Graphs Chapter Introduction¶
1. Graph Terminology and Implementation¶
Graphs provide the ultimate in data structure flexibility. A graph consists of a set of nodes, and a set of edges where an edge connects two nodes. Trees and lists can be viewed as special cases of graphs.
Graphs are used to model both real-world systems and abstract problems, and are the data structure of choice in many applications. Here is a small sampling of the types of problems that graphs are routinely used for.
Modeling connectivity in computer and communications networks.
Representing an abstract map as a set of locations with distances between locations. This can be used to compute shortest routes between locations such as in a GPS routefinder.
Modeling flow capacities in transportation networks to find which links create the bottlenecks.
Finding a path from a starting condition to a goal condition. This is a common way to model problems in artificial intelligence applications and computerized game players.
Modeling computer algorithms, to show transitions from one program state to another.
Finding an acceptable order for finishing subtasks in a complex activity, such as constructing large buildings.
Modeling relationships such as family trees, business or military organizations, and scientific taxonomies.
The rest of this module covers some basic graph terminology. The following modules will describe fundamental representations for graphs, provide a reference implementation, and cover core graph algorithms including traversal, topological sort, shortest paths algorithms, and algorithms to find the minimal-cost spanning tree. Besides being useful and interesting in their own right, these algorithms illustrate the use of many other data structures presented throughout the course.
A graph \(\mathbf{G} = (\mathbf{V}, \mathbf{E})\) consists of a set of vertices \(\mathbf{V}\) and a set of edges \(\mathbf{E}\), such that each edge in \(\mathbf{E}\) is a connection between a pair of vertices in \(\mathbf{V}\). 1 The number of vertices is written \(|\mathbf{V}|\), and the number of edges is written \(|\mathbf{E}|\). \(|\mathbf{E}|\) can range from zero to a maximum of \(|\mathbf{V}|^2 - |\mathbf{V}|\).
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Some graph applications require that a given pair of vertices can have multiple or parallel edges connecting them, or that a vertex can have an edge to itself. However, the applications discussed here do not require either of these special cases. To simplify our graph API, we will assume that there are no dupicate edges, and no edges that connect a node to itself.
A graph whose edges are not directed is called an undirected graph, as shown in part (a) of the following figure. A graph with edges directed from one vertex to another (as in (b)) is called a directed graph or digraph. A graph with labels associated with its vertices (as in (c)) is called a labeled graph. Associated with each edge may be a cost or weight. A graph whose edges have weights (as in (c)) is said to be a weighted graph.
Figure 1: Some types of graphs.
An edge connecting Vertices \(a\) and \(b\) is written \((a, b)\). Such an edge is said to be incident with Vertices \(a\) and \(b\). The two vertices are said to be adjacent. If the edge is directed from \(a\) to \(b\), then we say that \(a\) is adjacent to \(b\), and \(b\) is adjacent from \(a\). The degree of a vertex is the number of edges it is incident with. For example, Vertex \(e\) below has a degree of three.
In a directed graph, the out degree for a vertex is the number of neighbors adjacent from it (or the number of edges going out from it), while the in degree is the number of neighbors adjacent to it (or the number of edges coming in to it). In (c) above, the in degree of Vertex 1 is two, and its out degree is one.
A sequence of vertices \(v_1, v_2, ..., v_n\) forms a path of length \(n-1\) if there exist edges from \(v_i\) to \(v_{i+1}\) for \(1 \leq i < n\). A path is a simple path if all vertices on the path are distinct. The length of a path is the number of edges it contains. A cycle is a path of length three or more that connects some vertex \(v_1\) to itself. A cycle is a simple cycle if the path is simple, except for the first and last vertices being the same.
An undirected graph is a connected graph if there is at least one path from any vertex to any other. The maximally connected subgraphs of an undirected graph are called connected components. For example, this figure shows an undirected graph with three connected components.
A graph with relatively few edges is called a sparse graph, while a graph with many edges is called a dense graph. A graph containing all possible edges is said to be a complete graph. A subgraph \(\mathbf{S}\) is formed from graph \(\mathbf{G}\) by selecting a subset \(\mathbf{V}_s\) of \(\mathbf{G}\)’s vertices and a subset \(\mathbf{E}_s\) of \(\mathbf{G}\) ‘s edges such that for every edge \(e \in \mathbf{E}_s\), both vertices of \(e\) are in \(\mathbf{V}_s\). Any subgraph of \(V\) where all vertices in the graph connect to all other vertices in the subgraph is called a clique.
A graph without cycles is called an acyclic graph. Thus, a directed graph without cycles is called a directed acyclic graph or DAG.
A free tree is a connected, undirected graph with no simple cycles. An equivalent definition is that a free tree is connected and has \(|\mathbf{V}| - 1\) edges.
