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<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
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<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
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Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
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<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
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Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
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<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
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<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
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<span class="md-ellipsis">
7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
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<span class="md-ellipsis">
8.2 Building a Heap
8.2 Heap Construction Operation
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<span class="md-ellipsis">
8.3 Top-K Problem
8.3 Top-k Problem
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<span class="md-ellipsis">
10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
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<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
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10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
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<span class="md-ellipsis">
11.1 Sorting Algorithms
11.1 Sorting Algorithm
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<span class="md-ellipsis">
12.4 Hanoi Tower Problem
12.4 Hanota Problem
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<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
@@ -4443,15 +4443,15 @@ E &amp; = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline
G &amp; = \{ V, E \} \newline
\end{aligned}
\]</div>
<p>If we view vertices as nodes and edges as references (pointers) connecting the nodes, we can see graphs as a data structure extended from linked lists. As shown in Figure 9-1, <strong>compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex</strong>.</p>
<p>If we view vertices as nodes and edges as references (pointers) connecting them, we can regard a graph as an extension of the linked list data structure. As shown in Figure 9-1, <strong>compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex</strong>.</p>
<p><img alt="Relationships among linked lists, trees, and graphs" class="animation-figure" src="../graph.assets/linkedlist_tree_graph.png" /></p>
<p align="center"> Figure 9-1 &nbsp; Relationships among linked lists, trees, and graphs </p>
<h2 id="911-common-types-and-terminology-of-graphs">9.1.1 &nbsp; Common Types and Terminology of Graphs<a class="headerlink" href="#911-common-types-and-terminology-of-graphs" title="Permanent link">&para;</a></h2>
<p>Graphs can be divided into <u>undirected graphs</u> and <u>directed graphs</u> based on whether edges have direction, as shown in Figure 9-2.</p>
<ul>
<li>In undirected graphs, edges represent a "bidirectional" connection between two vertices, such as the "friend relationship" on WeChat or QQ.</li>
<li>In directed graphs, edges have directionality, meaning edges <span class="arithmatex">\(A \rightarrow B\)</span> and <span class="arithmatex">\(A \leftarrow B\)</span> are independent of each other, such as the "follow" and "be followed" relationships on Weibo or TikTok.</li>
<li>In undirected graphs, edges represent a "bidirectional" connection between two vertices, such as friendships on WeChat or QQ.</li>
<li>In directed graphs, edges have directionality, meaning edges <span class="arithmatex">\(A \rightarrow B\)</span> and <span class="arithmatex">\(A \leftarrow B\)</span> are independent of each other, such as following and follower relationships on Weibo or TikTok.</li>
</ul>
<p><img alt="Directed and undirected graphs" class="animation-figure" src="../graph.assets/directed_graph.png" /></p>
<p align="center"> Figure 9-2 &nbsp; Directed and undirected graphs </p>
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<p><img alt="Connected and disconnected graphs" class="animation-figure" src="../graph.assets/connected_graph.png" /></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 <u>weighted graphs</u> as shown in Figure 9-4. For example, in mobile games like "Honor of Kings", the system calculates the "intimacy" between players based on their shared game time, and such intimacy networks can be represented using weighted graphs.</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 "intimacy" between players based on how long they have played together, and such intimacy networks can be represented using weighted graphs.</p>
<p><img alt="Weighted and unweighted graphs" class="animation-figure" src="../graph.assets/weighted_graph.png" /></p>
<p align="center"> Figure 9-4 &nbsp; Weighted and unweighted graphs </p>
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<ul>
<li><u>Adjacency</u>: When two vertices are connected by an edge, 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 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> indicates how many edges point to the vertex, and <u>out-degree</u> indicates how many edges point out from the vertex.</li>
<li><u>Degree</u>: The number of edges a vertex has. For directed graphs, <u>in-degree</u> indicates how many edges point to the vertex, and <u>out-degree</u> indicates how many edges leave 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 uses undirected graphs as examples.</p>
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<ul>
