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 <!-- Page content -->
 <h1 id="41-array">4.1 &nbsp; Array<a class="headerlink" href="#41-array" title="Permanent link">&para;</a></h1>
-<p>An "array" is a linear data structure that operates as a lineup of similar items, stored together in a computer's memory in contiguous spaces. It's like a sequence that maintains organized storage. Each item in this lineup has its unique 'spot' known as an "index". Please refer to Figure 4-1 to observe how arrays work and grasp these key terms.</p>
+<p>An <u>array</u> is a linear data structure that operates as a lineup of similar items, stored together in a computer's memory in contiguous spaces. It's like a sequence that maintains organized storage. Each item in this lineup has its unique 'spot' known as an <u>index</u>. Please refer to Figure 4-1 to observe how arrays work and grasp these key terms.</p>
 <p><a class="glightbox" href="../array.assets/array_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array definition and storage method" class="animation-figure" src="../array.assets/array_definition.png" /></a></p>
 <p align="center"> Figure 4-1 &nbsp; Array definition and storage method </p>

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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <!-- Page content -->
 <h1 id="42-linked-list">4.2 &nbsp; Linked list<a class="headerlink" href="#42-linked-list" title="Permanent link">&para;</a></h1>
 <p>Memory space is a shared resource among all programs. In a complex system environment, available memory can be dispersed throughout the memory space. We understand that the memory allocated for an array must be continuous. However, for very large arrays, finding a sufficiently large contiguous memory space might be challenging. This is where the flexible advantage of linked lists becomes evident.</p>
-<p>A "linked list" is a linear data structure in which each element is a node object, and the nodes are interconnected through "references". These references hold the memory addresses of subsequent nodes, enabling navigation from one node to the next.</p>
+<p>A <u>linked list</u> is a linear data structure in which each element is a node object, and the nodes are interconnected through "references". These references hold the memory addresses of subsequent nodes, enabling navigation from one node to the next.</p>
 <p>The design of linked lists allows for their nodes to be distributed across memory locations without requiring contiguous memory addresses.</p>
 <p><a class="glightbox" href="../linked_list.assets/linkedlist_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linked list definition and storage method" class="animation-figure" src="../linked_list.assets/linkedlist_definition.png" /></a></p>
 <p align="center"> Figure 4-5 &nbsp; Linked list definition and storage method </p>

-<p>As shown in the figure, we see that the basic building block of a linked list is the "node" object. Each node comprises two key components: the node's "value" and a "reference" to the next node.</p>
+<p>As shown in the figure, we see that the basic building block of a linked list is the <u>node</u> object. Each node comprises two key components: the node's "value" and a "reference" to the next node.</p>
 <ul>
 <li>The first node in a linked list is the "head node", and the final one is the "tail node".</li>
 <li>The tail node points to "null", designated as <code>null</code> in Java, <code>nullptr</code> in C++, and <code>None</code> in Python.</li>
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <!-- Page content -->
 <h1 id="43-list">4.3 &nbsp; List<a class="headerlink" href="#43-list" title="Permanent link">&para;</a></h1>
-<p>A "list" is an abstract data structure concept that represents an ordered collection of elements, supporting operations such as element access, modification, addition, deletion, and traversal, without requiring users to consider capacity limitations. Lists can be implemented based on linked lists or arrays.</p>
+<p>A <u>list</u> is an abstract data structure concept that represents an ordered collection of elements, supporting operations such as element access, modification, addition, deletion, and traversal, without requiring users to consider capacity limitations. Lists can be implemented based on linked lists or arrays.</p>
 <ul>
 <li>A linked list inherently serves as a list, supporting operations for adding, deleting, searching, and modifying elements, with the flexibility to dynamically adjust its size.</li>
 <li>Arrays also support these operations, but due to their immutable length, they can be considered as a list with a length limit.</li>
 </ul>
 <p>When implementing lists using arrays, <strong>the immutability of length reduces the practicality of the list</strong>. This is because predicting the amount of data to be stored in advance is often challenging, making it difficult to choose an appropriate list length. If the length is too small, it may not meet the requirements; if too large, it may waste memory space.</p>
-<p>To solve this problem, we can implement lists using a "dynamic array." It inherits the advantages of arrays and can dynamically expand during program execution.</p>
+<p>To solve this problem, we can implement lists using a <u>dynamic array</u>. It inherits the advantages of arrays and can dynamically expand during program execution.</p>
 <p>In fact, <strong>many programming languages' standard libraries implement lists using dynamic arrays</strong>, such as Python's <code>list</code>, Java's <code>ArrayList</code>, C++'s <code>vector</code>, and C#'s <code>List</code>. In the following discussion, we will consider "list" and "dynamic array" as synonymous concepts.</p>
 <h2 id="431-common-list-operations">4.3.1 &nbsp; Common list operations<a class="headerlink" href="#431-common-list-operations" title="Permanent link">&para;</a></h2>
 <h3 id="1-initializing-a-list">1. &nbsp; Initializing a list<a class="headerlink" href="#1-initializing-a-list" title="Permanent link">&para;</a></h3>
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <p>In the first two sections of this chapter, we explored arrays and linked lists, two fundamental and important data structures, representing "continuous storage" and "dispersed storage" respectively.</p>
 <p>In fact, <strong>the physical structure largely determines the efficiency of a program's use of memory and cache</strong>, which in turn affects the overall performance of the algorithm.</p>
 <h2 id="441-computer-storage-devices">4.4.1 &nbsp; Computer storage devices<a class="headerlink" href="#441-computer-storage-devices" title="Permanent link">&para;</a></h2>
-<p>There are three types of storage devices in computers: "hard disk," "random-access memory (RAM)," and "cache memory." The following table shows their different roles and performance characteristics in computer systems.</p>
+<p>There are three types of storage devices in computers: <u>hard disk</u>, <u>random-access memory (RAM)</u>, and <u>cache memory</u>. The following table shows their different roles and performance characteristics in computer systems.</p>
 <p align="center"> Table 4-2 &nbsp; Computer storage devices </p>

 <div class="center-table">
@@ -3681,8 +3681,8 @@
 <p>On one hand, <strong>memory is limited and cannot be shared by multiple programs</strong>, so we hope that data structures can use space as efficiently as possible. The elements of an array are tightly packed without extra space for storing references (pointers) between linked list nodes, making them more space-efficient. However, arrays require allocating sufficient continuous memory space at once, which may lead to memory waste, and array expansion also requires additional time and space costs. In contrast, linked lists allocate and reclaim memory dynamically on a per-node basis, providing greater flexibility.</p>
 <p>On the other hand, during program execution, <strong>as memory is repeatedly allocated and released, the degree of fragmentation of free memory becomes higher</strong>, leading to reduced memory utilization efficiency. Arrays, due to their continuous storage method, are relatively less likely to cause memory fragmentation. In contrast, the elements of a linked list are dispersedly stored, and frequent insertion and deletion operations make memory fragmentation more likely.</p>
 <h2 id="443-cache-efficiency-of-data-structures">4.4.3 &nbsp; Cache efficiency of data structures<a class="headerlink" href="#443-cache-efficiency-of-data-structures" title="Permanent link">&para;</a></h2>
-<p>Although caches are much smaller in space capacity than memory, they are much faster and play a crucial role in program execution speed. Since the cache's capacity is limited and can only store a small part of frequently accessed data, when the CPU tries to access data not in the cache, a "cache miss" occurs, forcing the CPU to load the needed data from slower memory.</p>
-<p>Clearly, <strong>the fewer the cache misses, the higher the CPU's data read-write efficiency</strong>, and the better the program performance. The proportion of successful data retrieval from the cache by the CPU is called the "cache hit rate," a metric often used to measure cache efficiency.</p>
+<p>Although caches are much smaller in space capacity than memory, they are much faster and play a crucial role in program execution speed. Since the cache's capacity is limited and can only store a small part of frequently accessed data, when the CPU tries to access data not in the cache, a <u>cache miss</u> occurs, forcing the CPU to load the needed data from slower memory.</p>
+<p>Clearly, <strong>the fewer the cache misses, the higher the CPU's data read-write efficiency</strong>, and the better the program performance. The proportion of successful data retrieval from the cache by the CPU is called the <u>cache hit rate</u>, a metric often used to measure cache efficiency.</p>
 <p>To achieve higher efficiency, caches adopt the following data loading mechanisms.</p>
 <ul>
 <li><strong>Cache lines</strong>: Caches don't store and load data byte by byte but in units of cache lines. Compared to byte-by-byte transfer, the transmission of cache lines is more efficient.</li>
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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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>
@@ -3928,7 +3928,7 @@
 <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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 </details>
-<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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@@ -4612,7 +4612,7 @@
 <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>
@@ -4840,7 +4840,7 @@

