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<!-- Page content -->
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<h1 id="11-algorithms-are-everywhere">1.1 Algorithms are everywhere<a class="headerlink" href="#11-algorithms-are-everywhere" title="Permanent link">¶</a></h1>
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<p>When we hear the word "algorithm," we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.</p>
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<p>Before formally discussing algorithms, there's an interesting fact worth sharing: <strong>you have already unconsciously learned many algorithms and have become accustomed to applying them in your daily life</strong>. Here, I will give a few specific examples to prove this point.</p>
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<p><strong>Example 1: Looking Up a Dictionary</strong>. In an English dictionary, words are listed alphabetically. Suppose we're searching for a word that starts with the letter <span class="arithmatex">\(r\)</span>. This is typically done in the following way:</p>
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<p>When we hear the term "algorithm," we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.</p>
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<p>Before we start discussing about algorithms officially, there's an interesting fact worth sharing: <strong>you've learned many algorithms unconsciously and are used to applying them in your daily life</strong>. Here, I will give a few specific examples to prove this point.</p>
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<p><strong>Example 1: Looking Up a Dictionary</strong>. In an English dictionary, words are listed alphabetically. Assuming we're searching for a word that starts with the letter <span class="arithmatex">\(r\)</span>, this is typically done in the following way:</p>
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<ol>
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<li>Open the dictionary to about halfway and check the first letter on the page, let's say the letter is <span class="arithmatex">\(m\)</span>.</li>
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<li>Since <span class="arithmatex">\(r\)</span> comes after <span class="arithmatex">\(m\)</span> in the alphabet, we can ignore the first half of the dictionary and focus on the latter half.</li>
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<li>Open the dictionary to about halfway and check the first vocabulary of the page, let's say the letter starts with <span class="arithmatex">\(m\)</span>.</li>
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<li>Since <span class="arithmatex">\(r\)</span> comes after <span class="arithmatex">\(m\)</span> in the alphabet, the first half can be ignored and the search space is narrowed down to the second half.</li>
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<li>Repeat steps <code>1.</code> and <code>2.</code> until you find the page where the word starts with <span class="arithmatex">\(r\)</span>.</li>
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</ol>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:5"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1"><1></label><label for="__tabbed_1_2"><2></label><label for="__tabbed_1_3"><3></label><label for="__tabbed_1_4"><4></label><label for="__tabbed_1_5"><5></label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Process of Looking Up a Dictionary" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Process of looking up a dictionary" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" /></a></p>
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</div>
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<div class="tabbed-block">
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary Search in Dictionary Step 2" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary search in dictionary step 2" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" /></a></p>
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</div>
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<div class="tabbed-block">
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary Search in Dictionary Step 3" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary search in dictionary step 3" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" /></a></p>
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</div>
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<div class="tabbed-block">
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary Search in Dictionary Step 4" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary search in dictionary step 4" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" /></a></p>
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</div>
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<div class="tabbed-block">
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary Search in Dictionary Step 5" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary search in dictionary step 5" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" /></a></p>
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</div>
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</div>
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</div>
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<p align="center"> Figure 1-1 Process of Looking Up a Dictionary </p>
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<p align="center"> Figure 1-1 Process of looking up a dictionary </p>
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<p>This essential skill for elementary students, looking up a dictionary, is actually the famous "Binary Search" algorithm. From a data structure perspective, we can consider the dictionary as a sorted "array"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as "Binary Search."</p>
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<p><strong>Example 2: Organizing Playing Cards</strong>. When playing cards, we need to arrange the cards in our hand in ascending order, as shown in the following process.</p>
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<p>Looking up a dictionary, an essential skill for elementary school students is actually the famous "Binary Search" algorithm. From a data structure perspective, we can consider the dictionary as a sorted "array"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as the algorithm "Binary Search."</p>
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<p><strong>Example 2: Organizing Card Deck</strong>. When playing cards, we need to arrange the cards in our hands in ascending order, as shown in the following process.</p>
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<ol>
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<li>Divide the playing cards into "ordered" and "unordered" sections, assuming initially the leftmost card is already in order.</li>
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<li>Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order.</li>
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<li>Continue to repeat step <code>2.</code> until all cards are in order.</li>
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<li>Repeat step <code>2</code> until all cards are in order.</li>
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</ol>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/playing_cards_sorting.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Playing cards sorting process" class="animation-figure" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></a></p>
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<p align="center"> Figure 1-2 Playing cards sorting process </p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/playing_cards_sorting.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Process of sorting a deck of cards" class="animation-figure" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></a></p>
