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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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Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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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
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4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
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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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5.3 Double-Ended Queue
5.3 Deque
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Chapter 6. Hashing
Chapter 6. Hash Table
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Chapter 6. Hashing
Chapter 6. Hash Table
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7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
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8.2 Building a Heap
8.2 Heap Construction Operation
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8.3 Top-K Problem
8.3 Top-k Problem
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10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
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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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11.1 Sorting Algorithms
11.1 Sorting Algorithm
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12.4 Hanoi Tower Problem
12.4 Hanota Problem
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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>
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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>
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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">