 <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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 </details>
-<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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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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@@ -3608,10 +3608,10 @@
 <p>On one hand, <strong>it's difficult to eliminate interference from the testing environment</strong>. Hardware configurations can affect algorithm performance. For example, algorithm <code>A</code> might run faster than <code>B</code> on one computer, but the opposite result may occur on another computer with different configurations. This means we would need to test on a variety of machines to calculate average efficiency, which is impractical.</p>
 <p>On the other hand, <strong>conducting a full test is very resource-intensive</strong>. As the volume of input data changes, the efficiency of the algorithms may vary. For example, with smaller data volumes, algorithm <code>A</code> might run faster than <code>B</code>, but the opposite might be true with larger data volumes. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires significant computational resources.</p>
 <h2 id="212-theoretical-estimation">2.1.2 &nbsp; Theoretical estimation<a class="headerlink" href="#212-theoretical-estimation" title="Permanent link">&para;</a></h2>
-<p>Due to the significant limitations of actual testing, we can consider evaluating algorithm efficiency solely through calculations. This estimation method is known as "asymptotic complexity analysis," or simply "complexity analysis."</p>
+<p>Due to the significant limitations of actual testing, we can consider evaluating algorithm efficiency solely through calculations. This estimation method is known as <u>asymptotic complexity analysis</u>, or simply <u>complexity analysis</u>.</p>
 <p>Complexity analysis reflects the relationship between the time and space resources required for algorithm execution and the size of the input data. <strong>It describes the trend of growth in the time and space required by the algorithm as the size of the input data increases</strong>. This definition might sound complex, but we can break it down into three key points to understand it better.</p>
 <ul>
-<li>"Time and space resources" correspond to "time complexity" and "space complexity," respectively.</li>
+<li>"Time and space resources" correspond to <u>time complexity</u> and <u>space complexity</u>, respectively.</li>
 <li>"As the size of input data increases" means that complexity reflects the relationship between algorithm efficiency and the volume of input data.</li>
 <li>"The trend of growth in time and space" indicates that complexity analysis focuses not on the specific values of runtime or space occupied but on the "rate" at which time or space grows.</li>
 </ul>
@@ -3814,7 +3814,7 @@ aria-label="Footer"
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <!-- Page content -->
 <h1 id="24-space-complexity">2.4 &nbsp; Space complexity<a class="headerlink" href="#24-space-complexity" title="Permanent link">&para;</a></h1>
-<p>"Space complexity" is used to measure the growth trend of the memory space occupied by an algorithm as the amount of data increases. This concept is very similar to time complexity, except that "running time" is replaced with "occupied memory space".</p>
+<p><u>Space complexity</u> is used to measure the growth trend of the memory space occupied by an algorithm as the amount of data increases. This concept is very similar to time complexity, except that "running time" is replaced with "occupied memory space".</p>
 <h2 id="241-space-related-to-algorithms">2.4.1 &nbsp; Space related to algorithms<a class="headerlink" href="#241-space-related-to-algorithms" title="Permanent link">&para;</a></h2>
 <p>The memory space used by an algorithm during its execution mainly includes the following types.</p>
 <ul>
@@ -5955,7 +5955,7 @@ aria-label="Footer"
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <p><strong>Time Complexity</strong></p>
 <ul>
 <li>Time complexity measures the trend of an algorithm's running time with the increase in data volume, effectively assessing algorithm efficiency. However, it can fail in certain cases, such as with small input data volumes or when time complexities are the same, making it challenging to precisely compare the efficiency of algorithms.</li>
-<li>Worst-case time complexity is denoted using big O notation, representing the asymptotic upper bound, reflecting the growth level of the number of operations <span class="arithmatex">\(T(n)\)</span> as <span class="arithmatex">\(n\)</span> approaches infinity.</li>
+<li>Worst-case time complexity is denoted using big-<span class="arithmatex">\(O\)</span> notation, representing the asymptotic upper bound, reflecting the growth level of the number of operations <span class="arithmatex">\(T(n)\)</span> as <span class="arithmatex">\(n\)</span> approaches infinity.</li>
 <li>Calculating time complexity involves two steps: first counting the number of operations, then determining the asymptotic upper bound.</li>
 <li>Common time complexities, arranged from low to high, include <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n \log n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, <span class="arithmatex">\(O(2^n)\)</span>, and <span class="arithmatex">\(O(n!)\)</span>, among others.</li>
 <li>The time complexity of some algorithms is not fixed and depends on the distribution of input data. Time complexities are divided into worst, best, and average cases. The best case is rarely used because input data generally needs to meet strict conditions to achieve the best case.</li>
@@ -3618,7 +3618,7 @@
 <p><strong>Q</strong>: Is the space complexity of tail recursion <span class="arithmatex">\(O(1)\)</span>?</p>
 <p>Theoretically, the space complexity of a tail-recursive function can be optimized to <span class="arithmatex">\(O(1)\)</span>. However, most programming languages (such as Java, Python, C++, Go, C#) do not support automatic optimization of tail recursion, so it's generally considered to have a space complexity of <span class="arithmatex">\(O(n)\)</span>.</p>
 <p><strong>Q</strong>: What is the difference between the terms "function" and "method"?</p>
-<p>A "function" can be executed independently, with all parameters passed explicitly. A "method" is associated with an object and is implicitly passed to the object calling it, able to operate on the data contained within an instance of a class.</p>
+<p>A <u>function</u> can be executed independently, with all parameters passed explicitly. A <u>method</u> is associated with an object and is implicitly passed to the object calling it, able to operate on the data contained within an instance of a class.</p>
 <p>Here are some examples from common programming languages:</p>
 <ul>
 <li>C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with these structures are equivalent to methods in other programming languages.</li>
@@ -3816,7 +3816,7 @@ aria-label="Footer"
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <div class="arithmatex">\[
 T(n) = 3 + 2n
 \]</div>
-<p>Since <span class="arithmatex">\(T(n)\)</span> is a linear function, its growth trend is linear, and therefore, its time complexity is of linear order, denoted as <span class="arithmatex">\(O(n)\)</span>. This mathematical notation, known as "big-O notation," represents the "asymptotic upper bound" of the function <span class="arithmatex">\(T(n)\)</span>.</p>
+<p>Since <span class="arithmatex">\(T(n)\)</span> is a linear function, its growth trend is linear, and therefore, its time complexity is of linear order, denoted as <span class="arithmatex">\(O(n)\)</span>. This mathematical notation, known as <u>big-O notation</u>, represents the <u>asymptotic upper bound</u> of the function <span class="arithmatex">\(T(n)\)</span>.</p>
 <p>In essence, time complexity analysis is about finding the asymptotic upper bound of the "number of operations <span class="arithmatex">\(T(n)\)</span>". It has a precise mathematical definition.</p>
 <div class="admonition note">
 <p class="admonition-title">Asymptotic Upper Bound</p>
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 <!-- Page content -->
 <h1 id="34-character-encoding">3.4 &nbsp; Character encoding *<a class="headerlink" href="#34-character-encoding" title="Permanent link">&para;</a></h1>
-<p>In the computer system, all data is stored in binary form, and characters (represented by char) are no exception. To represent characters, we need to develop a "character set" that defines a one-to-one mapping between each character and binary numbers. With the character set, computers can convert binary numbers to characters by looking up the table.</p>
+<p>In the computer system, all data is stored in binary form, and <code>char</code> is no exception. To represent characters, we need to develop a "character set" that defines a one-to-one mapping between each character and binary numbers. With the character set, computers can convert binary numbers to characters by looking up the table.</p>
 <h2 id="341-ascii-character-set">3.4.1 &nbsp; ASCII character set<a class="headerlink" href="#341-ascii-character-set" title="Permanent link">&para;</a></h2>
-<p>The "ASCII code" is one of the earliest character sets, officially known as the American Standard Code for Information Interchange. It uses 7 binary digits (the lower 7 bits of a byte) to represent a character, allowing for a maximum of 128 different characters. As shown in Figure 3-6, ASCII includes uppercase and lowercase English letters, numbers 0 ~ 9, various punctuation marks, and certain control characters (such as newline and tab).</p>
+<p>The <u>ASCII code</u> is one of the earliest character sets, officially known as the American Standard Code for Information Interchange. It uses 7 binary digits (the lower 7 bits of a byte) to represent a character, allowing for a maximum of 128 different characters. As shown in Figure 3-6, ASCII includes uppercase and lowercase English letters, numbers 0 ~ 9, various punctuation marks, and certain control characters (such as newline and tab).</p>
 <p><a class="glightbox" href="../character_encoding.assets/ascii_table.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="ASCII code" class="animation-figure" src="../character_encoding.assets/ascii_table.png" /></a></p>
 <p align="center"> Figure 3-6 &nbsp; ASCII code </p>

