(0\)</span></li>
+<li>The last column has only the top square covered, denoted as <span class="arithmatex">\(1\)</span></li>
+<li>The last column has only the bottom square covered, denoted as <span class="arithmatex">\(2\)</span></li>
+<li>The last column is not covered, denoted as <span class="arithmatex">\(3\)</span></li>
+</ul>
+<p>The answer is <span class="arithmatex">\(f[n][0]\)</span>. Initially, <span class="arithmatex">\(f[0][0] = 1\)</span> and the rest <span class="arithmatex">\(f[0][j] = 0\)</span>.</p>
+<p>We consider tiling up to the <span class="arithmatex">\(i\)</span>-th column and look at the state transition equations:</p>
+<p>When <span class="arithmatex">\(j = 0\)</span>, the last column is fully covered. It can be transitioned from the previous column's states <span class="arithmatex">\(0, 1, 2, 3\)</span> by placing the corresponding tiles, i.e., <span class="arithmatex">\(f[i-1][0]\)</span> with a <code>1 x 2</code> tile, <span class="arithmatex">\(f[i-1][1]\)</span> with an <code>L'</code> tile, <span class="arithmatex">\(f[i-1][2]\)</span> with an <code>L'</code> tile, or <span class="arithmatex">\(f[i-1][3]\)</span> with two <code>2 x 1</code> tiles. Therefore, <span class="arithmatex">\(f[i][0] = \sum_{j=0}^3 f[i-1][j]\)</span>.</p>
+<p>When <span class="arithmatex">\(j = 1\)</span>, the last column has only the top square covered. It can be transitioned from the previous column's states <span class="arithmatex">\(2, 3\)</span> by placing a <code>2 x 1</code> tile or an <code>L</code> tile. Therefore, <span class="arithmatex">\(f[i][1] = f[i-1][2] + f[i-1][3]\)</span>.</p>
+<p>When <span class="arithmatex">\(j = 2\)</span>, the last column has only the bottom square covered. It can be transitioned from the previous column's states <span class="arithmatex">\(1, 3\)</span> by placing a <code>2 x 1</code> tile or an <code>L'</code> tile. Therefore, <span class="arithmatex">\(f[i][2] = f[i-1][1] + f[i-1][3]\)</span>.</p>
+<p>When <span class="arithmatex">\(j = 3\)</span>, the last column is not covered. It can be transitioned from the previous column's state <span class="arithmatex">\(0\)</span>. Therefore, <span class="arithmatex">\(f[i][3] = f[i-1][0]\)</span>.</p>
+<p>We can see that the state transition equations only involve the previous column's states, so we can use a rolling array to optimize the space complexity.</p>
+<p>Note that the values of the states can be very large, so we need to take modulo <span class="arithmatex">\(10^9 + 7\)</span>.</p>
+<p>The time complexity is <span class="arithmatex">\(O(n)\)</span>, and the space complexity is <span class="arithmatex">\(O(1)\)</span>. Where <span class="arithmatex">\(n\)</span> is the number of columns of the board.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:4"><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" /><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></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -86534,42 +86543,18 @@ <h3 id="solution-1">Solution 1</h3>
 <span class="normal"> 9</span>
 <span class="normal">10</span>
 <span class="normal">11</span>
-<span class="normal">12</span>
-<span class="normal">13</span>
-<span class="normal">14</span>
-<span class="normal">15</span>
-<span class="normal">16</span>
-<span class="normal">17</span>
-<span class="normal">18</span>
-<span class="normal">19</span>
-<span class="normal">20</span>
-<span class="normal">21</span>
-<span class="normal">22</span>
-<span class="normal">23</span>
-<span class="normal">24</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">class</span><span class="w"> </span><span class="nc">Solution</span><span class="p">:</span>
+<span class="normal">12</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">class</span><span class="w"> </span><span class="nc">Solution</span><span class="p">:</span>
     <span class="k">def</span><span class="w"> </span><span class="nf">numTilings</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="nd">@cache</span>
-        <span class="k">def</span><span class="w"> </span><span class="nf">dfs</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="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">n</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">0</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">n</span> <span class="ow">and</span> <span class="n">j</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">1</span>
-            <span class="n">ans</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                <span class="n">ans</span> <span class="o">=</span> <span class="p">(</span>
-                    <span class="n">dfs</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-                    <span class="o">+</span> <span class="n">dfs</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="o">+</span> <span class="n">dfs</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="o">+</span> <span class="n">dfs</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-                <span class="p">)</span>
-            <span class="k">elif</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
-                <span class="n">ans</span> <span class="o">=</span> <span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">dfs</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">ans</span> <span class="o">=</span> <span class="n">dfs</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">+</span> <span class="n">dfs</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">ans</span> <span class="o">%</span> <span class="n">mod</span>
-
+        <span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
         <span class="n">mod</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="mi">9</span> <span class="o">+</span> <span class="mi">7</span>
-        <span class="k">return</span> <span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+            <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">4</span>
+            <span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span> <span class="o">%</span> <span class="n">mod</span>
+            <span class="n">g</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span> <span class="o">%</span> <span class="n">mod</span>
+            <span class="n">g</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span> <span class="o">%</span> <span class="n">mod</span>
+            <span class="n">g</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+            <span class="n">f</span> <span class="o">=</span> <span class="n">g</span>
+        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
 </code></pre></div></td></tr></table></div>
 </div>
 <div class="tabbed-block">
@@ -86674,41 +86659,6 @@ <h3 id="solution-1">Solution 1</h3>
 </div>
 <!-- solution:end -->
 
-<!-- solution:start -->
-
-<h3 id="solution-2">Solution 2</h3>
-<div class="tabbed-set tabbed-alternate" data-tabs="2:1"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python3</label></div>
-<div class="tabbed-content">
-<div class="tabbed-block">
-<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"> 1</span>
-<span class="normal"> 2</span>
-<span class="normal"> 3</span>
-<span class="normal"> 4</span>
-<span class="normal"> 5</span>
-<span class="normal"> 6</span>
-<span class="normal"> 7</span>
-<span class="normal"> 8</span>
-<span class="normal"> 9</span>
-<span class="normal">10</span>
-<span class="normal">11</span>
-<span class="normal">12</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">class</span><span class="w"> </span><span class="nc">Solution</span><span class="p">:</span>
-    <span class="k">def</span><span class="w"> </span><span class="nf">numTilings</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-        <span class="n">mod</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="mi">9</span> <span class="o">+</span> <span class="mi">7</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">4</span>
-            <span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span> <span class="o">%</span> <span class="n">mod</span>
-            <span class="n">g</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span> <span class="o">%</span> <span class="n">mod</span>
-            <span class="n">g</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span> <span class="o">%</span> <span class="n">mod</span>
-            <span class="n">g</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">g</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</code></pre></div></td></tr></table></div>
-</div>
-</div>
-</div>
-<!-- solution:end -->
-
 <!-- problem:end -->