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@@ -4421,7 +4421,7 @@ T(n) = 3 + 2n
<p class="admonition-title">Asymptotic Upper Bound</p>
<p>If there exist positive real numbers <span class="arithmatex">\(c\)</span> and <span class="arithmatex">\(n_0\)</span> such that for all <span class="arithmatex">\(n &gt; n_0\)</span>, <span class="arithmatex">\(T(n) \leq c \cdot f(n)\)</span>, then <span class="arithmatex">\(f(n)\)</span> is considered an asymptotic upper bound of <span class="arithmatex">\(T(n)\)</span>, denoted as <span class="arithmatex">\(T(n) = O(f(n))\)</span>.</p>
</div>
<p>As illustrated below, calculating the asymptotic upper bound involves finding a function <span class="arithmatex">\(f(n)\)</span> such that, as <span class="arithmatex">\(n\)</span> approaches infinity, <span class="arithmatex">\(T(n)\)</span> and <span class="arithmatex">\(f(n)\)</span> have the same growth order, differing only by a constant factor <span class="arithmatex">\(c\)</span>.</p>
<p>As shown in Figure 2-8, calculating the asymptotic upper bound involves finding a function <span class="arithmatex">\(f(n)\)</span> such that, as <span class="arithmatex">\(n\)</span> approaches infinity, <span class="arithmatex">\(T(n)\)</span> and <span class="arithmatex">\(f(n)\)</span> have the same growth order, differing only by a constant factor <span class="arithmatex">\(c\)</span>.</p>
<p><a class="glightbox" href="../time_complexity.assets/asymptotic_upper_bound.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Asymptotic upper bound of a function" class="animation-figure" src="../time_complexity.assets/asymptotic_upper_bound.png" /></a></p>
<p align="center"> Figure 2-8 &nbsp; Asymptotic upper bound of a function </p>
@@ -4694,7 +4694,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</table>
</div>
<h2 id="234-common-types-of-time-complexity">2.3.4 &nbsp; Common types of time complexity<a class="headerlink" href="#234-common-types-of-time-complexity" title="Permanent link">&para;</a></h2>
<p>Let's consider the input data size as <span class="arithmatex">\(n\)</span>. The common types of time complexities are illustrated below, arranged from lowest to highest:</p>
<p>Let's consider the input data size as <span class="arithmatex">\(n\)</span>. The common types of time complexities are shown in Figure 2-9, arranged from lowest to highest:</p>
<div class="arithmatex">\[
\begin{aligned}
&amp; 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!) \newline
@@ -4719,7 +4719,14 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">constant</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Constant complexity */</span>
<a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</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="p">{</span>
<a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100000</span><span class="p">;</span>
<a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></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">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
<a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
<a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></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-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -4793,7 +4800,13 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">linear</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Linear complexity */</span>
<a id="__codelineno-67-2" name="__codelineno-67-2" href="#__codelineno-67-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</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="p">{</span>
<a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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-67-4" name="__codelineno-67-4" href="#__codelineno-67-4"></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>
<a id="__codelineno-67-5" name="__codelineno-67-5" href="#__codelineno-67-5"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
<a id="__codelineno-67-6" name="__codelineno-67-6" href="#__codelineno-67-6"></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-67-7" name="__codelineno-67-7" href="#__codelineno-67-7"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -4866,7 +4879,15 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">arrayTraversal</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Linear complexity (traversing an array) */</span>
<a id="__codelineno-81-2" name="__codelineno-81-2" href="#__codelineno-81-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">arrayTraversal</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w"> </span><span class="c1">// Loop count is proportional to the length of the array</span>
<a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></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">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
<a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></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-81-9" name="__codelineno-81-9" href="#__codelineno-81-9"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -4944,7 +4965,17 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">quadratic</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a><span class="cm">/* Quadratic complexity */</span>
<a id="__codelineno-95-2" name="__codelineno-95-2" href="#__codelineno-95-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</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="p">{</span>
<a id="__codelineno-95-3" name="__codelineno-95-3" href="#__codelineno-95-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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-95-4" name="__codelineno-95-4" href="#__codelineno-95-4"></a><span class="w"> </span><span class="c1">// Loop count is squared in relation to the data size n</span>
<a id="__codelineno-95-5" name="__codelineno-95-5" href="#__codelineno-95-5"></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-95-6" name="__codelineno-95-6" href="#__codelineno-95-6"></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">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-95-7" name="__codelineno-95-7" href="#__codelineno-95-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