-<p>However, <strong>ASCII can only represent English characters</strong>. With the globalization of computers, a character set called "EASCII" was developed to represent more languages. It expands from the 7-bit structure of ASCII to 8 bits, enabling the representation of 256 characters.</p>
+<p>However, <strong>ASCII can only represent English characters</strong>. With the globalization of computers, a character set called <u>EASCII</u> was developed to represent more languages. It expands from the 7-bit structure of ASCII to 8 bits, enabling the representation of 256 characters.</p>
 <p>Globally, various region-specific EASCII character sets have been introduced. The first 128 characters of these sets are consistent with the ASCII, while the remaining 128 characters are defined differently to accommodate the requirements of different languages.</p>
 <h2 id="342-gbk-character-set">3.4.2 &nbsp; GBK character set<a class="headerlink" href="#342-gbk-character-set" title="Permanent link">&para;</a></h2>
-<p>Later, it was found that <strong>EASCII still could not meet the character requirements of many languages</strong>. For instance, there are nearly a hundred thousand Chinese characters, with several thousand used regularly. In 1980, the Standardization Administration of China released the "GB2312" character set, which included 6763 Chinese characters, essentially fulfilling the computer processing needs for the Chinese language.</p>
-<p>However, GB2312 could not handle some rare and traditional characters. The "GBK" character set expands GB2312 and includes 21886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented with one byte, while Chinese characters use two bytes.</p>
+<p>Later, it was found that <strong>EASCII still could not meet the character requirements of many languages</strong>. For instance, there are nearly a hundred thousand Chinese characters, with several thousand used regularly. In 1980, the Standardization Administration of China released the <u>GB2312</u> character set, which included 6763 Chinese characters, essentially fulfilling the computer processing needs for the Chinese language.</p>
+<p>However, GB2312 could not handle some rare and traditional characters. The <u>GBK</u> character set expands GB2312 and includes 21886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented with one byte, while Chinese characters use two bytes.</p>
 <h2 id="343-unicode-character-set">3.4.3 &nbsp; Unicode character set<a class="headerlink" href="#343-unicode-character-set" title="Permanent link">&para;</a></h2>
 <p>With the rapid evolution of computer technology and a plethora of character sets and encoding standards, numerous problems arose. On the one hand, these character sets generally only defined characters for specific languages and could not function properly in multilingual environments. On the other hand, the existence of multiple character set standards for the same language caused garbled text when information was exchanged between computers using different encoding standards.</p>
 <p>Researchers of that era thought: <strong>What if a comprehensive character set encompassing all global languages and symbols was developed? Wouldn't this resolve the issues associated with cross-linguistic environments and garbled text?</strong> Inspired by this idea, the extensive character set, Unicode, was born.</p>
-<p>"Unicode" is referred to as "统一码" (Unified Code) in Chinese, theoretically capable of accommodating over a million characters. It aims to incorporate characters from all over the world into a single set, providing a universal character set for processing and displaying various languages and reducing the issues of garbled text due to different encoding standards.</p>
+<p><u>Unicode</u> is referred to as "统一码" (Unified Code) in Chinese, theoretically capable of accommodating over a million characters. It aims to incorporate characters from all over the world into a single set, providing a universal character set for processing and displaying various languages and reducing the issues of garbled text due to different encoding standards.</p>
 <p>Since its release in 1991, Unicode has continually expanded to include new languages and characters. As of September 2022, Unicode contains 149,186 characters, including characters, symbols, and even emojis from various languages. In the vast Unicode character set, commonly used characters occupy 2 bytes, while some rare characters may occupy 3 or even 4 bytes.</p>
 <p>Unicode is a universal character set that assigns a number (called a "code point") to each character, <strong>but it does not specify how these character code points should be stored in a computer system</strong>. One might ask: How does a system interpret Unicode code points of varying lengths within a text? For example, given a 2-byte code, how does the system determine if it represents a single 2-byte character or two 1-byte characters?</p>
-<p>A straightforward solution to this problem is to store all characters as equal-length encodings. As shown in Figure 3-7, each character in "Hello" occupies 1 byte, while each character in "算法" (algorithm) occupies 2 bytes. We could encode all characters in "Hello 算法" as 2 bytes by padding the higher bits with zeros. This method would enable the system to interpret a character every 2 bytes, recovering the content of the phrase.</p>
+<p><strong>A straightforward solution to this problem is to store all characters as equal-length encodings</strong>. As shown in Figure 3-7, each character in "Hello" occupies 1 byte, while each character in "算法" (algorithm) occupies 2 bytes. We could encode all characters in "Hello 算法" as 2 bytes by padding the higher bits with zeros. This method would enable the system to interpret a character every 2 bytes, recovering the content of the phrase.</p>
 <p><a class="glightbox" href="../character_encoding.assets/unicode_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Unicode encoding example" class="animation-figure" src="../character_encoding.assets/unicode_hello_algo.png" /></a></p>
 <p align="center"> Figure 3-7 &nbsp; Unicode encoding example </p>

