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<h1 id="22-iteration-and-recursion">2.2 &nbsp; Iteration and recursion<a class="headerlink" href="#22-iteration-and-recursion" title="Permanent link">&para;</a></h1>
<p>In algorithms, the repeated execution of a task is quite common and is closely related to the analysis of complexity. Therefore, before delving into the concepts of time complexity and space complexity, let's first explore how to implement repetitive tasks in programming. This involves understanding two fundamental programming control structures: iteration and recursion.</p>
<h2 id="221-iteration">2.2.1 &nbsp; Iteration<a class="headerlink" href="#221-iteration" title="Permanent link">&para;</a></h2>
<p>"Iteration" is a control structure for repeatedly performing a task. In iteration, a program repeats a block of code as long as a certain condition is met until this condition is no longer satisfied.</p>
<p><u>Iteration</u> is a control structure for repeatedly performing a task. In iteration, a program repeats a block of code as long as a certain condition is met until this condition is no longer satisfied.</p>
<h3 id="1-for-loops">1. &nbsp; For loops<a class="headerlink" href="#1-for-loops" title="Permanent link">&para;</a></h3>
<p>The <code>for</code> loop is one of the most common forms of iteration, and <strong>it's particularly suitable when the number of iterations is known in advance</strong>.</p>
<p>The following function uses a <code>for</code> loop to perform a summation of <span class="arithmatex">\(1 + 2 + \dots + n\)</span>, with the sum being stored in the variable <code>res</code>. It's important to note that in Python, <code>range(a, b)</code> creates an interval that is inclusive of <code>a</code> but exclusive of <code>b</code>, meaning it iterates over the range from <span class="arithmatex">\(a\)</span> up to <span class="arithmatex">\(b−1\)</span>.</p>
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<p><a class="glightbox" href="../iteration_and_recursion.assets/iteration.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Flowchart of the sum function" class="animation-figure" src="../iteration_and_recursion.assets/iteration.png" /></a></p>
<p align="center"> Figure 2-1 &nbsp; Flowchart of the sum function </p>
<p>The number of operations in this summation function is proportional to the size of the input data <span class="arithmatex">\(n\)</span>, or in other words, it has a "linear relationship." This "linear relationship" is what time complexity describes. This topic will be discussed in more detail in the next section.</p>
<p>The number of operations in this summation function is proportional to the size of the input data <span class="arithmatex">\(n\)</span>, or in other words, it has a linear relationship. <strong>This "linear relationship" is what time complexity describes</strong>. This topic will be discussed in more detail in the next section.</p>
<h3 id="2-while-loops">2. &nbsp; While loops<a class="headerlink" href="#2-while-loops" title="Permanent link">&para;</a></h3>
<p>Similar to <code>for</code> loops, <code>while</code> loops are another approach for implementing iteration. In a <code>while</code> loop, the program checks a condition at the beginning of each iteration; if the condition is true, the execution continues, otherwise, the loop ends.</p>
<p>Below we use a <code>while</code> loop to implement the sum <span class="arithmatex">\(1 + 2 + \dots + n\)</span>.</p>
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<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20while_loop%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22while%20%E5%BE%AA%E7%8E%AF%22%22%22%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20i%20%3D%201%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%9D%A1%E4%BB%B6%E5%8F%98%E9%87%8F%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%E6%B1%82%E5%92%8C%201,%202,%20...,%20n-1,%20n%0A%20%20%20%20while%20i%20%3C%3D%20n%3A%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20i%0A%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20%E6%9B%B4%E6%96%B0%E6%9D%A1%E4%BB%B6%E5%8F%98%E9%87%8F%0A%20%20%20%20return%20res%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%20res%20%3D%20while_loop%28n%29%0A%20%20%20%20print%28f%22%5Cnwhile%20%E5%BE%AA%E7%8E%AF%E7%9A%84%E6%B1%82%E5%92%8C%E7%BB%93%E6%9E%9C%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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<p><strong><code>While</code> loops provide more flexibility than <code>for</code> loops</strong>, especially since they allow for custom initialization and modification of the condition variable at each step.</p>
<p><strong><code>while</code> loops provide more flexibility than <code>for</code> loops</strong>, especially since they allow for custom initialization and modification of the condition variable at each step.</p>
<p>For example, in the following code, the condition variable <span class="arithmatex">\(i\)</span> is updated twice each round, which would be inconvenient to implement with a <code>for</code> loop.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="3:14"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><input id="__tabbed_3_14" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label><label for="__tabbed_3_14">Zig</label></div>
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<p>In such cases, the number of operations of the function is proportional to <span class="arithmatex">\(n^2\)</span>, meaning the algorithm's runtime and the size of the input data <span class="arithmatex">\(n\)</span> has a 'quadratic relationship.'</p>
<p>We can further increase the complexity by adding more nested loops, each level of nesting effectively "increasing the dimension," which raises the time complexity to "cubic," "quartic," and so on.</p>
<h2 id="222-recursion">2.2.2 &nbsp; Recursion<a class="headerlink" href="#222-recursion" title="Permanent link">&para;</a></h2>
<p>"Recursion" is an algorithmic strategy where a function solves a problem by calling itself. It primarily involves two phases:</p>
<p><u>Recursion</u> is an algorithmic strategy where a function solves a problem by calling itself. It primarily involves two phases:</p>
<ol>
<li><strong>Calling</strong>: This is where the program repeatedly calls itself, often with progressively smaller or simpler arguments, moving towards the "termination condition."</li>
<li><strong>Returning</strong>: Upon triggering the "termination condition," the program begins to return from the deepest recursive function, aggregating the results of each layer.</li>
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<p>In practice, the depth of recursion allowed by programming languages is usually limited, and excessively deep recursion can lead to stack overflow errors.</p>
<h3 id="2-tail-recursion">2. &nbsp; Tail recursion<a class="headerlink" href="#2-tail-recursion" title="Permanent link">&para;</a></h3>
<p>Interestingly, <strong>if a function performs its recursive call as the very last step before returning,</strong> it can be optimized by the compiler or interpreter to be as space-efficient as iteration. This scenario is known as "tail recursion."</p>
<p>Interestingly, <strong>if a function performs its recursive call as the very last step before returning,</strong> it can be optimized by the compiler or interpreter to be as space-efficient as iteration. This scenario is known as <u>tail recursion</u>.</p>
<ul>
<li><strong>Regular recursion</strong>: In standard recursion, when the function returns to the previous level, it continues to execute more code, requiring the system to save the context of the previous call.</li>
<li><strong>Tail recursion</strong>: Here, the recursive call is the final operation before the function returns. This means that upon returning to the previous level, no further actions are needed, so the system does not need to save the context of the previous level.</li>
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<p>Observing the above code, we see that it recursively calls two functions within itself, <strong>meaning that one call generates two branching calls</strong>. As illustrated below, this continuous recursive calling eventually creates a "recursion tree" with a depth of <span class="arithmatex">\(n\)</span>.</p>
<p>Observing the above code, we see that it recursively calls two functions within itself, <strong>meaning that one call generates two branching calls</strong>. As illustrated below, this continuous recursive calling eventually creates a <u>recursion tree</u> with a depth of <span class="arithmatex">\(n\)</span>.</p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Fibonacci sequence recursion tree" class="animation-figure" src="../iteration_and_recursion.assets/recursion_tree.png" /></a></p>
<p align="center"> Figure 2-6 &nbsp; Fibonacci sequence recursion tree </p>
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