<li>In simple graphs, vertices cannot connect to themselves, so the elements on the main diagonal of the adjacency matrix are meaningless.</li>
<li>For undirected graphs, edges in both directions are equivalent, so the adjacency matrix is symmetric about the main diagonal.</li>
<li>Replacing the elements of the adjacency matrix from <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(0\)</span> to weights allows representation of weighted graphs.</li>
<li>Replacing the <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(0\)</span> entries in the adjacency matrix with weights allows it to represent weighted graphs.</li>
</ul>
<p>When using adjacency matrices to represent graphs, we can directly access matrix elements to obtain edges, resulting in highly efficient addition, deletion, lookup, and modification operations, 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 significant memory.</p>
<h3 id="2-adjacency-list">2. &nbsp; Adjacency List<a class="headerlink" href="#2-adjacency-list" title="Permanent link">&para;</a></h3>
@@ -4494,8 +4494,8 @@ G &amp; = \{ V, E \} \newline
<p><img alt="Adjacency list representation of a graph" class="animation-figure" src="../graph.assets/adjacency_list.png" /></p>
<p align="center"> Figure 9-6 &nbsp; Adjacency list representation of a graph </p>
<p>Adjacency lists only store edges that actually exist, and the total number of edges is typically much less than <span class="arithmatex">\(n^2\)</span>, making them more space-efficient. However, finding edges in an adjacency list requires traversing the linked list, so its time efficiency is inferior to that of adjacency matrices.</p>
<p>Observing Figure 9-6, <strong>the structure of adjacency lists is very similar to "chaining" in hash tables, so we can adopt similar methods to optimize efficiency</strong>. For example, when linked lists are long, they can be converted to AVL trees or red-black trees, thereby optimizing time efficiency from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(\log n)\)</span>; linked lists can also be converted to hash tables, thereby reducing time complexity to <span class="arithmatex">\(O(1)\)</span>.</p>
<p>Adjacency lists only store edges that actually exist, and the total number of edges is typically much less than <span class="arithmatex">\(n^2\)</span>, making them more space-efficient. However, finding edges in an adjacency list requires traversing the linked list, so it is less time-efficient than an adjacency matrix.</p>
<p>As shown in Figure 9-6, <strong>the structure of adjacency lists is very similar to separate chaining in hash tables, so we can use similar methods to improve efficiency</strong>. For example, when a linked list becomes long, it can be converted into an AVL tree or red-black tree, improving the time complexity from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(\log n)\)</span>; it can also be converted into a hash table, reducing the time complexity to <span class="arithmatex">\(O(1)\)</span>.</p>
<h2 id="913-common-applications-of-graphs">9.1.3 &nbsp; Common Applications of Graphs<a class="headerlink" href="#913-common-applications-of-graphs" title="Permanent link">&para;</a></h2>
<p>As shown in Table 9-1, many real-world systems can be modeled using graphs, and corresponding problems can be reduced to graph computation problems.</p>
<p align="center"> Table 9-1 &nbsp; Common graphs in real life </p>
en/chapter_graph/graph_operations/index.html
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
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<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
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<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
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<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
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Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
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<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
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<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
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<span class="md-ellipsis">
7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
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<span class="md-ellipsis">
8.2 Building a Heap
8.2 Heap Construction Operation
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<span class="md-ellipsis">
8.3 Top-K Problem
8.3 Top-k Problem
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<span class="md-ellipsis">
10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
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<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
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<span class="md-ellipsis">
10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
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<span class="md-ellipsis">
11.1 Sorting Algorithms
11.1 Sorting Algorithm
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<span class="md-ellipsis">
12.4 Hanoi Tower Problem
12.4 Hanota Problem
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<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
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<!-- Page content -->
<h1 id="92-basic-operations-on-graphs">9.2 &nbsp; Basic Operations on Graphs<a class="headerlink" href="#92-basic-operations-on-graphs" title="Permanent link">&para;</a></h1>
<p>Basic operations on graphs can be divided into operations on "edges" and operations on "vertices". Under the two representation methods of "adjacency matrix" and "adjacency list", the implementation methods differ.</p>