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 <li><strong>Linear data structures</strong>: Arrays, Linked Lists, Stacks, Queues, Hash Tables.</li>
 <li><strong>Non-linear data structures</strong>: Trees, Heaps, Graphs, Hash Tables.</li>
 </ul>
-<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_logic_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linear and non-linear data structures" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></a></p>
-<p align="center"> Figure 3-1 &nbsp; Linear and non-linear data structures </p>
-
 <p>Non-linear data structures can be further divided into tree structures and network structures.</p>
 <ul>
-<li><strong>Linear structures</strong>: Arrays, linked lists, queues, stacks, and hash tables, where elements have a one-to-one sequential relationship.</li>
 <li><strong>Tree structures</strong>: Trees, Heaps, Hash Tables, where elements have a one-to-many relationship.</li>
 <li><strong>Network structures</strong>: Graphs, where elements have a many-to-many relationships.</li>
 </ul>
+<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_logic_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Linear and non-linear data structures" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></a></p>
+<p align="center"> Figure 3-1 &nbsp; Linear and non-linear data structures </p>
+
 <h2 id="312-physical-structure-contiguous-and-dispersed">3.1.2 &nbsp; Physical structure: contiguous and dispersed<a class="headerlink" href="#312-physical-structure-contiguous-and-dispersed" title="Permanent link">&para;</a></h2>
 <p><strong>During the execution of an algorithm, the data being processed is stored in memory</strong>. Figure 3-2 shows a computer memory stick where each black square is a physical memory space. We can think of memory as a vast Excel spreadsheet, with each cell capable of storing a certain amount of data.</p>
 <p><strong>The system accesses the data at the target location by means of a memory address</strong>. As shown in Figure 3-2, the computer assigns a unique identifier to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data stored in memory.</p>
@@ -3623,9 +3622,11 @@
 <p><a class="glightbox" href="../classification_of_data_structure.assets/classification_phisical_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Contiguous space storage and dispersed space storage" class="animation-figure" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></a></p>
 <p align="center"> Figure 3-3 &nbsp; Contiguous space storage and dispersed space storage </p>

-<p><strong>It is worth noting that all data structures are implemented based on arrays, linked lists, or a combination of both</strong>. For example, stacks and queues can be implemented using either arrays or linked lists; while implementations of hash tables may involve both arrays and linked lists.
- <strong>Array-based implementations</strong>: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, Matrices, Tensors (arrays with dimensions <span class="arithmatex">\(\geq 3\)</span>).
- <strong>Linked-list-based implementations</strong>: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, etc.</p>
+<p><strong>It is worth noting that all data structures are implemented based on arrays, linked lists, or a combination of both</strong>. For example, stacks and queues can be implemented using either arrays or linked lists; while implementations of hash tables may involve both arrays and linked lists.</p>
+<ul>
+<li><strong>Array-based implementations</strong>: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, Matrices, Tensors (arrays with dimensions <span class="arithmatex">\(\geq 3\)</span>).</li>
+<li><strong>Linked-list-based implementations</strong>: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, etc.</li>
+</ul>
 <p>Data structures implemented based on arrays are also called “Static Data Structures,” meaning their length cannot be changed after initialization. Conversely, those based on linked lists are called “Dynamic Data Structures,” which can still adjust their size during program execution.</p>
 <div class="admonition tip">
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 <p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Conversions between sign-magnitude, one's complement, and two's complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
 <p align="center"> Figure 3-4 &nbsp; Conversions between sign-magnitude, one's complement, and two's complement </p>

-<p>Although sign-magnitude is the most intuitive, it has limitations. For one, <strong>negative numbers in sign-magnitude cannot be directly used in calculations</strong>. For example, in sign-magnitude, calculating <span class="arithmatex">\(1 + (-2)\)</span> results in <span class="arithmatex">\(-3\)</span>, which is incorrect.</p>
+<p>Although <u>sign-magnitude</u> is the most intuitive, it has limitations. For one, <strong>negative numbers in sign-magnitude cannot be directly used in calculations</strong>. For example, in sign-magnitude, calculating <span class="arithmatex">\(1 + (-2)\)</span> results in <span class="arithmatex">\(-3\)</span>, which is incorrect.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 &amp; 1 + (-2) \newline
@@ -3616,7 +3616,7 @@
 &amp; \rightarrow -3
 \end{aligned}
 \]</div>
-<p>To address this, computers introduced the <strong>one's complement</strong>. If we convert to one's complement and calculate <span class="arithmatex">\(1 + (-2)\)</span>, then convert the result back to sign-magnitude, we get the correct result of <span class="arithmatex">\(-1\)</span>.</p>
+<p>To address this, computers introduced the <u>one's complement</u>. If we convert to one's complement and calculate <span class="arithmatex">\(1 + (-2)\)</span>, then convert the result back to sign-magnitude, we get the correct result of <span class="arithmatex">\(-1\)</span>.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 &amp; 1 + (-2) \newline
@@ -3634,7 +3634,7 @@
 -0 &amp; \rightarrow 1000 \; 0000
 \end{aligned}
 \]</div>
-<p>Like sign-magnitude, one's complement also suffers from the positive and negative zero ambiguity. Therefore, computers further introduced the <strong>two's complement</strong>. Let's observe the conversion process for negative zero in sign-magnitude, one's complement, and two's complement:</p>
+<p>Like sign-magnitude, one's complement also suffers from the positive and negative zero ambiguity. Therefore, computers further introduced the <u>two's complement</u>. Let's observe the conversion process for negative zero in sign-magnitude, one's complement, and two's complement:</p>
 <div class="arithmatex">\[
 \begin{aligned}
 -0 \rightarrow \; &amp; 1000 \; 0000 \; \text{(Sign-magnitude)} \newline
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 <h2 id="1453-coin-change-problem-ii">14.5.3 &nbsp; Coin change problem II<a class="headerlink" href="#1453-coin-change-problem-ii" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given <span class="arithmatex">\(n\)</span> types of coins, where the denomination of the <span class="arithmatex">\(i^{th}\)</span> type of coin is <span class="arithmatex">\(coins[i - 1]\)</span>, and the target amount is <span class="arithmatex">\(amt\)</span>. <strong>Each type of coin can be selected multiple times</strong>, <strong>ask how many combinations of coins can make up the target amount</strong>. See the example below.</p>
+<p>Given <span class="arithmatex">\(n\)</span> types of coins, where the denomination of the <span class="arithmatex">\(i^{th}\)</span> type of coin is <span class="arithmatex">\(coins[i - 1]\)</span>, and the target amount is <span class="arithmatex">\(amt\)</span>. Each type of coin can be selected multiple times, <strong>ask how many combinations of coins can make up the target amount</strong>. See the example below.</p>
 </div>
 <p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_ii_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example data for Coin Change Problem II" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_ii_example.png" /></a></p>
 <p align="center"> Figure 14-26 &nbsp; Example data for Coin Change Problem II </p>
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 <!-- Page content -->
 <h1 id="91-graph">9.1 &nbsp; Graph<a class="headerlink" href="#91-graph" title="Permanent link">&para;</a></h1>
-<p>A "graph" is a type of nonlinear data structure, consisting of "vertices" and "edges". A graph <span class="arithmatex">\(G\)</span> can be abstractly represented as a collection of a set of vertices <span class="arithmatex">\(V\)</span> and a set of edges <span class="arithmatex">\(E\)</span>. The following example shows a graph containing 5 vertices and 7 edges.</p>
+<p>A <u>graph</u> is a type of nonlinear data structure, consisting of <u>vertices</u> and <u>edges</u>. A graph <span class="arithmatex">\(G\)</span> can be abstractly represented as a collection of a set of vertices <span class="arithmatex">\(V\)</span> and a set of edges <span class="arithmatex">\(E\)</span>. The following example shows a graph containing 5 vertices and 7 edges.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 V &amp; = \{ 1, 2, 3, 4, 5 \} \newline
@@ -3670,7 +3670,7 @@ G &amp; = \{ V, E \} \newline
 <p align="center"> Figure 9-1 &nbsp; Relationship between linked lists, trees, and graphs </p>

