deploy

2026-04-27 20:11:01 +08:00 · 2023-08-27 00:50:10 +08:00
parent 3bb4725cbb
commit 8350023f50
12 changed files with 239 additions and 230 deletions
--- a/chapter_dynamic_programming/dp_solution_pipeline/index.html
+++ b/chapter_dynamic_programming/dp_solution_pipeline/index.html
@@ -2909,7 +2909,7 @@
        
          <li class="md-nav__item">
  <a href="#4" class="md-nav__link">
-    4. &nbsp; 状态压缩
+    4. &nbsp; 空间优化
  </a>
  
 </li>
@@ -3484,7 +3484,7 @@
        
          <li class="md-nav__item">
  <a href="#4" class="md-nav__link">
-    4. &nbsp; 状态压缩
+    4. &nbsp; 空间优化
  </a>
  
 </li>
@@ -4311,13 +4311,13 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </div>
 <p align="center"> 图 14-16 &nbsp; 最小路径和的动态规划过程 </p>

-<h3 id="4">4. &nbsp; 状态压缩<a class="headerlink" href="#4" title="Permanent link">&para;</a></h3>
+<h3 id="4">4. &nbsp; 空间优化<a class="headerlink" href="#4" title="Permanent link">&para;</a></h3>
 <p>由于每个格子只与其左边和上边的格子有关，因此我们可以只用一个单行数组来实现 <span class="arithmatex">\(dp\)</span> 表。</p>
 <p>请注意，因为数组 <code>dp</code> 只能表示一行的状态，所以我们无法提前初始化首列状态，而是在遍历每行中更新它。</p>
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 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* 最小路径和：状态压缩后的动态规划 */</span>
+<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* 最小路径和：空间优化后的动态规划 */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><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">grid</span><span class="p">.</span><span class="na">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
@@ -4341,7 +4341,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* 最小路径和：状态压缩后的动态规划 */</span>
+<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* 最小路径和：空间优化后的动态规划 */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDPComp</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><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">grid</span><span class="p">.</span><span class="n">size</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
@@ -4366,7 +4366,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">min_path_sum.py</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="k">def</span> <span class="nf">min_path_sum_dp_comp</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</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>
-<a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;最小路径和：状态压缩后的动态规划&quot;&quot;&quot;</span>
+<a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;最小路径和：空间优化后的动态规划&quot;&quot;&quot;</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a>    <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a>    <span class="c1"># 初始化 dp 表</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span>
@@ -4385,7 +4385,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="cm">/* 最小路径和：状态压缩后的动态规划 */</span>
+<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="cm">/* 最小路径和：空间优化后的动态规划 */</span>
 <a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">minPathSumDPComp</span><span class="p">(</span><span class="nx">grid</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a><span class="w">    </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">grid</span><span class="p">),</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
@@ -4421,7 +4421,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* 最小路径和：状态压缩后的动态规划 */</span>
+<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* 最小路径和：空间优化后的动态规划 */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><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">grid</span><span class="p">.</span><span class="n">Length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">].</span><span class="n">Length</span><span class="p">;</span>
 <a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
@@ -4445,7 +4445,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* 最小路径和：状态压缩后的动态规划 */</span>
+<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* 最小路径和：空间优化后的动态规划 */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span> <span class="nf">minPathSumDPComp</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span> <span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span> <span class="p">-&gt;</span> <span class="nb">Int</span> <span class="p">{</span>
 <a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a>    <span class="kd">let</span> <span class="nv">n</span> <span class="p">=</span> <span class="n">grid</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a>    <span class="kd">let</span> <span class="nv">m</span> <span class="p">=</span> <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="bp">count</span>
@@ -4470,7 +4470,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.zig</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="c1">// 最小路径和：状态压缩后的动态规划</span>
+<div class="highlight"><span class="filename">min_path_sum.zig</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="c1">// 最小路径和：空间优化后的动态规划</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">minPathSumDPComp</span><span class="p">(</span><span class="kr">comptime</span><span class="w"> </span><span class="n">grid</span><span class="o">:</span><span class="w"> </span><span class="n">anytype</span><span class="p">)</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="kr">comptime</span><span class="w"> </span><span class="kr">var</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">grid</span><span class="p">.</span><span class="n">len</span><span class="p">;</span>
 <a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="kr">comptime</span><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="n">len</span><span class="p">;</span>
@@ -4494,7 +4494,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* 最小路径和：状态压缩后的动态规划 */</span>
+<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* 最小路径和：空间优化后的动态规划 */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">minPathSumDPComp</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><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">grid</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">].</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">  </span><span class="c1">// 初始化 dp 表</span>
@@ -4517,7 +4517,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* 最小路径和：状态压缩后的动态规划 */</span>
+<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* 最小路径和：空间优化后的动态规划 */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="k">fn</span> <span class="nf">min_path_sum_dp_comp</span><span class="p">(</span><span class="n">grid</span>: <span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="kt">i32</span> <span class="p">{</span>
 <a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">grid</span><span class="p">.</span><span class="n">len</span><span class="p">(),</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="n">len</span><span class="p">());</span>
 <a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>