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en/chapter_computational_complexity/space_complexity/index.html

@@ -3744,7 +3744,7 @@
 <li><strong>Stack frame space</strong>: Used to save the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns.</li>
 <li><strong>Instruction space</strong>: Used to store compiled program instructions, which are usually negligible in actual statistics.</li>
 </ul>
-<p>When analyzing the space complexity of a program, <strong>we typically count the Temporary Data, Stack Frame Space, and Output Data</strong>, as shown in the Figure 2-15 .</p>
+<p>When analyzing the space complexity of a program, <strong>we typically count the Temporary Data, Stack Frame Space, and Output Data</strong>, as shown in Figure 2-15.</p>
 <p><a class="glightbox" href="../space_complexity.assets/space_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Space types used in algorithms" class="animation-figure" src="../space_complexity.assets/space_types.png" /></a></p>
 <p align="center"> Figure 2-15 &nbsp; Space types used in algorithms </p>
 
@@ -5036,7 +5036,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
 <p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20linear%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E7%BA%BF%E6%80%A7%E9%98%B6%22%22%22%0A%20%20%20%20%23%20%E9%95%BF%E5%BA%A6%E4%B8%BA%20n%20%E7%9A%84%E5%88%97%E8%A1%A8%E5%8D%A0%E7%94%A8%20O%28n%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20nums%20%3D%20%5B0%5D%20*%20n%0A%20%20%20%20%23%20%E9%95%BF%E5%BA%A6%E4%B8%BA%20n%20%E7%9A%84%E5%93%88%E5%B8%8C%E8%A1%A8%E5%8D%A0%E7%94%A8%20O%28n%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20hmap%20%3D%20dict%5Bint,%20str%5D%28%29%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20hmap%5Bi%5D%20%3D%20str%28i%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E7%BA%BF%E6%80%A7%E9%98%B6%0A%20%20%20%20linear%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
 <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20linear%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E7%BA%BF%E6%80%A7%E9%98%B6%22%22%22%0A%20%20%20%20%23%20%E9%95%BF%E5%BA%A6%E4%B8%BA%20n%20%E7%9A%84%E5%88%97%E8%A1%A8%E5%8D%A0%E7%94%A8%20O%28n%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20nums%20%3D%20%5B0%5D%20*%20n%0A%20%20%20%20%23%20%E9%95%BF%E5%BA%A6%E4%B8%BA%20n%20%E7%9A%84%E5%93%88%E5%B8%8C%E8%A1%A8%E5%8D%A0%E7%94%A8%20O%28n%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20hmap%20%3D%20dict%5Bint,%20str%5D%28%29%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20hmap%5Bi%5D%20%3D%20str%28i%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E7%BA%BF%E6%80%A7%E9%98%B6%0A%20%20%20%20linear%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
 </details>
-<p>As shown below, this function's recursive depth is <span class="arithmatex">\(n\)</span>, meaning there are <span class="arithmatex">\(n\)</span> instances of unreturned <code>linear_recur()</code> function, using <span class="arithmatex">\(O(n)\)</span> size of stack frame space:</p>
+<p>As shown in Figure 2-17, this function's recursive depth is <span class="arithmatex">\(n\)</span>, meaning there are <span class="arithmatex">\(n\)</span> instances of unreturned <code>linear_recur()</code> function, using <span class="arithmatex">\(O(n)\)</span> size of stack frame space:</p>
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@@ -5409,7 +5409,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
 <p><div style="height: 405px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20quadratic%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E5%B9%B3%E6%96%B9%E9%98%B6%22%22%22%0A%20%20%20%20%23%20%E4%BA%8C%E7%BB%B4%E5%88%97%E8%A1%A8%E5%8D%A0%E7%94%A8%20O%28n%5E2%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20num_matrix%20%3D%20%5B%5B0%5D%20*%20n%20for%20_%20in%20range%28n%29%5D%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E5%B9%B3%E6%96%B9%E9%98%B6%0A%20%20%20%20quadratic%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
 <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20quadratic%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22%E5%B9%B3%E6%96%B9%E9%98%B6%22%22%22%0A%20%20%20%20%23%20%E4%BA%8C%E7%BB%B4%E5%88%97%E8%A1%A8%E5%8D%A0%E7%94%A8%20O%28n%5E2%29%20%E7%A9%BA%E9%97%B4%0A%20%20%20%20num_matrix%20%3D%20%5B%5B0%5D%20*%20n%20for%20_%20in%20range%28n%29%5D%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20%E5%B9%B3%E6%96%B9%E9%98%B6%0A%20%20%20%20quadratic%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
 </details>
-<p>As shown below, the recursive depth of this function is <span class="arithmatex">\(n\)</span>, and in each recursive call, an array is initialized with lengths <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(n-1\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span>, averaging <span class="arithmatex">\(n/2\)</span>, thus overall occupying <span class="arithmatex">\(O(n^2)\)</span> space:</p>
+<p>As shown in Figure 2-18, the recursive depth of this function is <span class="arithmatex">\(n\)</span>, and in each recursive call, an array is initialized with lengths <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(n-1\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span>, averaging <span class="arithmatex">\(n/2\)</span>, thus overall occupying <span class="arithmatex">\(O(n^2)\)</span> space:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="8:14"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><input id="__tabbed_8_9" name="__tabbed_8" type="radio" /><input id="__tabbed_8_10" name="__tabbed_8" type="radio" /><input id="__tabbed_8_11" name="__tabbed_8" type="radio" /><input id="__tabbed_8_12" name="__tabbed_8" type="radio" /><input id="__tabbed_8_13" name="__tabbed_8" type="radio" /><input id="__tabbed_8_14" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Python</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Java</label><label for="__tabbed_8_4">C#</label><label for="__tabbed_8_5">Go</label><label for="__tabbed_8_6">Swift</label><label for="__tabbed_8_7">JS</label><label for="__tabbed_8_8">TS</label><label for="__tabbed_8_9">Dart</label><label for="__tabbed_8_10">Rust</label><label for="__tabbed_8_11">C</label><label for="__tabbed_8_12">Kotlin</label><label for="__tabbed_8_13">Ruby</label><label for="__tabbed_8_14">Zig</label></div>
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@@ -5581,7 +5581,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
 <p align="center"> Figure 2-18 &nbsp; Recursive function generating quadratic order space complexity </p>
 
 <h3 id="4-exponential-order-o2n">4. &nbsp; Exponential order <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-exponential-order-o2n" title="Permanent link">&para;</a></h3>
-<p>Exponential order is common in binary trees. Observe the below image, a "full binary tree" with <span class="arithmatex">\(n\)</span> levels has <span class="arithmatex">\(2^n - 1\)</span> nodes, occupying <span class="arithmatex">\(O(2^n)\)</span> space:</p>
+<p>Exponential order is common in binary trees. Observe Figure 2-19, a "full binary tree" with <span class="arithmatex">\(n\)</span> levels has <span class="arithmatex">\(2^n - 1\)</span> nodes, occupying <span class="arithmatex">\(O(2^n)\)</span> space:</p>
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