 <h2 id="911-common-types-of-graphs">9.1.1 &nbsp; Common types of graphs<a class="headerlink" href="#911-common-types-of-graphs" title="Permanent link">&para;</a></h2>
-<p>Based on whether edges have direction, graphs can be divided into "undirected graphs" and "directed graphs", as shown in Figure 9-2.</p>
+<p>Based on whether edges have direction, graphs can be divided into <u>undirected graphs</u> and <u>directed graphs</u>, as shown in Figure 9-2.</p>
 <ul>
 <li>In undirected graphs, edges represent a "bidirectional" connection between two vertices, for example, the "friendship" in WeChat or QQ.</li>
 <li>In directed graphs, edges have directionality, that is, the edges <span class="arithmatex">\(A \rightarrow B\)</span> and <span class="arithmatex">\(A \leftarrow B\)</span> are independent of each other, for example, the "follow" and "be followed" relationship on Weibo or TikTok.</li>
@@ -3678,7 +3678,7 @@ G &amp; = \{ V, E \} \newline
 <p><a class="glightbox" href="../graph.assets/directed_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Directed and undirected graphs" class="animation-figure" src="../graph.assets/directed_graph.png" /></a></p>
 <p align="center"> Figure 9-2 &nbsp; Directed and undirected graphs </p>

-<p>Based on whether all vertices are connected, graphs can be divided into "connected graphs" and "disconnected graphs", as shown in Figure 9-3.</p>
+<p>Based on whether all vertices are connected, graphs can be divided into <u>connected graphs</u> and <u>disconnected graphs</u>, as shown in Figure 9-3.</p>
 <ul>
 <li>For connected graphs, it is possible to reach any other vertex starting from a certain vertex.</li>
 <li>For disconnected graphs, there is at least one vertex that cannot be reached from a certain starting vertex.</li>
@@ -3686,20 +3686,20 @@ G &amp; = \{ V, E \} \newline
 <p><a class="glightbox" href="../graph.assets/connected_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Connected and disconnected graphs" class="animation-figure" src="../graph.assets/connected_graph.png" /></a></p>
 <p align="center"> Figure 9-3 &nbsp; Connected and disconnected graphs </p>

-<p>We can also add a "weight" variable to edges, resulting in "weighted graphs" as shown in Figure 9-4. For example, in mobile games like "Honor of Kings", the system calculates the "closeness" between players based on shared gaming time, and this closeness network can be represented with a weighted graph.</p>
+<p>We can also add a weight variable to edges, resulting in <u>weighted graphs</u> as shown in Figure 9-4. For example, in mobile games like "Honor of Kings", the system calculates the "closeness" between players based on shared gaming time, and this closeness network can be represented with a weighted graph.</p>
 <p><a class="glightbox" href="../graph.assets/weighted_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Weighted and unweighted graphs" class="animation-figure" src="../graph.assets/weighted_graph.png" /></a></p>
 <p align="center"> Figure 9-4 &nbsp; Weighted and unweighted graphs </p>

 <p>Graph data structures include the following commonly used terms.</p>
 <ul>
-<li>"Adjacency": When there is an edge connecting two vertices, these two vertices are said to be "adjacent". In Figure 9-4, the adjacent vertices of vertex 1 are vertices 2, 3, and 5.</li>
-<li>"Path": The sequence of edges passed from vertex A to vertex B is called a "path" from A to B. In Figure 9-4, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4.</li>
-<li>"Degree": The number of edges a vertex has. For directed graphs, "in-degree" refers to how many edges point to the vertex, and "out-degree" refers to how many edges point out from the vertex.</li>
+<li><u>Adjacency</u>: When there is an edge connecting two vertices, these two vertices are said to be "adjacent". In Figure 9-4, the adjacent vertices of vertex 1 are vertices 2, 3, and 5.</li>
+<li><u>Path</u>: The sequence of edges passed from vertex A to vertex B is called a path from A to B. In Figure 9-4, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4.</li>
+<li><u>Degree</u>: The number of edges a vertex has. For directed graphs, <u>in-degree</u> refers to how many edges point to the vertex, and <u>out-degree</u> refers to how many edges point out from the vertex.</li>
 </ul>
 <h2 id="912-representation-of-graphs">9.1.2 &nbsp; Representation of graphs<a class="headerlink" href="#912-representation-of-graphs" title="Permanent link">&para;</a></h2>
 <p>Common representations of graphs include "adjacency matrices" and "adjacency lists". The following examples use undirected graphs.</p>
 <h3 id="1-adjacency-matrix">1. &nbsp; Adjacency matrix<a class="headerlink" href="#1-adjacency-matrix" title="Permanent link">&para;</a></h3>
-<p>Let the number of vertices in the graph be <span class="arithmatex">\(n\)</span>, the "adjacency matrix" uses an <span class="arithmatex">\(n \times n\)</span> matrix to represent the graph, where each row (column) represents a vertex, and the matrix elements represent edges, with <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(0\)</span> indicating whether there is an edge between two vertices.</p>
+<p>Let the number of vertices in the graph be <span class="arithmatex">\(n\)</span>, the <u>adjacency matrix</u> uses an <span class="arithmatex">\(n \times n\)</span> matrix to represent the graph, where each row (column) represents a vertex, and the matrix elements represent edges, with <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(0\)</span> indicating whether there is an edge between two vertices.</p>
 <p>As shown in Figure 9-5, let the adjacency matrix be <span class="arithmatex">\(M\)</span>, and the list of vertices be <span class="arithmatex">\(V\)</span>, then the matrix element <span class="arithmatex">\(M[i, j] = 1\)</span> indicates there is an edge between vertex <span class="arithmatex">\(V[i]\)</span> and vertex <span class="arithmatex">\(V[j]\)</span>, conversely <span class="arithmatex">\(M[i, j] = 0\)</span> indicates there is no edge between the two vertices.</p>
 <p><a class="glightbox" href="../graph.assets/adjacency_matrix.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Representation of a graph with an adjacency matrix" class="animation-figure" src="../graph.assets/adjacency_matrix.png" /></a></p>
 <p align="center"> Figure 9-5 &nbsp; Representation of a graph with an adjacency matrix </p>
@@ -3712,7 +3712,7 @@ G &amp; = \{ V, E \} \newline
 </ul>
 <p>When representing graphs with adjacency matrices, it is possible to directly access matrix elements to obtain edges, thus operations of addition, deletion, lookup, and modification are very efficient, all with a time complexity of <span class="arithmatex">\(O(1)\)</span>. However, the space complexity of the matrix is <span class="arithmatex">\(O(n^2)\)</span>, which consumes more memory.</p>
 <h3 id="2-adjacency-list">2. &nbsp; Adjacency list<a class="headerlink" href="#2-adjacency-list" title="Permanent link">&para;</a></h3>
-<p>The "adjacency list" uses <span class="arithmatex">\(n\)</span> linked lists to represent the graph, with each linked list node representing a vertex. The <span class="arithmatex">\(i\)</span>-th linked list corresponds to vertex <span class="arithmatex">\(i\)</span> and contains all adjacent vertices (vertices connected to that vertex). Figure 9-6 shows an example of a graph stored using an adjacency list.</p>
+<p>The <u>adjacency list</u> uses <span class="arithmatex">\(n\)</span> linked lists to represent the graph, with each linked list node representing a vertex. The <span class="arithmatex">\(i\)</span>-th linked list corresponds to vertex <span class="arithmatex">\(i\)</span> and contains all adjacent vertices (vertices connected to that vertex). Figure 9-6 shows an example of a graph stored using an adjacency list.</p>
 <p><a class="glightbox" href="../graph.assets/adjacency_list.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Representation of a graph with an adjacency list" class="animation-figure" src="../graph.assets/adjacency_list.png" /></a></p>
 <p align="center"> Figure 9-6 &nbsp; Representation of a graph with an adjacency list </p>

