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103 lines
7.2 KiB
Markdown
103 lines
7.2 KiB
Markdown
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comments: true
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---
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# 9.1 Graph
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A <u>graph</u> is a type of nonlinear data structure, consisting of <u>vertices</u> and <u>edges</u>. A graph $G$ can be abstractly represented as a collection of a set of vertices $V$ and a set of edges $E$. The following example shows a graph containing 5 vertices and 7 edges.
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$$
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\begin{aligned}
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V & = \{ 1, 2, 3, 4, 5 \} \newline
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E & = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline
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G & = \{ V, E \} \newline
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\end{aligned}
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$$
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If vertices are viewed as nodes and edges as references (pointers) connecting the nodes, graphs can be seen as a data structure that extends from linked lists. As shown in Figure 9-1, **compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) are more complex due to their higher degree of freedom**.
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{ class="animation-figure" }
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<p align="center"> Figure 9-1 Relationship between linked lists, trees, and graphs </p>
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## 9.1.1 Common types of graphs
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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.
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- In undirected graphs, edges represent a "bidirectional" connection between two vertices, for example, the "friendship" in WeChat or QQ.
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- In directed graphs, edges have directionality, that is, the edges $A \rightarrow B$ and $A \leftarrow B$ are independent of each other, for example, the "follow" and "be followed" relationship on Weibo or TikTok.
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{ class="animation-figure" }
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<p align="center"> Figure 9-2 Directed and undirected graphs </p>
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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.
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- For connected graphs, it is possible to reach any other vertex starting from a certain vertex.
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- For disconnected graphs, there is at least one vertex that cannot be reached from a certain starting vertex.
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{ class="animation-figure" }
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<p align="center"> Figure 9-3 Connected and disconnected graphs </p>
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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.
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{ class="animation-figure" }
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<p align="center"> Figure 9-4 Weighted and unweighted graphs </p>
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Graph data structures include the following commonly used terms.
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- <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.
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- <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.
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- <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.
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## 9.1.2 Representation of graphs
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Common representations of graphs include "adjacency matrices" and "adjacency lists". The following examples use undirected graphs.
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### 1. Adjacency matrix
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Let the number of vertices in the graph be $n$, the <u>adjacency matrix</u> uses an $n \times n$ matrix to represent the graph, where each row (column) represents a vertex, and the matrix elements represent edges, with $1$ or $0$ indicating whether there is an edge between two vertices.
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As shown in Figure 9-5, let the adjacency matrix be $M$, and the list of vertices be $V$, then the matrix element $M[i, j] = 1$ indicates there is an edge between vertex $V[i]$ and vertex $V[j]$, conversely $M[i, j] = 0$ indicates there is no edge between the two vertices.
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{ class="animation-figure" }
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<p align="center"> Figure 9-5 Representation of a graph with an adjacency matrix </p>
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Adjacency matrices have the following characteristics.
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- A vertex cannot be connected to itself, so the elements on the main diagonal of the adjacency matrix are meaningless.
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- For undirected graphs, edges in both directions are equivalent, thus the adjacency matrix is symmetric about the main diagonal.
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- By replacing the elements of the adjacency matrix from $1$ and $0$ to weights, it can represent weighted graphs.
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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 $O(1)$. However, the space complexity of the matrix is $O(n^2)$, which consumes more memory.
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### 2. Adjacency list
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The <u>adjacency list</u> uses $n$ linked lists to represent the graph, with each linked list node representing a vertex. The $i$-th linked list corresponds to vertex $i$ and contains all adjacent vertices (vertices connected to that vertex). Figure 9-6 shows an example of a graph stored using an adjacency list.
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{ class="animation-figure" }
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<p align="center"> Figure 9-6 Representation of a graph with an adjacency list </p>
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The adjacency list only stores actual edges, and the total number of edges is often much less than $n^2$, making it more space-efficient. However, finding edges in the adjacency list requires traversing the linked list, so its time efficiency is not as good as that of the adjacency matrix.
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Observing Figure 9-6, **the structure of the adjacency list is very similar to the "chaining" in hash tables, hence we can use similar methods to optimize efficiency**. For example, when the linked list is long, it can be transformed into an AVL tree or red-black tree, thus optimizing the time efficiency from $O(n)$ to $O(\log n)$; the linked list can also be transformed into a hash table, thus reducing the time complexity to $O(1)$.
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## 9.1.3 Common applications of graphs
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As shown in Table 9-1, many real-world systems can be modeled with graphs, and corresponding problems can be reduced to graph computing problems.
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<p align="center"> Table 9-1 Common graphs in real life </p>
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<div class="center-table" markdown>
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| | Vertices | Edges | Graph Computing Problem |
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| --------------- | ---------------- | --------------------------------------------- | -------------------------------- |
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| Social Networks | Users | Friendships | Potential Friend Recommendations |
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| Subway Lines | Stations | Connectivity Between Stations | Shortest Route Recommendations |
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| Solar System | Celestial Bodies | Gravitational Forces Between Celestial Bodies | Planetary Orbit Calculations |
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</div>
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