<a id="__codelineno-95-8" name="__codelineno-95-8" href="#__codelineno-95-8"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-95-9" name="__codelineno-95-9" href="#__codelineno-95-9"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-95-10" name="__codelineno-95-10" href="#__codelineno-95-10"></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-95-11" name="__codelineno-95-11" href="#__codelineno-95-11"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -5032,7 +5063,24 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-109-1" name="__codelineno-109-1" href="#__codelineno-109-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">bubbleSort</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-109-1" name="__codelineno-109-1" href="#__codelineno-109-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
<a id="__codelineno-109-2" name="__codelineno-109-2" href="#__codelineno-109-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-109-3" name="__codelineno-109-3" href="#__codelineno-109-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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">// Counter</span>
<a id="__codelineno-109-4" name="__codelineno-109-4" href="#__codelineno-109-4"></a><span class="w"> </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
<a id="__codelineno-109-5" name="__codelineno-109-5" href="#__codelineno-109-5"></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="n">nums</span><span class="p">.</span><span class="n">size</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><span class="w"> </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="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-109-6" name="__codelineno-109-6" href="#__codelineno-109-6"></a><span class="w"> </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the right end of the range</span>
<a id="__codelineno-109-7" name="__codelineno-109-7" href="#__codelineno-109-7"></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">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</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-109-8" name="__codelineno-109-8" href="#__codelineno-109-8"></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-109-9" name="__codelineno-109-9" href="#__codelineno-109-9"></a><span class="w"> </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
<a id="__codelineno-109-10" name="__codelineno-109-10" href="#__codelineno-109-10"></a><span class="w"> </span><span class="kt">int</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-109-11" name="__codelineno-109-11" href="#__codelineno-109-11"></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-109-12" name="__codelineno-109-12" href="#__codelineno-109-12"></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-109-13" name="__codelineno-109-13" href="#__codelineno-109-13"></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">// Element swap includes 3 individual operations</span>
<a id="__codelineno-109-14" name="__codelineno-109-14" href="#__codelineno-109-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-109-15" name="__codelineno-109-15" href="#__codelineno-109-15"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-109-16" name="__codelineno-109-16" href="#__codelineno-109-16"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-109-17" name="__codelineno-109-17" href="#__codelineno-109-17"></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-109-18" name="__codelineno-109-18" href="#__codelineno-109-18"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -5122,7 +5170,19 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-123-1" name="__codelineno-123-1" href="#__codelineno-123-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">exponential</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-123-1" name="__codelineno-123-1" href="#__codelineno-123-1"></a><span class="cm">/* Exponential complexity (loop implementation) */</span>
<a id="__codelineno-123-2" name="__codelineno-123-2" href="#__codelineno-123-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</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="p">{</span>
<a id="__codelineno-123-3" name="__codelineno-123-3" href="#__codelineno-123-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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">base</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-123-4" name="__codelineno-123-4" href="#__codelineno-123-4"></a><span class="w"> </span><span class="c1">// Cells split into two every round, forming the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
<a id="__codelineno-123-5" name="__codelineno-123-5" href="#__codelineno-123-5"></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-123-6" name="__codelineno-123-6" href="#__codelineno-123-6"></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">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">base</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-123-7" name="__codelineno-123-7" href="#__codelineno-123-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
<a id="__codelineno-123-8" name="__codelineno-123-8" href="#__codelineno-123-8"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-123-9" name="__codelineno-123-9" href="#__codelineno-123-9"></a><span class="w"> </span><span class="n">base</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-123-10" name="__codelineno-123-10" href="#__codelineno-123-10"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-123-11" name="__codelineno-123-11" href="#__codelineno-123-11"></a><span class="w"> </span><span class="c1">// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1</span>
<a id="__codelineno-123-12" name="__codelineno-123-12" href="#__codelineno-123-12"></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-123-13" name="__codelineno-123-13" href="#__codelineno-123-13"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -5202,7 +5262,12 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-137-1" name="__codelineno-137-1" href="#__codelineno-137-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">expRecur</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-137-1" name="__codelineno-137-1" href="#__codelineno-137-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