<p>Basic operations on graphs can be divided into operations on "edges" and operations on "vertices". Their implementations differ depending on whether the graph is represented as an "adjacency matrix" or an "adjacency list".</p>
<h2 id="921-implementation-based-on-adjacency-matrix">9.2.1 &nbsp; Implementation Based on Adjacency Matrix<a class="headerlink" href="#921-implementation-based-on-adjacency-matrix" title="Permanent link">&para;</a></h2>
<p>Given an undirected graph with <span class="arithmatex">\(n\)</span> vertices, the various operations are implemented as shown in Figure 9-7.</p>
<ul>
<li><strong>Adding or removing an edge</strong>: Directly modify the specified edge in the adjacency matrix, using <span class="arithmatex">\(O(1)\)</span> time. Since it is an undirected graph, both directions of the edge need to be updated simultaneously.</li>
<li><strong>Adding a vertex</strong>: Add a row and a column at the end of the adjacency matrix and fill them all with <span class="arithmatex">\(0\)</span>s, using <span class="arithmatex">\(O(n)\)</span> time.</li>
<li><strong>Removing a vertex</strong>: Delete a row and a column in the adjacency matrix. The worst case occurs when removing the first row and column, requiring <span class="arithmatex">\((n-1)^2\)</span> elements to be "moved up and to the left", thus using <span class="arithmatex">\(O(n^2)\)</span> time.</li>
<li><strong>Initialization</strong>: Pass in <span class="arithmatex">\(n\)</span> vertices, initialize a vertex list <code>vertices</code> of length <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> time; initialize an adjacency matrix <code>adjMat</code> of size <span class="arithmatex">\(n \times n\)</span>, using <span class="arithmatex">\(O(n^2)\)</span> time.</li>
<li><strong>Initialization</strong>: Given <span class="arithmatex">\(n\)</span> vertices, initialize a vertex list <code>vertices</code> of length <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> time; initialize an adjacency matrix <code>adjMat</code> of size <span class="arithmatex">\(n \times n\)</span>, using <span class="arithmatex">\(O(n^2)\)</span> time.</li>
</ul>
<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">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label></div>
<div class="tabbed-content">
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<ul>
<li><strong>Adding an edge</strong>: Add the edge at the end of the corresponding vertex's linked list, using <span class="arithmatex">\(O(1)\)</span> time. Since it is an undirected graph, edges in both directions need to be added simultaneously.</li>
<li><strong>Removing an edge</strong>: Find and remove the specified edge in the corresponding vertex's linked list, using <span class="arithmatex">\(O(m)\)</span> time. In an undirected graph, edges in both directions need to be removed simultaneously.</li>
<li><strong>Adding a vertex</strong>: Add a linked list in the adjacency list and set the new vertex as the head node of the list, using <span class="arithmatex">\(O(1)\)</span> time.</li>
<li><strong>Adding a vertex</strong>: Add a linked list to the adjacency list, with the new vertex as the head node, using <span class="arithmatex">\(O(1)\)</span> time.</li>
<li><strong>Removing a vertex</strong>: Traverse the entire adjacency list and remove all edges containing the specified vertex, using <span class="arithmatex">\(O(n + m)\)</span> time.</li>
<li><strong>Initialization</strong>: Create <span class="arithmatex">\(n\)</span> vertices and <span class="arithmatex">\(2m\)</span> edges in the adjacency list, using <span class="arithmatex">\(O(n + m)\)</span> time.</li>
</ul>
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</div>
<p align="center"> Figure 9-8 &nbsp; Initialization, adding and removing edges, adding and removing vertices in adjacency list </p>
<p>The following is the adjacency list code implementation. Compared to Figure 9-8, the actual code has the following differences.</p>
<p>The following code shows the adjacency list implementation. Compared with Figure 9-8, the actual code differs in the following ways.</p>
<ul>
<li>For convenience in adding and removing vertices, and to simplify the code, we use lists (dynamic arrays) instead of linked lists.</li>
<li>A hash table is used to store the adjacency list, where <code>key</code> is the vertex instance and <code>value</code> is the list (linked list) of adjacent vertices for that vertex.</li>
</ul>
<p>Additionally, we use the <code>Vertex</code> class to represent vertices in the adjacency list. The reason for this is: if we used list indices to distinguish different vertices as with adjacency matrices, then to delete the vertex at index <span class="arithmatex">\(i\)</span>, we would need to traverse the entire adjacency list and decrement all indices greater than <span class="arithmatex">\(i\)</span> by <span class="arithmatex">\(1\)</span>, which is very inefficient. However, if each vertex is a unique <code>Vertex</code> instance, deleting a vertex does not require modifying other vertices.</p>
<p>Additionally, we use the <code>Vertex</code> class to represent vertices in the adjacency list for the following reason: if we used list indices to distinguish different vertices, as with adjacency matrices, then to delete the vertex at index <span class="arithmatex">\(i\)</span>, we would need to traverse the entire adjacency list and decrement all indices greater than <span class="arithmatex">\(i\)</span> by <span class="arithmatex">\(1\)</span>, which is very inefficient. However, if each vertex is a unique <code>Vertex</code> instance, deleting one vertex does not require modifying the others.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
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<h2 id="923-efficiency-comparison">9.2.3 &nbsp; Efficiency Comparison<a class="headerlink" href="#923-efficiency-comparison" title="Permanent link">&para;</a></h2>
