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elements from <code>b</code>.</p><p>Time complexity: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(m + n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span>, where <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo separator="true">,</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m, n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">n</span></span></span></span></span> denote the length of
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<code>a</code> and <code>b</code> respectively.</p><p>Space complexity: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(m + n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span>, where <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo separator="true">,</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m, n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">n</span></span></span></span></span> denote the length of
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<code>a</code> and <code>b</code> respectively, since we construct new list.</p></li></ul><h2 class="anchor anchorWithStickyNavbar_LWe7" id="example-1">Example #1<a href="#example-1" class="hash-link" aria-label="Direct link to Example #1" title="Direct link to Example #1"></a></h2><p>Let us assume function that uses divide & conquer strategy to return indices at which we can find specific element in any list.</p><div class="language-py codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-py codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv codeBlockLinesWithNumbering_o6Pm"><span class="token-line codeLine_lJS_" style="color:#000000"><span class="codeLineNumber_Tfdd"></span><span class="codeLineContent_feaV"><span class="token keyword" style="color:rgb(0, 0, 255)">def</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">recursive_find_in_list</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token plain"></span></span><br></span><span class="token-line codeLine_lJS_" style="color:#000000"><span class="codeLineNumber_Tfdd"></span><span class="codeLineContent_feaV"><span class="token plain"> values</span><span class="token punctuation" style="color:rgb(4, 81, 165)">:</span><span class="token plain"> List</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token plain">Any</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> key</span><span class="token punctuation" style="color:rgb(4, 81, 165)">:</span><span class="token plain"> Any</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> lower</span><span class="token punctuation" style="color:rgb(4, 81, 165)">:</span><span class="token plain"> </span><span class="token builtin" style="color:rgb(0, 112, 193)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> upper</span><span class="token punctuation" style="color:rgb(4, 81, 165)">:</span><span class="token plain"> </span><span class="token builtin" style="color:rgb(0, 112, 193)">int</span><span class="token plain"></span></span><br></span><span class="token-line codeLine_lJS_" style="color:#000000"><span class="codeLineNumber_Tfdd"></span><span class="codeLineContent_feaV"><span class="token plain"></span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(0, 0, 0)">-</span><span class="token operator" style="color:rgb(0, 0, 0)">></span><span class="token plain"> List</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token builtin" style="color:rgb(0, 112, 193)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">:</span><span class="token plain"></span></span><br></span><span class="token-line codeLine_lJS_" style="color:#000000"><span class="codeLineNumber_Tfdd"></span><span class="codeLineContent_feaV"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">if</span><span class="token plain"> lower </span><span class="token operator" style="color:rgb(0, 0, 0)">==</span><span class="token plain"> upper</span><span class="token punctuation" style="color:rgb(4, 81, 165)">:</span><span class="token plain"></span></span><br></span><span class="token-line codeLine_lJS_" style="color:#000000"><span class="codeLineNumber_Tfdd"></span><span class="codeLineContent_feaV"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">return</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">
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<img loading="lazy" alt="Rendered construction of the list" src="/assets/images/construction_dark-fcb91acb746c787329bf9bd3cb54c5b7.png#gh-dark-mode-only" width="851" height="276" class="img_ev3q"></p><p>Let us assume that you extend the result with the list that you get from the recursive call.</p><ul><li><p>B iterates through 1, 2 and 3; returns <code>[1, 2, 3]</code></p></li><li><p>C iterates through 4, 5 and 6; returns <code>[4, 5, 6]</code></p></li><li><p>D iterates through 7, 8 and 9; returns <code>[7, 8, 9]</code></p></li><li><p>now we return those lists to the calls from A), so each of the <code>extend</code> calls iterates through:</p><ul><li>1, 2, 3 that was returned from B</li><li>4, 5, 6 that was returned from C</li><li>7, 8, 9 that was returned from D</li></ul><p>and returns <code>[1, 2, 3, 4, 5, 6, 7, 8, 9]</code></p></li></ul><p>If the recursion had bigger depth and/or more elements, it would iterate through them more than twice, therefore it does not take constant time to do nor some constant multiple of the input, since it traverses all of the elements in each of the levels.</p><h2 class="anchor anchorWithStickyNavbar_LWe7" id="implementation-of-extend">Implementation of <code>extend</code><a href="#implementation-of-extend" class="hash-link" aria-label="Direct link to implementation-of-extend" title="Direct link to implementation-of-extend"></a></h2><p>There is an example of dynamic array:</p><ul><li><a href="/files/ib002/time-complexity/extend/dynlist.h" target="_blank" rel="noopener noreferrer">interface (<code>dynlist.h</code>)</a></li><li><a href="/files/ib002/time-complexity/extend/dynlist.c" target="_blank" rel="noopener noreferrer">implementation (<code>dynlist.c</code>)</a></li></ul><p>For the sake of <em>Algorithms and Data Structures I</em> we consider <code>APPEND</code> operation, i.e. adding the element to the end of the list, to have time complexity <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span> (<strong>amortized</strong>; which is out of the scope of IB002).</p><p>If we have a look at the <code>extend</code> implementation in this dynamic array example:</p><div class="language-c codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-c codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv codeBlockLinesWithNumbering_o6Pm"><span class="token-line codeLine_lJS_" style="color:#000000"><span class="codeLineNumber_Tfdd"></span><span class="codeLineContent_feaV"><span class="token keyword" style="color:rgb(0, 0, 255)">void</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">dynamic_array_extend</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token keyword" style="color:rgb(0, 0, 255)">struct</span><span class="token plain"> </span><span class="token class-name" style="color:rgb(38, 127, 153)">dynamic_array_t</span><span class="token plain"> </span><span class="token operator" style="color:rgb(0, 0, 0)">*</span><span class="token plain">arr</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">struct</span><span class="token plain"> </span><span class="token class-name" style="color:rgb(38, 127, 153)">dynamic_array_t</span><span class="token plain"> </span><span clas
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