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 <h1 id="93-graph-traversal">9.3 &nbsp; Graph traversal<a class="headerlink" href="#93-graph-traversal" title="Permanent link">&para;</a></h1>
 <p>Trees represent a "one-to-many" relationship, while graphs have a higher degree of freedom and can represent any "many-to-many" relationship. Therefore, we can consider trees as a special case of graphs. Clearly, <strong>tree traversal operations are also a special case of graph traversal operations</strong>.</p>
-<p>Both graphs and trees require the application of search algorithms to implement traversal operations. Graph traversal can be divided into two types: "Breadth-First Search (BFS)" and "Depth-First Search (DFS)".</p>
+<p>Both graphs and trees require the application of search algorithms to implement traversal operations. Graph traversal can be divided into two types: <u>Breadth-First Search (BFS)</u> and <u>Depth-First Search (DFS)</u>.</p>
 <h2 id="931-breadth-first-search">9.3.1 &nbsp; Breadth-first search<a class="headerlink" href="#931-breadth-first-search" title="Permanent link">&para;</a></h2>
 <p><strong>Breadth-first search is a near-to-far traversal method, starting from a certain node, always prioritizing the visit to the nearest vertices and expanding outwards layer by layer</strong>. As shown in Figure 9-9, starting from the top left vertex, first traverse all adjacent vertices of that vertex, then traverse all adjacent vertices of the next vertex, and so on, until all vertices have been visited.</p>
 <p><a class="glightbox" href="../graph_traversal.assets/graph_bfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Breadth-first traversal of a graph" class="animation-figure" src="../graph_traversal.assets/graph_bfs.png" /></a></p>
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 <p align="center"> Figure 15-4 &nbsp; Value per unit weight of the item </p>

 <h3 id="1-greedy-strategy-determination">1. &nbsp; Greedy strategy determination<a class="headerlink" href="#1-greedy-strategy-determination" title="Permanent link">&para;</a></h3>
-<p>Maximizing the total value of the items in the knapsack essentially means maximizing the value per unit weight. From this, the greedy strategy shown in Figure 15-5 can be deduced.</p>
+<p>Maximizing the total value of the items in the knapsack <strong>essentially means maximizing the value per unit weight</strong>. From this, the greedy strategy shown in Figure 15-5 can be deduced.</p>
 <ol>
 <li>Sort the items by their unit value from high to low.</li>
 <li>Iterate over all items, <strong>greedily choosing the item with the highest unit value in each round</strong>.</li>
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 </ol>
 <p>There are mainly two methods for improving the structure of hash tables: "Separate Chaining" and "Open Addressing".</p>
 <h2 id="621-separate-chaining">6.2.1 &nbsp; Separate chaining<a class="headerlink" href="#621-separate-chaining" title="Permanent link">&para;</a></h2>
-<p>In the original hash table, each bucket can store only one key-value pair. "Separate chaining" transforms individual elements into a linked list, with key-value pairs as list nodes, storing all colliding key-value pairs in the same list. Figure 6-5 shows an example of a hash table with separate chaining.</p>
+<p>In the original hash table, each bucket can store only one key-value pair. <u>Separate chaining</u> transforms individual elements into a linked list, with key-value pairs as list nodes, storing all colliding key-value pairs in the same list. Figure 6-5 shows an example of a hash table with separate chaining.</p>
 <p><a class="glightbox" href="../hash_collision.assets/hash_table_chaining.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Separate chaining hash table" class="animation-figure" src="../hash_collision.assets/hash_table_chaining.png" /></a></p>
 <p align="center"> Figure 6-5 &nbsp; Separate chaining hash table </p>

@@ -5176,7 +5176,7 @@
 </details>
 <p>It's worth noting that when the list is very long, the query efficiency <span class="arithmatex">\(O(n)\)</span> is poor. <strong>At this point, the list can be converted to an "AVL tree" or "Red-Black tree"</strong> to optimize the time complexity of the query operation to <span class="arithmatex">\(O(\log n)\)</span>.</p>
 <h2 id="622-open-addressing">6.2.2 &nbsp; Open addressing<a class="headerlink" href="#622-open-addressing" title="Permanent link">&para;</a></h2>
-<p>"Open addressing" does not introduce additional data structures but uses "multiple probes" to handle hash collisions. The probing methods mainly include linear probing, quadratic probing, and double hashing.</p>
+<p><u>Open addressing</u> does not introduce additional data structures but uses "multiple probes" to handle hash collisions. The probing methods mainly include linear probing, quadratic probing, and double hashing.</p>
 <p>Let's use linear probing as an example to introduce the mechanism of open addressing hash tables.</p>
 <h3 id="1-linear-probing">1. &nbsp; Linear probing<a class="headerlink" href="#1-linear-probing" title="Permanent link">&para;</a></h3>
 <p>Linear probing uses a fixed-step linear search for probing, differing from ordinary hash tables.</p>
@@ -5193,7 +5193,7 @@
 <p><a class="glightbox" href="../hash_collision.assets/hash_table_open_addressing_deletion.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Query issues caused by deletion in open addressing" class="animation-figure" src="../hash_collision.assets/hash_table_open_addressing_deletion.png" /></a></p>
 <p align="center"> Figure 6-7 &nbsp; Query issues caused by deletion in open addressing </p>