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<p align="center"> Figure 1-2 Process of sorting a deck of cards </p>
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<p>The above method of organizing playing cards is essentially the "Insertion Sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.</p>
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<p><strong>Example 3: Making Change</strong>. Suppose we buy goods worth <span class="arithmatex">\(69\)</span> yuan at a supermarket and give the cashier <span class="arithmatex">\(100\)</span> yuan, then the cashier needs to give us <span class="arithmatex">\(31\)</span> yuan in change. They would naturally complete the thought process as shown in Figure 1-3.</p>
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<p>The above method of organizing playing cards is practically the "Insertion Sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.</p>
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<p><strong>Example 3: Making Change</strong>. Assume making a purchase of <span class="arithmatex">\(69\)</span> at a supermarket. If you give the cashier <span class="arithmatex">\(100\)</span>, they will need to provide you with <span class="arithmatex">\(31\)</span> in change. This process can be clearly understood as illustrated in Figure 1-3.</p>
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<ol>
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<li>The options are currencies smaller than <span class="arithmatex">\(31\)</span>, including <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(5\)</span>, <span class="arithmatex">\(10\)</span>, and <span class="arithmatex">\(20\)</span>.</li>
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<li>The options are currencies valued below <span class="arithmatex">\(31\)</span>, including <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(5\)</span>, <span class="arithmatex">\(10\)</span>, and <span class="arithmatex">\(20\)</span>.</li>
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<li>Take out the largest <span class="arithmatex">\(20\)</span> from the options, leaving <span class="arithmatex">\(31 - 20 = 11\)</span>.</li>
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<li>Take out the largest <span class="arithmatex">\(10\)</span> from the remaining options, leaving <span class="arithmatex">\(11 - 10 = 1\)</span>.</li>
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<li>Take out the largest <span class="arithmatex">\(1\)</span> from the remaining options, leaving <span class="arithmatex">\(1 - 1 = 0\)</span>.</li>
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<li>Complete the change-making, with the solution being <span class="arithmatex">\(20 + 10 + 1 = 31\)</span>.</li>
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<li>Complete change-making, the solution is <span class="arithmatex">\(20 + 10 + 1 = 31\)</span>.</li>
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</ol>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/greedy_change.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Change making process" class="animation-figure" src="../algorithms_are_everywhere.assets/greedy_change.png" /></a></p>
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<p align="center"> Figure 1-3 Change making process </p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/greedy_change.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Process of making change" class="animation-figure" src="../algorithms_are_everywhere.assets/greedy_change.png" /></a></p>
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<p align="center"> Figure 1-3 Process of making change </p>
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<p>In the above steps, we make the best choice at each step (using the largest denomination possible), ultimately resulting in a feasible change-making plan. From the perspective of data structures and algorithms, this method is essentially a "Greedy" algorithm.</p>
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<p>From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers, solving various complex issues more efficiently.</p>
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<p>In the steps described, we choose the best option at each stage by utilizing the largest denomination available, which leads to an effective change-making strategy. From a data structures and algorithms perspective, this approach is known as a "Greedy" algorithm.</p>
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<p>From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers and solve various complex issues in a more efficient way.</p>
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<div class="admonition tip">
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<p class="admonition-title">Tip</p>
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<p>If concepts such as data structures, algorithms, arrays, and binary search still seem somewhat obscure, I encourage you to continue reading. This book will gently guide you into the realm of understanding data structures and algorithms.</p>
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<p>If you are still confused about concepts like data structures, algorithms, arrays, and binary searches, I encourage you to keep reading. This book will gently guide you into the realm of understanding data structures and algorithms.</p>
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</div>
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<!-- Source file information -->
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<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.2/es5/tex-mml-chtml.min.js"></script>
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</style> <script src="../assets/javascripts/glightbox.min.js"></script></head>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.2/es5/tex-mml-chtml.min.js"></script>
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<script>document$.subscribe(() => {const lightbox = GLightbox({"touchNavigation": true, "loop": false, "zoomable": true, "draggable": false, "openEffect": "zoom", "closeEffect": "zoom", "slideEffect": "none"});})</script></body>
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<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
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html.glightbox-open { overflow: initial; height: 100%; }
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.gslide-title { margin-top: 0px; user-select: text; }
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</style> <script src="../../assets/javascripts/glightbox.min.js"></script></head>
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html.glightbox-open { overflow: initial; height: 100%; }
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.gslide-title { margin-top: 0px; user-select: text; }
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<label class="md-nav__link md-nav__link--active" for="__toc">
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<span class="md-ellipsis">
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1.3 Summary
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
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<label class="md-nav__title" for="__toc">
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<span class="md-nav__icon md-icon"></span>
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Table of contents
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</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#1-q-a" class="md-nav__link">
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<span class="md-ellipsis">
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1. Q & A