<a id="__codelineno-137-2" name="__codelineno-137-2" href="#__codelineno-137-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">expRecur</span><span class="p">(</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="p">{</span>
<a id="__codelineno-137-3" name="__codelineno-137-3" href="#__codelineno-137-3"></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">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-137-4" name="__codelineno-137-4" href="#__codelineno-137-4"></a><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-137-5" name="__codelineno-137-5" href="#__codelineno-137-5"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">expRecur</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="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">expRecur</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="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-137-6" name="__codelineno-137-6" href="#__codelineno-137-6"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -5277,7 +5342,15 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-151-1" name="__codelineno-151-1" href="#__codelineno-151-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">logarithmic</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-151-1" name="__codelineno-151-1" href="#__codelineno-151-1"></a><span class="cm">/* Logarithmic complexity (loop implementation) */</span>
<a id="__codelineno-151-2" name="__codelineno-151-2" href="#__codelineno-151-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</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="p">{</span>
<a id="__codelineno-151-3" name="__codelineno-151-3" href="#__codelineno-151-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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-151-4" name="__codelineno-151-4" href="#__codelineno-151-4"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</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="p">{</span>
<a id="__codelineno-151-5" name="__codelineno-151-5" href="#__codelineno-151-5"></a><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </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-151-6" name="__codelineno-151-6" href="#__codelineno-151-6"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
<a id="__codelineno-151-7" name="__codelineno-151-7" href="#__codelineno-151-7"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-151-8" name="__codelineno-151-8" href="#__codelineno-151-8"></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-151-9" name="__codelineno-151-9" href="#__codelineno-151-9"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -5353,7 +5426,12 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified Count (o.O)}
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-165-1" name="__codelineno-165-1" href="#__codelineno-165-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">logRecur</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-165-1" name="__codelineno-165-1" href="#__codelineno-165-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
<a id="__codelineno-165-2" name="__codelineno-165-2" href="#__codelineno-165-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logRecur</span><span class="p">(</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="p">{</span>
<a id="__codelineno-165-3" name="__codelineno-165-3" href="#__codelineno-165-3"></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>
<a id="__codelineno-165-4" name="__codelineno-165-4" href="#__codelineno-165-4"></a><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-165-5" name="__codelineno-165-5" href="#__codelineno-165-5"></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-165-6" name="__codelineno-165-6" href="#__codelineno-165-6"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -5436,7 +5514,16 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-179-1" name="__codelineno-179-1" href="#__codelineno-179-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">linearLogRecur</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-179-1" name="__codelineno-179-1" href="#__codelineno-179-1"></a><span class="cm">/* Linear logarithmic complexity */</span>
<a id="__codelineno-179-2" name="__codelineno-179-2" href="#__codelineno-179-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linearLogRecur</span><span class="p">(</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="p">{</span>
<a id="__codelineno-179-3" name="__codelineno-179-3" href="#__codelineno-179-3"></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>
<a id="__codelineno-179-4" name="__codelineno-179-4" href="#__codelineno-179-4"></a><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-179-5" name="__codelineno-179-5" href="#__codelineno-179-5"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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-179-6" name="__codelineno-179-6" href="#__codelineno-179-6"></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-179-7" name="__codelineno-179-7" href="#__codelineno-179-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
<a id="__codelineno-179-8" name="__codelineno-179-8" href="#__codelineno-179-8"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-179-9" name="__codelineno-179-9" href="#__codelineno-179-9"></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-179-10" name="__codelineno-179-10" href="#__codelineno-179-10"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -5524,7 +5611,17 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-193-1" name="__codelineno-193-1" href="#__codelineno-193-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">factorialRecur</span><span class="p">}</span>
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-193-1" name="__codelineno-193-1" href="#__codelineno-193-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
<a id="__codelineno-193-2" name="__codelineno-193-2" href="#__codelineno-193-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">factorialRecur</span><span class="p">(</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="p">{</span>
<a id="__codelineno-193-3" name="__codelineno-193-3" href="#__codelineno-193-3"></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">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