-<p>To solve this problem, we can use a "lazy deletion" mechanism: instead of directly removing elements from the hash table, <strong>use a constant <code>TOMBSTONE</code> to mark the bucket</strong>. In this mechanism, both <code>None</code> and <code>TOMBSTONE</code> represent empty buckets and can hold key-value pairs. However, when linear probing encounters <code>TOMBSTONE</code>, it should continue traversing since there may still be key-value pairs below it.</p>
+<p>To solve this problem, we can use a <u>lazy deletion</u> mechanism: instead of directly removing elements from the hash table, <strong>use a constant <code>TOMBSTONE</code> to mark the bucket</strong>. In this mechanism, both <code>None</code> and <code>TOMBSTONE</code> represent empty buckets and can hold key-value pairs. However, when linear probing encounters <code>TOMBSTONE</code>, it should continue traversing since there may still be key-value pairs below it.</p>
 <p>However, <strong>lazy deletion may accelerate the degradation of hash table performance</strong>. Every deletion operation produces a delete mark, and as <code>TOMBSTONE</code> increases, so does the search time, as linear probing may have to skip multiple <code>TOMBSTONE</code> to find the target element.</p>
 <p>Therefore, consider recording the index of the first <code>TOMBSTONE</code> encountered during linear probing and swapping the target element found with this <code>TOMBSTONE</code>. The advantage of this is that each time a query or addition is performed, the element is moved to a bucket closer to the ideal position (starting point of probing), thereby optimizing the query efficiency.</p>
 <p>The code below implements an open addressing (linear probing) hash table with lazy deletion. To make fuller use of the hash table space, we treat the hash table as a "circular array," continuing to traverse from the beginning when the end of the array is passed.</p>
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 <!-- Page content -->
 <h1 id="61-hash-table">6.1 &nbsp; Hash table<a class="headerlink" href="#61-hash-table" title="Permanent link">&para;</a></h1>
-<p>A "hash table", also known as a "hash map", achieves efficient element querying by establishing a mapping between keys and values. Specifically, when we input a <code>key</code> into the hash table, we can retrieve the corresponding <code>value</code> in <span class="arithmatex">\(O(1)\)</span> time.</p>
+<p>A <u>hash table</u> achieves efficient element querying by establishing a mapping between keys and values. Specifically, when we input a <code>key</code> into the hash table, we can retrieve the corresponding <code>value</code> in <span class="arithmatex">\(O(1)\)</span> time.</p>
 <p>As shown in Figure 6-1, given <span class="arithmatex">\(n\)</span> students, each with two pieces of data: "name" and "student number". If we want to implement a query feature that returns the corresponding name when given a student number, we can use the hash table shown in Figure 6-1.</p>
 <p><a class="glightbox" href="../hash_map.assets/hash_table_lookup.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Abstract representation of a hash table" class="animation-figure" src="../hash_map.assets/hash_table_lookup.png" /></a></p>
 <p align="center"> Figure 6-1 &nbsp; Abstract representation of a hash table </p>
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 </details>
 <h2 id="612-simple-implementation-of-hash-table">6.1.2 &nbsp; Simple implementation of hash table<a class="headerlink" href="#612-simple-implementation-of-hash-table" title="Permanent link">&para;</a></h2>
-<p>First, let's consider the simplest case: <strong>implementing a hash table using just an array</strong>. In the hash table, each empty slot in the array is called a "bucket", and each bucket can store one key-value pair. Therefore, the query operation involves finding the bucket corresponding to the <code>key</code> and retrieving the <code>value</code> from it.</p>
-<p>So, how do we locate the appropriate bucket based on the <code>key</code>? This is achieved through a "hash function". The role of the hash function is to map a larger input space to a smaller output space. In a hash table, the input space is all possible keys, and the output space is all buckets (array indices). In other words, input a <code>key</code>, <strong>and we can use the hash function to determine the storage location of the corresponding key-value pair in the array</strong>.</p>
+<p>First, let's consider the simplest case: <strong>implementing a hash table using just an array</strong>. In the hash table, each empty slot in the array is called a <u>bucket</u>, and each bucket can store one key-value pair. Therefore, the query operation involves finding the bucket corresponding to the <code>key</code> and retrieving the <code>value</code> from it.</p>
+<p>So, how do we locate the appropriate bucket based on the <code>key</code>? This is achieved through a <u>hash function</u>. The role of the hash function is to map a larger input space to a smaller output space. In a hash table, the input space is all possible keys, and the output space is all buckets (array indices). In other words, input a <code>key</code>, <strong>and we can use the hash function to determine the storage location of the corresponding key-value pair in the array</strong>.</p>
 <p>The calculation process of the hash function for a given <code>key</code> is divided into the following two steps:</p>
 <ol>
 <li>Calculate the hash value using a certain hash algorithm <code>hash()</code>.</li>
@@ -5362,7 +5362,7 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="m">12836</span><span class="w"> </span>%<span class="w"> </span><span class="nv">100</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">36</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="m">20336</span><span class="w"> </span>%<span class="w"> </span><span class="nv">100</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">36</span>
 </code></pre></div>
-<p>As shown in Figure 6-3, both student numbers point to the same name, which is obviously incorrect. This situation where multiple inputs correspond to the same output is known as "hash collision".</p>
+<p>As shown in Figure 6-3, both student numbers point to the same name, which is obviously incorrect. This situation where multiple inputs correspond to the same output is known as <u>hash collision</u>.</p>
 <p><a class="glightbox" href="../hash_map.assets/hash_collision.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example of hash collision" class="animation-figure" src="../hash_map.assets/hash_collision.png" /></a></p>
 <p align="center"> Figure 6-3 &nbsp; Example of hash collision </p>

@@ -5372,7 +5372,7 @@
 <p align="center"> Figure 6-4 &nbsp; Hash table expansion </p>

 <p>Similar to array expansion, resizing a hash table requires migrating all key-value pairs from the original hash table to the new one, which is time-consuming. Furthermore, since the capacity <code>capacity</code> of the hash table changes, we need to recalculate the storage positions of all key-value pairs using the hash function, which adds to the computational overhead of the resizing process. Therefore, programming languages often reserve a sufficiently large capacity for the hash table to prevent frequent resizing.</p>
-<p>The "load factor" is an important concept for hash tables. It is defined as the ratio of the number of elements in the hash table to the number of buckets. It is used to measure the severity of hash collisions and <strong>is often used as a trigger for resizing the hash table</strong>. For example, in Java, when the load factor exceeds <span class="arithmatex">\(0.75\)</span>, the system will resize the hash table to twice its original size.</p>
+<p>The <u>load factor</u> is an important concept for hash tables. It is defined as the ratio of the number of elements in the hash table to the number of buckets. It is used to measure the severity of hash collisions and <strong>is often used as a trigger for resizing the hash table</strong>. For example, in Java, when the load factor exceeds <span class="arithmatex">\(0.75\)</span>, the system will resize the hash table to twice its original size.</p>

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 <!-- Page content -->
 <h1 id="81-heap">8.1 &nbsp; Heap<a class="headerlink" href="#81-heap" title="Permanent link">&para;</a></h1>
-<p>A "heap" is a complete binary tree that satisfies specific conditions and can be mainly divided into two types, as shown in Figure 8-1.</p>
+<p>A <u>heap</u> is a complete binary tree that satisfies specific conditions and can be mainly divided into two types, as shown in Figure 8-1.</p>
 <ul>
-<li>"Min heap": The value of any node <span class="arithmatex">\(\leq\)</span> the values of its child nodes.</li>
-<li>"Max heap": The value of any node <span class="arithmatex">\(\geq\)</span> the values of its child nodes.</li>
+<li><u>Min heap</u>: The value of any node <span class="arithmatex">\(\leq\)</span> the values of its child nodes.</li>
+<li><u>Max heap</u>: The value of any node <span class="arithmatex">\(\geq\)</span> the values of its child nodes.</li>
 </ul>
 <p><a class="glightbox" href="../heap.assets/min_heap_and_max_heap.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Min heap and max heap" class="animation-figure" src="../heap.assets/min_heap_and_max_heap.png" /></a></p>
 <p align="center"> Figure 8-1 &nbsp; Min heap and max heap </p>
@@ -3708,7 +3708,7 @@
 <li>For max heaps (min heaps), the value of the heap top element (root node) is the largest (smallest).</li>
 </ul>
 <h2 id="811-common-operations-on-heaps">8.1.1 &nbsp; Common operations on heaps<a class="headerlink" href="#811-common-operations-on-heaps" title="Permanent link">&para;</a></h2>
-<p>It should be noted that many programming languages provide a "priority queue," which is an abstract data structure defined as a queue with priority sorting.</p>
+<p>It should be noted that many programming languages provide a <u>priority queue</u>, which is an abstract data structure defined as a queue with priority sorting.</p>
 <p>In fact, <strong>heaps are often used to implement priority queues, with max heaps equivalent to priority queues where elements are dequeued in descending order</strong>. From a usage perspective, we can consider "priority queue" and "heap" as equivalent data structures. Therefore, this book does not make a special distinction between the two, uniformly referring to them as "heap."</p>
 <p>Common operations on heaps are shown in Table 8-1, and the method names depend on the programming language.</p>
 <p align="center"> Table 8-1 &nbsp; Efficiency of Heap Operations </p>
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 </details>
 <h3 id="3-inserting-an-element-into-the-heap">3. &nbsp; Inserting an element into the heap<a class="headerlink" href="#3-inserting-an-element-into-the-heap" title="Permanent link">&para;</a></h3>
-<p>Given an element <code>val</code>, we first add it to the bottom of the heap. After addition, since <code>val</code> may be larger than other elements in the heap, the heap's integrity might be compromised, <strong>thus it's necessary to repair the path from the inserted node to the root node</strong>. This operation is called "heapifying".</p>
+<p>Given an element <code>val</code>, we first add it to the bottom of the heap. After addition, since <code>val</code> may be larger than other elements in the heap, the heap's integrity might be compromised, <strong>thus it's necessary to repair the path from the inserted node to the root node</strong>. This operation is called <u>heapifying</u>.</p>
 <p>Considering starting from the node inserted, <strong>perform heapify from bottom to top</strong>. As shown in Figure 8-3, we compare the value of the inserted node with its parent node, and if the inserted node is larger, we swap them. Then continue this operation, repairing each node in the heap from bottom to top until passing the root node or encountering a node that does not need to be swapped.</p>
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+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <h1 id="12-what-is-an-algorithm">1.2 &nbsp; What is an algorithm<a class="headerlink" href="#12-what-is-an-algorithm" title="Permanent link">&para;</a></h1>
 <h2 id="121-definition-of-an-algorithm">1.2.1 &nbsp; Definition of an algorithm<a class="headerlink" href="#121-definition-of-an-algorithm" title="Permanent link">&para;</a></h2>
-<p>An "algorithm" is a set of instructions or steps to solve a specific problem within a finite amount of time. It has the following characteristics:</p>
+<p>An <u>algorithm</u> is a set of instructions or steps to solve a specific problem within a finite amount of time. It has the following characteristics:</p>
 <ul>
 <li>The problem is clearly defined, including unambiguous definitions of input and output.</li>
 <li>The algorithm is feasible, meaning it can be completed within a finite number of steps, time, and memory space.</li>
 <li>Each step has a definitive meaning. The output is consistently the same under the same inputs and conditions.</li>
 </ul>
 <h2 id="122-definition-of-a-data-structure">1.2.2 &nbsp; Definition of a data structure<a class="headerlink" href="#122-definition-of-a-data-structure" title="Permanent link">&para;</a></h2>
-<p>A "data structure" is a way of organizing and storing data in a computer, with the following design goals:</p>
+<p>A <u>data structure</u> is a way of organizing and storing data in a computer, with the following design goals:</p>
 <ul>
 <li>Minimize space occupancy to save computer memory.</li>
 <li>Make data operations as fast as possible, covering data access, addition, deletion, updating, etc.</li>
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 <h1 id="53-double-ended-queue">5.3 &nbsp; Double-ended queue<a class="headerlink" href="#53-double-ended-queue" title="Permanent link">&para;</a></h1>
-<p>In a queue, we can only delete elements from the head or add elements to the tail. As shown in the following diagram, a "double-ended queue (deque)" offers more flexibility, allowing the addition or removal of elements at both the head and the tail.</p>
+<p>In a queue, we can only delete elements from the head or add elements to the tail. As shown in the following diagram, a <u>double-ended queue (deque)</u> offers more flexibility, allowing the addition or removal of elements at both the head and the tail.</p>
 <p><a class="glightbox" href="../deque.assets/deque_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Operations in double-ended queue" class="animation-figure" src="../deque.assets/deque_operations.png" /></a></p>
 <p align="center"> Figure 5-7 &nbsp; Operations in double-ended queue </p>

