deploy

chapter_computational_complexity/time_complexity/index.html

@@ -5021,7 +5021,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
 <a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">count</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-69-5" name="__codelineno-69-5" href="#__codelineno-69-5"></a><span class="w">    </span><span class="kr">const</span><span class="w"> </span><span class="n">size</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100</span><span class="n">_000</span><span class="p">;</span>
 <a id="__codelineno-69-6" name="__codelineno-69-6" href="#__codelineno-69-6"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">i</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-69-7" name="__codelineno-69-7" href="#__codelineno-69-7"></a><span class="w">    </span><span class="k">while</span><span class="p">(</span><span class="n">i</span><span class="o">&lt;</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="o">:</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">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-69-7" name="__codelineno-69-7" href="#__codelineno-69-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="o">:</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">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-69-8" name="__codelineno-69-8" href="#__codelineno-69-8"></a><span class="w">        </span><span class="n">count</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-69-9" name="__codelineno-69-9" href="#__codelineno-69-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-69-10" name="__codelineno-69-10" href="#__codelineno-69-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
@@ -5856,7 +5856,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.zig</span><pre><span></span><code><a id="__codelineno-125-1" name="__codelineno-125-1" href="#__codelineno-125-1"></a><span class="c1">// 平方阶（冒泡排序）</span>
 <a id="__codelineno-125-2" name="__codelineno-125-2" href="#__codelineno-125-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">bubbleSort</span><span class="p">(</span><span class="n">nums</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kt">i32</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-125-3" name="__codelineno-125-3" href="#__codelineno-125-3"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">count</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</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="c1">// 计数器 </span>
+<a id="__codelineno-125-3" name="__codelineno-125-3" href="#__codelineno-125-3"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">count</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</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="c1">// 计数器</span>
 <a id="__codelineno-125-4" name="__codelineno-125-4" href="#__codelineno-125-4"></a><span class="w">    </span><span class="c1">// 外循环：未排序区间为 [0, i]</span>
 <a id="__codelineno-125-5" name="__codelineno-125-5" href="#__codelineno-125-5"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">i</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@as</span><span class="p">(</span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="nb">@intCast</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">))</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-125-6" name="__codelineno-125-6" href="#__codelineno-125-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">:</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">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5865,10 +5865,10 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
 <a id="__codelineno-125-9" name="__codelineno-125-9" href="#__codelineno-125-9"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </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="p">{</span>
 <a id="__codelineno-125-10" name="__codelineno-125-10" href="#__codelineno-125-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</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="p">{</span>
 <a id="__codelineno-125-11" name="__codelineno-125-11" href="#__codelineno-125-11"></a><span class="w">                </span><span class="c1">// 交换 nums[j] 与 nums[j + 1]</span>
-<a id="__codelineno-125-12" name="__codelineno-125-12" href="#__codelineno-125-12"></a><span class="w">                </span><span class="kr">var</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
+<a id="__codelineno-125-12" name="__codelineno-125-12" href="#__codelineno-125-12"></a><span class="w">                </span><span class="kr">const</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-125-13" name="__codelineno-125-13" href="#__codelineno-125-13"></a><span class="w">                </span><span class="n">nums</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">nums</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>
 <a id="__codelineno-125-14" name="__codelineno-125-14" href="#__codelineno-125-14"></a><span class="w">                </span><span class="n">nums</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">tmp</span><span class="p">;</span>
-<a id="__codelineno-125-15" name="__codelineno-125-15" href="#__codelineno-125-15"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w">  </span><span class="c1">// 元素交换包含 3 个单元操作</span>
+<a id="__codelineno-125-15" name="__codelineno-125-15" href="#__codelineno-125-15"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// 元素交换包含 3 个单元操作</span>
 <a id="__codelineno-125-16" name="__codelineno-125-16" href="#__codelineno-125-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-125-17" name="__codelineno-125-17" href="#__codelineno-125-17"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-125-18" name="__codelineno-125-18" href="#__codelineno-125-18"></a><span class="w">    </span><span class="p">}</span>
@@ -6430,14 +6430,12 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
 <div class="highlight"><span class="filename">time_complexity.zig</span><pre><span></span><code><a id="__codelineno-167-1" name="__codelineno-167-1" href="#__codelineno-167-1"></a><span class="c1">// 对数阶（循环实现）</span>