<a id="__codelineno-193-4" name="__codelineno-193-4" href="#__codelineno-193-4"></a><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-193-5" name="__codelineno-193-5" href="#__codelineno-193-5"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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-193-6" name="__codelineno-193-6" href="#__codelineno-193-6"></a><span class="w"> </span><span class="c1">// From 1 split into n</span>
<a id="__codelineno-193-7" name="__codelineno-193-7" href="#__codelineno-193-7"></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">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-193-8" name="__codelineno-193-8" href="#__codelineno-193-8"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorialRecur</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>
<a id="__codelineno-193-9" name="__codelineno-193-9" href="#__codelineno-193-9"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-193-10" name="__codelineno-193-10" href="#__codelineno-193-10"></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-193-11" name="__codelineno-193-11" href="#__codelineno-193-11"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@@ -5620,9 +5717,30 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
</code></pre></div>
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<div class="tabbed-block">
<div class="highlight"><span class="filename">worst_best_time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-207-1" name="__codelineno-207-1" href="#__codelineno-207-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">randomNumbers</span><span class="p">}</span>
<a id="__codelineno-207-2" name="__codelineno-207-2" href="#__codelineno-207-2"></a>
<a id="__codelineno-207-3" name="__codelineno-207-3" href="#__codelineno-207-3"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">findOne</span><span class="p">}</span>
<div class="highlight"><span class="filename">worst_best_time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-207-1" name="__codelineno-207-1" href="#__codelineno-207-1"></a><span class="cm">/* Generate an array with elements {1, 2, ..., n} in a randomly shuffled order */</span>
<a id="__codelineno-207-2" name="__codelineno-207-2" href="#__codelineno-207-2"></a><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">randomNumbers</span><span class="p">(</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="p">{</span>
<a id="__codelineno-207-3" name="__codelineno-207-3" href="#__codelineno-207-3"></a><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
<a id="__codelineno-207-4" name="__codelineno-207-4" href="#__codelineno-207-4"></a><span class="w"> </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
<a id="__codelineno-207-5" name="__codelineno-207-5" href="#__codelineno-207-5"></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-207-6" name="__codelineno-207-6" href="#__codelineno-207-6"></a><span class="w"> </span><span class="n">nums</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">i</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-207-7" name="__codelineno-207-7" href="#__codelineno-207-7"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-207-8" name="__codelineno-207-8" href="#__codelineno-207-8"></a><span class="w"> </span><span class="c1">// Generate a random seed using system time</span>
<a id="__codelineno-207-9" name="__codelineno-207-9" href="#__codelineno-207-9"></a><span class="w"> </span><span class="kt">unsigned</span><span class="w"> </span><span class="n">seed</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">chrono</span><span class="o">::</span><span class="n">system_clock</span><span class="o">::</span><span class="n">now</span><span class="p">().</span><span class="n">time_since_epoch</span><span class="p">().</span><span class="n">count</span><span class="p">();</span>
<a id="__codelineno-207-10" name="__codelineno-207-10" href="#__codelineno-207-10"></a><span class="w"> </span><span class="c1">// Randomly shuffle array elements</span>
<a id="__codelineno-207-11" name="__codelineno-207-11" href="#__codelineno-207-11"></a><span class="w"> </span><span class="n">shuffle</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">(),</span><span class="w"> </span><span class="n">default_random_engine</span><span class="p">(</span><span class="n">seed</span><span class="p">));</span>
<a id="__codelineno-207-12" name="__codelineno-207-12" href="#__codelineno-207-12"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
<a id="__codelineno-207-13" name="__codelineno-207-13" href="#__codelineno-207-13"></a><span class="p">}</span>
<a id="__codelineno-207-14" name="__codelineno-207-14" href="#__codelineno-207-14"></a>
<a id="__codelineno-207-15" name="__codelineno-207-15" href="#__codelineno-207-15"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
<a id="__codelineno-207-16" name="__codelineno-207-16" href="#__codelineno-207-16"></a><span class="kt">int</span><span class="w"> </span><span class="n">findOne</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-207-17" name="__codelineno-207-17" href="#__codelineno-207-17"></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">nums</span><span class="p">.</span><span class="n">size</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-207-18" name="__codelineno-207-18" href="#__codelineno-207-18"></a><span class="w"> </span><span class="c1">// When element 1 is at the start of the array, achieve best time complexity O(1)</span>
<a id="__codelineno-207-19" name="__codelineno-207-19" href="#__codelineno-207-19"></a><span class="w"> </span><span class="c1">// When element 1 is at the end of the array, achieve worst time complexity O(n)</span>
<a id="__codelineno-207-20" name="__codelineno-207-20" href="#__codelineno-207-20"></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">i</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-207-21" name="__codelineno-207-21" href="#__codelineno-207-21"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-207-22" name="__codelineno-207-22" href="#__codelineno-207-22"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-207-23" name="__codelineno-207-23" href="#__codelineno-207-23"></a><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-207-24" name="__codelineno-207-24" href="#__codelineno-207-24"></a><span class="p">}</span>
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