<p>Assuming the graph has <span class="arithmatex">\(n\)</span> vertices and <span class="arithmatex">\(m\)</span> edges, Table 9-2 compares the time efficiency and space efficiency of adjacency matrices and adjacency lists. Note that the adjacency list (linked list) corresponds to the implementation in this text, while the adjacency list (hash table) refers specifically to the implementation where all linked lists are replaced with hash tables.</p>
<p>Assuming the graph has <span class="arithmatex">\(n\)</span> vertices and <span class="arithmatex">\(m\)</span> edges, Table 9-2 compares the time efficiency and space efficiency of adjacency matrices and adjacency lists. Note that the adjacency list (linked list) corresponds to the implementation used in this section, while the adjacency list (hash table) refers specifically to the implementation where all linked lists are replaced with hash tables.</p>
<p align="center"> Table 9-2 &nbsp; Comparison of adjacency matrix and adjacency list </p>
<div class="center-table">
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
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<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
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<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
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<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
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Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
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<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
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<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
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Chapter 6. Hashing
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<p>Trees represent "one-to-many" relationships, while graphs have a higher degree of freedom and can represent any "many-to-many" relationships. Therefore, we can view 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 methods can also be divided into two types: <u>breadth-first traversal</u> and <u>depth-first traversal</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 that, starting from a certain node, always prioritizes visiting the nearest vertices and expands outward 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><strong>Breadth-first search proceeds from near to far: starting from a given node, it always visits the nearest vertices first and expands outward 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><img alt="Breadth-first search of a graph" class="animation-figure" src="../graph_traversal.assets/graph_bfs.png" /></p>
<p align="center"> Figure 9-9 &nbsp; Breadth-first search of a graph </p>
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<p>To prevent revisiting vertices, we use a hash set <code>visited</code> to record which nodes have been visited.</p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>A hash set can be viewed as a hash table that stores only <code>key</code> without storing <code>value</code>. It can perform addition, deletion, lookup, and modification operations on <code>key</code> in <span class="arithmatex">\(O(1)\)</span> time complexity. Based on the uniqueness of <code>key</code>, hash sets are typically used for data deduplication and similar scenarios.</p>
<p>A hash set can be viewed as a hash table that stores only <code>key</code> without storing <code>value</code>. It supports insertion, deletion, lookup, and update operations on <code>key</code> in <span class="arithmatex">\(O(1)\)</span> time. Based on the uniqueness of <code>key</code>, hash sets are typically used for data deduplication and similar scenarios.</p>
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<p>The algorithm flow of depth-first search is shown in Figure 9-12.</p>
<ul>
<li><strong>Straight dashed lines represent downward recursion</strong>, indicating that a new recursive method has been initiated to visit a new vertex.</li>
<li><strong>Curved dashed lines represent upward backtracking</strong>, indicating that this recursive method has returned to the position where it was initiated.</li>
<li><strong>Curved dashed lines represent upward backtracking</strong>, indicating that this recursive call has returned to the point where it was made.</li>
</ul>
<p>To deepen understanding, it is recommended to combine Figure 9-12 with the code to mentally simulate (or draw out) the entire DFS process, including when each recursive method is initiated and when it returns.</p>
<p>To deepen understanding, it is recommended to combine Figure 9-12 with the code to mentally simulate (or draw out) the entire DFS process, including when each recursive call begins and when it returns.</p>
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<p class="admonition-title">Is the depth-first traversal sequence unique?</p>
<p>Similar to breadth-first search, the order of depth-first traversal sequences is also not unique. Given a certain vertex, exploring in any direction first is valid, meaning the order of adjacent vertices can be arbitrarily shuffled, all being depth-first search.</p>
<p>Similar to breadth-first search, depth-first traversal sequences are also not unique. Given a vertex, any exploration direction may be chosen first; that is, the order of adjacent vertices can be arbitrarily rearranged and still constitute depth-first search.</p>