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 <h1 id="52-queue">5.2 &nbsp; Queue<a class="headerlink" href="#52-queue" title="Permanent link">&para;</a></h1>
-<p>"Queue" is a linear data structure that follows the First-In-First-Out (FIFO) rule. As the name suggests, a queue simulates the phenomenon of lining up, where newcomers join the queue at the rear, and the person at the front leaves the queue first.</p>
+<p>A <u>queue</u> is a linear data structure that follows the First-In-First-Out (FIFO) rule. As the name suggests, a queue simulates the phenomenon of lining up, where newcomers join the queue at the rear, and the person at the front leaves the queue first.</p>
 <p>As shown in Figure 5-4, we call the front of the queue the "head" and the back the "tail." The operation of adding elements to the rear of the queue is termed "enqueue," and the operation of removing elements from the front is termed "dequeue."</p>
 <p><a class="glightbox" href="../queue.assets/queue_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Queue's first-in-first-out rule" class="animation-figure" src="../queue.assets/queue_operations.png" /></a></p>
 <p align="center"> Figure 5-4 &nbsp; Queue's first-in-first-out rule </p>
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 <h1 id="51-stack">5.1 &nbsp; Stack<a class="headerlink" href="#51-stack" title="Permanent link">&para;</a></h1>
-<p>A "Stack" is a linear data structure that follows the principle of Last-In-First-Out (LIFO).</p>
+<p>A <u>stack</u> is a linear data structure that follows the principle of Last-In-First-Out (LIFO).</p>
 <p>We can compare a stack to a pile of plates on a table. To access the bottom plate, one must first remove the plates on top. By replacing the plates with various types of elements (such as integers, characters, objects, etc.), we obtain the data structure known as a stack.</p>
 <p>As shown in Figure 5-1, we refer to the top of the pile of elements as the "top of the stack" and the bottom as the "bottom of the stack." The operation of adding elements to the top of the stack is called "push," and the operation of removing the top element is called "pop."</p>
 <p><a class="glightbox" href="../stack.assets/stack_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Stack's last-in-first-out rule" class="animation-figure" src="../stack.assets/stack_operations.png" /></a></p>
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+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <p><strong>Q</strong>: A double-ended queue seems like two stacks joined together. What are its uses?</p>
 <p>A double-ended queue, which is a combination of a stack and a queue or two stacks joined together, exhibits both stack and queue logic. Thus, it can implement all applications of stacks and queues while offering more flexibility.</p>
 <p><strong>Q</strong>: How exactly are undo and redo implemented?</p>
-<p>Undo and redo operations are implemented using two stacks: Stack A for undo and Stack B for redo.</p>
+<p>Undo and redo operations are implemented using two stacks: Stack <code>A</code> for undo and Stack <code>B</code> for redo.</p>
 <ol>
-<li>Each time a user performs an operation, it is pushed onto Stack A, and Stack B is cleared.</li>
-<li>When the user executes an "undo", the most recent operation is popped from Stack A and pushed onto Stack B.</li>
-<li>When the user executes a "redo", the most recent operation is popped from Stack B and pushed back onto Stack A.</li>
+<li>Each time a user performs an operation, it is pushed onto Stack <code>A</code>, and Stack <code>B</code> is cleared.</li>
+<li>When the user executes an "undo", the most recent operation is popped from Stack <code>A</code> and pushed onto Stack <code>B</code>.</li>
+<li>When the user executes a "redo", the most recent operation is popped from Stack <code>B</code> and pushed back onto Stack <code>A</code>.</li>
 </ol>

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 <p><a class="glightbox" href="../avl_tree.assets/avltree_degradation_from_inserting_node.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Degradation of an AVL tree after inserting nodes" class="animation-figure" src="../avl_tree.assets/avltree_degradation_from_inserting_node.png" /></a></p>
 <p align="center"> Figure 7-25 &nbsp; Degradation of an AVL tree after inserting nodes </p>

-<p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the "AVL Tree" in their paper "An algorithm for the organization of information". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at <span class="arithmatex">\(O(\log n)\)</span> level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.</p>
+<p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the <u>AVL Tree</u> in their paper "An algorithm for the organization of information". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at <span class="arithmatex">\(O(\log n)\)</span> level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.</p>
 <h2 id="751-common-terminology-in-avl-trees">7.5.1 &nbsp; Common terminology in AVL trees<a class="headerlink" href="#751-common-terminology-in-avl-trees" title="Permanent link">&para;</a></h2>
-<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a "balanced binary search tree".</p>
+<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a <u>balanced binary search tree</u>.</p>
 <h3 id="1-node-height">1. &nbsp; Node height<a class="headerlink" href="#1-node-height" title="Permanent link">&para;</a></h3>
 <p>Since the operations related to AVL trees require obtaining node heights, we need to add a <code>height</code> variable to the node class:</p>
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 <h3 id="2-node-balance-factor">2. &nbsp; Node balance factor<a class="headerlink" href="#2-node-balance-factor" title="Permanent link">&para;</a></h3>
-<p>The "balance factor" of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as <span class="arithmatex">\(0\)</span>. We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:</p>
+<p>The <u>balance factor</u> of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as <span class="arithmatex">\(0\)</span>. We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:</p>
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@@ -6634,7 +6634,7 @@ aria-label="Footer"
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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