 <a id="__codelineno-167-2" name="__codelineno-167-2" href="#__codelineno-167-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">logarithmic</span><span class="p">(</span><span class="n">n</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</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-167-3" name="__codelineno-167-3" href="#__codelineno-167-3"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">count</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-167-4" name="__codelineno-167-4" href="#__codelineno-167-4"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">n_var</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
-<a id="__codelineno-167-5" name="__codelineno-167-5" href="#__codelineno-167-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n_var</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-167-6" name="__codelineno-167-6" href="#__codelineno-167-6"></a><span class="w">    </span><span class="p">{</span>
-<a id="__codelineno-167-7" name="__codelineno-167-7" href="#__codelineno-167-7"></a><span class="w">        </span><span class="n">n_var</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n_var</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
-<a id="__codelineno-167-8" name="__codelineno-167-8" href="#__codelineno-167-8"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-167-9" name="__codelineno-167-9" href="#__codelineno-167-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-167-10" name="__codelineno-167-10" href="#__codelineno-167-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
-<a id="__codelineno-167-11" name="__codelineno-167-11" href="#__codelineno-167-11"></a><span class="p">}</span>
+<a id="__codelineno-167-4" name="__codelineno-167-4" href="#__codelineno-167-4"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">n_var</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
+<a id="__codelineno-167-5" name="__codelineno-167-5" href="#__codelineno-167-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n_var</span><span class="w"> </span><span class="o">&gt;</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="p">(</span><span class="n">n_var</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@divTrunc</span><span class="p">(</span><span class="n">n_var</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-167-6" name="__codelineno-167-6" href="#__codelineno-167-6"></a><span class="w">        </span><span class="n">count</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-167-7" name="__codelineno-167-7" href="#__codelineno-167-7"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-167-8" name="__codelineno-167-8" href="#__codelineno-167-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
+<a id="__codelineno-167-9" name="__codelineno-167-9" href="#__codelineno-167-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 </div>
@@ -6571,7 +6569,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
 <div class="highlight"><span class="filename">time_complexity.zig</span><pre><span></span><code><a id="__codelineno-181-1" name="__codelineno-181-1" href="#__codelineno-181-1"></a><span class="c1">// 对数阶（递归实现）</span>
 <a id="__codelineno-181-2" name="__codelineno-181-2" href="#__codelineno-181-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">logRecur</span><span class="p">(</span><span class="n">n</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</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-181-3" name="__codelineno-181-3" href="#__codelineno-181-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-181-4" name="__codelineno-181-4" href="#__codelineno-181-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">logRecur</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">2</span><span class="p">)</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-181-4" name="__codelineno-181-4" href="#__codelineno-181-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">logRecur</span><span class="p">(</span><span class="nb">@divTrunc</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">))</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-181-5" name="__codelineno-181-5" href="#__codelineno-181-5"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -6766,7 +6764,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 <div class="highlight"><span class="filename">time_complexity.zig</span><pre><span></span><code><a id="__codelineno-195-1" name="__codelineno-195-1" href="#__codelineno-195-1"></a><span class="c1">// 线性对数阶</span>
 <a id="__codelineno-195-2" name="__codelineno-195-2" href="#__codelineno-195-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</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-195-3" name="__codelineno-195-3" href="#__codelineno-195-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-195-4" name="__codelineno-195-4" href="#__codelineno-195-4"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">count</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">linearLogRecur</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">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">linearLogRecur</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">2</span><span class="p">);</span>
+<a id="__codelineno-195-4" name="__codelineno-195-4" href="#__codelineno-195-4"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">count</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="nb">@divTrunc</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="nb">@divTrunc</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">));</span>
 <a id="__codelineno-195-5" name="__codelineno-195-5" href="#__codelineno-195-5"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">i</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-195-6" name="__codelineno-195-6" href="#__codelineno-195-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</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="o">:</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">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-195-7" name="__codelineno-195-7" href="#__codelineno-195-7"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>