<p>Taking tree traversal as an example, "root <span class="arithmatex">\(\rightarrow\)</span> left <span class="arithmatex">\(\rightarrow\)</span> right", "left <span class="arithmatex">\(\rightarrow\)</span> root <span class="arithmatex">\(\rightarrow\)</span> right", and "left <span class="arithmatex">\(\rightarrow\)</span> right <span class="arithmatex">\(\rightarrow\)</span> root" correspond to pre-order, in-order, and post-order traversals, respectively. They represent three different traversal priorities, yet all three belong to depth-first search.</p>
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<h3 id="2-complexity-analysis_1">2. &nbsp; Complexity Analysis<a class="headerlink" href="#2-complexity-analysis_1" title="Permanent link">&para;</a></h3>
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<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<ul>
<li>Graphs consist of vertices and edges and can be represented as a set of vertices and a set of edges.</li>
<li>Compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex.</li>
<li>Directed graphs have edges with directionality, connected graphs have all vertices reachable from any vertex, and weighted graphs have edges that each contain a weight variable.</li>
<li>Compared with the linear relationships modeled by linked lists and the divide-and-conquer relationships modeled by trees, the network relationships modeled by graphs offer much greater flexibility and are therefore more complex.</li>
<li>In directed graphs, edges have direction; in connected graphs, every vertex is reachable from any other vertex; and in weighted graphs, each edge carries a weight.</li>
<li>Adjacency matrices use matrices to represent graphs, where each row (column) represents a vertex, and matrix elements represent edges, using <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(0\)</span> to indicate whether two vertices have an edge or not. Adjacency matrices are highly efficient for addition, deletion, lookup, and modification operations, but consume significant space.</li>
<li>Adjacency lists use multiple linked lists to represent graphs, where the <span class="arithmatex">\(i\)</span>-th linked list corresponds to vertex <span class="arithmatex">\(i\)</span> and stores all adjacent vertices of that vertex. Adjacency lists are more space-efficient than adjacency matrices, but have lower time efficiency because they require traversing linked lists to find edges.</li>
<li>Adjacency lists use multiple linked lists to represent a graph: the <span class="arithmatex">\(i\)</span>-th linked list corresponds to vertex <span class="arithmatex">\(i\)</span> and stores all vertices adjacent to it. Compared with adjacency matrices, adjacency lists use less space, but edge lookups are less efficient because the linked list must be traversed.</li>
<li>When linked lists in adjacency lists become too long, they can be converted to red-black trees or hash tables, thereby improving lookup efficiency.</li>
<li>From an algorithmic perspective, adjacency matrices embody "trading space for time", while adjacency lists embody "trading time for space".</li>
<li>Graphs can be used to model various real-world systems, such as social networks and subway lines.</li>
<li>Trees are a special case of graphs, and tree traversal is a special case of graph traversal.</li>
<li>Breadth-first search of graphs is a near-to-far, layer-by-layer expansion search method, typically implemented using a queue.</li>
<li>Depth-first search of graphs is a search method that prioritizes going as far as possible and backtracks when no path remains, commonly implemented using recursion.</li>
<li>Breadth-first search in graphs explores from near to far, expanding layer by layer, and is typically implemented with a queue.</li>
<li>Depth-first search in graphs follows a path as deep as possible and backtracks when it can go no farther, and is commonly implemented with recursion.</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>: Is a path defined as a sequence of vertices or a sequence of edges?</p>
<p>The definitions in different language versions of Wikipedia are inconsistent: the English version states "a path is a sequence of edges", while the Chinese version states "a path is a sequence of vertices". The following is the original English text: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p>
<p>In this text, a path is viewed as a sequence of edges, not a sequence of vertices. This is because there may be multiple edges connecting two vertices, in which case each edge corresponds to a path.</p>
<p><strong>Q</strong>: In a disconnected graph, will there be unreachable vertices?</p>
<p>In a disconnected graph, starting from a certain vertex, at least one vertex cannot be reached. Traversing a disconnected graph requires setting multiple starting points to traverse all connected components of the graph.</p>
<p><strong>Q</strong>: In an adjacency list, is there a requirement for the order of "all vertices connected to that vertex"?</p>
<p>It can be in any order. However, in practical applications, it may be necessary to sort according to specified rules, such as the order in which vertices were added, or the order of vertex values, which helps quickly find vertices "with certain extreme values".</p>
<p>In a disconnected graph, if you start from one vertex, at least one other vertex will be unreachable. To traverse a disconnected graph, you need multiple starting points so that all connected components are covered.</p>
<p><strong>Q</strong>: In an adjacency list, is there any required ordering for the vertices adjacent to a given vertex?</p>
<p>They can appear in any order. In practice, however, they may need to be sorted according to specific rules, such as the order in which vertices were added or the order of vertex values, which helps when quickly finding a vertex with some extreme value.</p>
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