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</span>
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</a>
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</li>
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</ul>
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1. Q & A
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<li>The principle of looking up a word in a dictionary is consistent with the binary search algorithm. The binary search algorithm embodies the important algorithmic concept of divide and conquer.</li>
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<li>The process of organizing playing cards is very similar to the insertion sort algorithm. The insertion sort algorithm is suitable for sorting small datasets.</li>
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<li>The steps of making change in currency essentially follow the greedy algorithm, where each step involves making the best possible choice at the moment.</li>
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<li>An algorithm is a set of instructions or steps used to solve a specific problem within a finite amount of time, while a data structure is the way data is organized and stored in a computer.</li>
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<li>An algorithm is a set of step-by-step instructions for solving a specific problem within a finite time, while a data structure defines how data is organized and stored in a computer.</li>
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<li>Data structures and algorithms are closely linked. Data structures are the foundation of algorithms, and algorithms are the stage to utilize the functions of data structures.</li>
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<li>We can liken data structures and algorithms to building blocks. The blocks represent data, the shape and connection method of the blocks represent data structures, and the steps of assembling the blocks correspond to algorithms.</li>
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<li>We can compare data structures and algorithms to assembling building blocks. The blocks represent data, the shape and connection method of the blocks represent data structures, and the steps of assembling the blocks correspond to algorithms.</li>
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</ul>
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<h3 id="1-q-a">1. Q & A<a class="headerlink" href="#1-q-a" title="Permanent link">¶</a></h3>
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<p><strong>Q</strong>：As a programmer, I’ve rarely needed to implement algorithms manually in my daily work. Most commonly used algorithms are already built into programming languages and libraries, ready to use. Does this suggest that the problems we encounter in our work haven’t yet reached the level of complexity that demands custom algorithm design?</p>
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<p>If specific work skills are like the "moves" in martial arts, then fundamental subjects are more like "internal strength".</p>
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<p>I believe the significance of learning algorithms (and other fundamental subjects) isn’t necessarily to implement them from scratch at work, but to enable more professional decision-making and problem-solving based on a solid understanding of the concepts. This, in turn, raises the overall quality of our work. For example, every programming language provides a built-in sorting function:</p>
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<ul>
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<li>If we have not learned data structures and algorithms, then given any data, we might just give it to this sorting function. It runs smoothly, has good performance, and seems to have no problems.</li>
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<li>However, if we’ve studied algorithms, we understand that the time complexity of a built-in sorting function is typically <span class="arithmatex">\(O(n \log n)\)</span>. Moreover, if the data consists of integers with a fixed number of digits (such as student IDs), we can apply a more efficient approach like radix sort, reducing the time complexity to O(nk) , where k is the number of digits. When handling large volumes of data, the time saved can turn into significant value — lowering costs, improving user experience, and enhancing system performance.</li>
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<p>In engineering, many problems are difficult to solve optimally; most are addressed with ‘near-optimal’ solutions. The difficulty of a problem depends not only on its inherent complexity but also on the knowledge and experience of the person tackling it. The deeper one’s expertise and experience, the more thorough the analysis, and the more elegantly the problem can be solved.</p>
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@@ -3619,13 +3617,13 @@
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||||
<p><strong>Designing data structures is a balancing act, often requiring trade-offs</strong>. If you want to improve in one aspect, you often need to compromise in another. Here are two examples:</p>
|
||||
<ul>
|
||||
<li>Compared to arrays, linked lists offer more convenience in data addition and deletion but sacrifice data access speed.</li>
|
||||
<li>Graphs, compared to linked lists, provide richer logical information but require more memory space.</li>
|
||||
<li>Compared with linked lists, graphs provide richer logical information but require more memory space.</li>
|
||||
</ul>
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<h2 id="123-relationship-between-data-structures-and-algorithms">1.2.3 Relationship between data structures and algorithms<a class="headerlink" href="#123-relationship-between-data-structures-and-algorithms" title="Permanent link">¶</a></h2>
|
||||
<p>As shown in Figure 1-4, data structures and algorithms are highly related and closely integrated, specifically in the following three aspects:</p>
|
||||
<ul>
|
||||
<li>Data structures are the foundation of algorithms. They provide structured data storage and methods for manipulating data for algorithms.</li>
|
||||
<li>Algorithms are the stage where data structures come into play. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved.</li>
|
||||
<li>Algorithms inject vitality into data structures. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved.</li>
|
||||
<li>Algorithms can often be implemented based on different data structures, but their execution efficiency can vary greatly. Choosing the right data structure is key.</li>
|
||||
</ul>
|
||||
<p><a class="glightbox" href="../what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Relationship between data structures and algorithms" class="animation-figure" src="../what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png" /></a></p>
|
||||
@@ -3669,7 +3667,7 @@
|
||||
<p>It's worth noting that data structures and algorithms are independent of programming languages. For this reason, this book is able to provide implementations in multiple programming languages.</p>
|
||||
<div class="admonition tip">
|
||||
<p class="admonition-title">Conventional Abbreviation</p>
|
||||
<p>In real-life discussions, we often refer to "Data Structures and Algorithms" simply as "Algorithms". For example, the well-known LeetCode algorithm problems actually test both data structure and algorithm knowledge.</p>
|
||||
<p>In real-life discussions, we often refer to "Data Structures and Algorithms" simply as "Algorithms". For example, the well-known LeetCode algorithm questions actually test knowledge of both data structures and algorithms.</p>
|
||||
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Reference in New Issue
Block a user