1.1. Graph Representations¶
There are two commonly used methods for representing graphs. The adjacency matrix for a graph is a \(|\mathbf{V}| \times |\mathbf{V}|\) array. We typically label the vertices from \(v_0\) through \(v_{|\mathbf{V}|-1}\). Row \(i\) of the adjacency matrix contains entries for Vertex \(v_i\). Column \(j\) in row \(i\) is marked if there is an edge from \(v_i\) to \(v_j\) and is not marked otherwise. The space requirements for the adjacency matrix are \(\Theta(|\mathbf{V}|^2)\).
The second common representation for graphs is the adjacency list. The adjacency list is an array of linked lists. The array is \(|\mathbf{V}|\) items long, with position \(i\) storing a pointer to the linked list of edges for Vertex \(v_i\). This linked list represents the edges by the vertices that are adjacent to Vertex \(v_i\).
Here is an example of the two representations on a directed graph. The entry for Vertex 0 stores 1 and 4 because there are two edges in the graph leaving Vertex 0, with one going to Vertex 1 and one going to Vertex 4. The list for Vertex 2 stores an entry for Vertex 4 because there is an edge from Vertex 2 to Vertex 4, but no entry for Vertex 3 because this edge comes into Vertex 2 rather than going out.
Figure 7: Representing a directed graph.
Both the adjacency matrix and the adjacency list can be used to store directed or undirected graphs. Each edge of an undirected graph connecting Vertices \(u\) and \(v\) is represented by two directed edges: one from \(u\) to \(v\) and one from \(v\) to \(u\). Here is an example of the two representations on an undirected graph. We see that there are twice as many edge entries in both the adjacency matrix and the adjacency list. For example, for the undirected graph, the list for Vertex 2 stores an entry for both Vertex 3 and Vertex 4.
Figure 8: Representing an undirected graph.
The storage requirements for the adjacency list depend on both the number of edges and the number of vertices in the graph. There must be an array entry for each vertex (even if the vertex is not adjacent to any other vertex and thus has no elements on its linked list), and each edge must appear on one of the lists. Thus, the cost is \(\Theta(|\mathbf{V}| + |\mathbf{E}|)\).
Sometimes we want to store weights or distances with each each edge, such as in Figure 1 (c). This is easy with the adjacency matrix, where we will just store values for the weights in the matrix. In Figures 7 and 8 we store a value of “1” at each position just to show that the edge exists. That could have been done using a single bit, but since bit manipulation is typically complicated in most programming languages, an implementation might store a byte or an integer at each matrix position. For a weighted graph, we would need to store at each position in the matrix enough space to represent the weight, which might typically be an integer.
The adjacency list needs to explicitly store a weight with each edge. In the adjacency list shown below, each linked list node is shown storing two values. The first is the index for the neighbor at the end of the associated edge. The second is the value for the weight. As with the adjacency matrix, this value requires space to represent, typically an integer.
Which graph representation is more space efficient depends on the number of edges in the graph. The adjacency list stores information only for those edges that actually appear in the graph, while the adjacency matrix requires space for each potential edge, whether it exists or not. However, the adjacency matrix requires no overhead for pointers, which can be a substantial cost, especially if the only information stored for an edge is one bit to indicate its existence. As the graph becomes denser, the adjacency matrix becomes relatively more space efficient. Sparse graphs are likely to have their adjacency list representation be more space efficient.
Example 1
Assume that a vertex index requires two bytes, a pointer requires four bytes, and an edge weight requires two bytes. Then, each link node in the adjacency list needs \(2 + 2 + 4 = 8\) bytes. The adjacency matrix for the directed graph above requires \(2 |\mathbf{V}^2| = 50\) bytes while the adjacency list requires \(4 |\mathbf{V}| + 8 |\mathbf{E}| = 68\) bytes. For the undirected version of the graph above, the adjacency matrix requires the same space as before, while the adjacency list requires \(4 |\mathbf{V}| + 8 |\mathbf{E}| = 116\) bytes (because there are now 12 edges represented instead of 6).
The adjacency matrix often requires a higher asymptotic cost for an algorithm than would result if the adjacency list were used. The reason is that it is common for a graph algorithm to visit each neighbor of each vertex. Using the adjacency list, only the actual edges connecting a vertex to its neighbors are examined. However, the adjacency matrix must look at each of its \(|\mathbf{V}|\) potential edges, yielding a total cost of \(\Theta(|\mathbf{V}^2|)\) time when the algorithm might otherwise require only \(\Theta(|\mathbf{V}| + |\mathbf{E}|)\) time. This is a considerable disadvantage when the graph is sparse, but not when the graph is closer to full.
