9.1 Graphs¶
A graph is a nonlinear data structure made up of vertices and edges. We can represent a graph \(G\) abstractly using a set of vertices \(V\) and a set of edges \(E\). The following example describes a graph with 5 vertices and 7 edges.
If we think of vertices as nodes and edges as references (pointers) connecting those nodes, a graph can be viewed as an extension of a linked list. As Figure 9-1 shows, network relationships (graphs) allow more freedom than linear relationships (linked lists) or divide-and-conquer relationships (trees), making them more complex.
Figure 9-1 Relationships between linked lists, trees, and graphs
9.1.1 Common graph types and terminology¶
Depending on whether their edges have a direction, graphs are classified as undirected graphs or directed graphs, as shown in Figure 9-2.
- In an undirected graph, an edge represents a two-way connection between two vertices, such as a friendship on WeChat or QQ.
- In a directed graph, edges have a direction: the edges \(A \rightarrow B\) and \(A \leftarrow B\) are independent of each other. Examples include following and being followed on Weibo or Douyin.
Figure 9-2 Directed and undirected graphs
Depending on whether all vertices are connected, graphs are classified as connected graphs or disconnected graphs, as shown in Figure 9-3.
- In a connected graph, every other vertex can be reached from any starting vertex.
- In a disconnected graph, at least one vertex cannot be reached from a given starting vertex.
Figure 9-3 Connected and disconnected graphs
We can also assign a weight to each edge to obtain a weighted graph, as shown in Figure 9-4. For example, mobile games such as Honor of Kings calculate an affinity score between players based on the time they spend playing together. A weighted graph can represent this network of affinities.
Figure 9-4 Weighted and unweighted graphs
The following terms are commonly used when discussing graphs.
- Adjacency: Two vertices are adjacent if an edge connects them. In Figure 9-4, the vertices adjacent to vertex 1 are 2, 3, and 5.
- Path: A sequence of edges followed 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 forms a path from vertex 1 to vertex 4.
- Degree: The number of edges incident to a vertex. In a directed graph, the in-degree is the number of edges pointing to the vertex, and the out-degree is the number of edges pointing away from it.
9.1.2 Representing a graph¶
Two common ways to represent a graph are an adjacency matrix and an adjacency list. We will use undirected graphs in the following examples.
1. Adjacency matrix¶
For a graph with \(n\) vertices, an adjacency matrix represents the graph using an \(n \times n\) matrix. Each row (or column) represents a vertex, and the matrix entries represent edges: \(1\) or \(0\) indicates whether an edge exists between two vertices.
As shown in Figure 9-5, let \(M\) be the adjacency matrix and \(V\) the vertex list. The entry \(M[i, j] = 1\) means that an edge connects vertices \(V[i]\) and \(V[j]\), whereas \(M[i, j] = 0\) means that no edge connects them.
Figure 9-5 Representing a graph with an adjacency matrix
An adjacency matrix has the following properties.
- In a simple graph, a vertex cannot connect to itself, so the entries on the main diagonal do not represent meaningful connections.
- In an undirected graph, edges in the two directions are equivalent, so the adjacency matrix is symmetric about its main diagonal.
- Replacing the \(1\) and \(0\) entries with weights allows the matrix to represent a weighted graph.
With an adjacency matrix, we can access a matrix entry directly to retrieve an edge. Adding, removing, looking up, and modifying edges are therefore efficient, each taking \(O(1)\) time. However, the matrix requires \(O(n^2)\) space and uses a substantial amount of memory.
2. Adjacency list¶
An adjacency list represents a graph using \(n\) linked lists whose nodes represent vertices. The \(i\)-th linked list corresponds to vertex \(i\) and stores all of its adjacent vertices (the vertices connected to it). Figure 9-6 shows an example of a graph stored as an adjacency list.
Figure 9-6 Representing a graph with an adjacency list
An adjacency list stores only edges that actually exist. Since the total number of edges is usually much smaller than \(n^2\), it saves space. However, finding an edge requires traversing a linked list, so lookups are slower than with an adjacency matrix.
As Figure 9-6 shows, an adjacency list closely resembles separate chaining in a hash table, so similar techniques can improve its efficiency. For example, a long linked list can be replaced with an AVL tree or a red-black tree to improve lookup time from \(O(n)\) to \(O(\log n)\). Replacing it with a hash table can further reduce the time complexity to \(O(1)\).
9.1.3 Common applications of graphs¶
As shown in Table 9-1, many real-world systems can be modeled as graphs, and their associated problems can be reduced to graph computation problems.
Table 9-1 Common graphs in everyday life
| Vertices | Edges | Graph computation problem | |
|---|---|---|---|
| Social network | Users | Friendships | Recommending potential friends |
| Subway network | Stations | Connections between stations | Recommending the shortest route |
| Solar system | Celestial bodies | Gravitational interactions between bodies | Calculating planetary orbits |