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@@ -3693,7 +3693,7 @@ 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.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 最小路径和：暴力搜索 */</span>
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">gridCols</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[][</span><span class="n">gridCols</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="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</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="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="c1">// 若为左上角单元格，则终止搜索</span>
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</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="p">{</span>
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="k">return</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="mi">0</span><span class="p">];</span>
@@ -3703,10 +3703,10 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w"> </span><span class="c1">// 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价</span>
<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">gridCols</span><span class="p">,</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">gridCols</span><span class="p">,</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</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">1</span><span class="p">);</span>
<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</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">1</span><span class="p">);</span>
<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w"> </span><span class="c1">// 返回从左上角到 (i, j) 的最小路径代价</span>
<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</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">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</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">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="p">}</span>
</code></pre></div>
</div>
@@ -3987,7 +3987,7 @@ 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.c</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 最小路径和：记忆化搜索 */</span>
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">gridCols</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[][</span><span class="n">gridCols</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mem</span><span class="p">[][</span><span class="n">gridCols</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="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</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="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="c1">// 若为左上角单元格，则终止搜索</span>
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</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="p">{</span>
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w"> </span><span class="k">return</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="mi">0</span><span class="p">];</span>
@@ -4001,10 +4001,10 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w"> </span><span class="c1">// 左边和上边单元格的最小路径代价</span>
<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">gridCols</span><span class="p">,</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">gridCols</span><span class="p">,</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</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">1</span><span class="p">);</span>
<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</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">1</span><span class="p">);</span>
<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w"> </span><span class="c1">// 记录并返回左上角到 (i, j) 的最小路径代价</span>
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w"> </span><span class="n">mem</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="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</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">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w"> </span><span class="n">mem</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="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</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">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="p">}</span>
</code></pre></div>
@@ -4301,26 +4301,34 @@ 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.c</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* 最小路径和：动态规划 */</span>
<a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">gridCols</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[][</span><span class="n">gridCols</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w"> </span><span class="c1">// 初始化 dp 表</span>
<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w"> </span><span class="n">dp</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="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="mi">0</span><span class="p">];</span>
<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w"> </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><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">1</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">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</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">j</span><span class="p">];</span>
<a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w"> </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><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">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</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="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</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">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w"> </span><span class="c1">// 状态转移：其余行列</span>
<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><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">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</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="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><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">1</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">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w"> </span><span class="n">dp</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="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</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">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="p">}</span>
<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><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">i</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">i</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="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w"> </span><span class="n">dp</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="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w"> </span><span class="n">dp</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="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="mi">0</span><span class="p">];</span>
<a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w"> </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><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">1</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">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</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">j</span><span class="p">];</span>
<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w"> </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><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">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</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="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</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">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
<a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w"> </span><span class="c1">// 状态转移：其余行列</span>
<a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><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">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</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="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><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">1</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">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w"> </span><span class="n">dp</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="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</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">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-22" name="__codelineno-34-22" href="#__codelineno-34-22"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-34-24" name="__codelineno-34-24" href="#__codelineno-34-24"></a><span class="w"> </span><span class="c1">// 释放内存</span>
<a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a><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">i</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">i</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="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-34-26" name="__codelineno-34-26" href="#__codelineno-34-26"></a><span class="w"> </span><span class="n">free</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
<a id="__codelineno-34-27" name="__codelineno-34-27" href="#__codelineno-34-27"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-34-28" name="__codelineno-34-28" href="#__codelineno-34-28"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
<a id="__codelineno-34-29" name="__codelineno-34-29" href="#__codelineno-34-29"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -4640,9 +4648,9 @@ 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.c</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="nf">minPathSumDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">gridCols</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[][</span><span class="n">gridCols</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</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="nf">minPathSumDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</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="c1">// 初始化 dp 表</span>
<a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</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="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w"> </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</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="mi">0</span><span class="p">];</span>
<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><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">1</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">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4654,11 +4662,14 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
<a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</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">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
<a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w"> </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><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">1</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">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">])</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">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">])</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">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-46-17" name="__codelineno-46-17" href="#__codelineno-46-17"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-46-19" name="__codelineno-46-19" href="#__codelineno-46-19"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-46-20" name="__codelineno-46-20" href="#__codelineno-46-20"></a><span class="p">}</span>
<a id="__codelineno-46-19" name="__codelineno-46-19" href="#__codelineno-46-19"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-46-20" name="__codelineno-46-20" href="#__codelineno-46-20"></a><span class="w"> </span><span class="c1">// 释放内存</span>
<a id="__codelineno-46-21" name="__codelineno-46-21" href="#__codelineno-46-21"></a><span class="w"> </span><span class="n">free</span><span class="p">(</span><span class="n">dp</span><span class="p">);</span>
<a id="__codelineno-46-22" name="__codelineno-46-22" href="#__codelineno-46-22"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
<a id="__codelineno-46-23" name="__codelineno-46-23" href="#__codelineno-46-23"></a><span class="p">}</span>
</code></pre></div>
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