12.2: A Graph as a Collection of Lists

Adjacency list representations of graphs take a more vertex-centric approach. There are many possible implementations of adjacency lists. In this section, we present a simple one. At the end of the section, we discuss different possibilities. In an adjacency list representation, the graph \(G=(V,E)\) is represented as an array, \(\mathtt{adj}\), of lists. The list \(\mathtt{adj[i]}\) contains a list of all the vertices adjacent to vertex \(\mathtt{i}\). That is, it contains every index \(\mathtt{j}\) such that \(\mathtt{(i,j)}\in E\).

    int n;
    List<Integer>[] adj;
    AdjacencyLists(int n0) {
        n = n0;
        adj = (List<Integer>[])new List[n];
        for (int i = 0; i < n; i++) 
            adj[i] = new ArrayStack<Integer>(Integer.class);
    }

(An example is shown in Figure \(\PageIndex{1}\).) In this particular implementation, we represent each list in \(\mathtt{adj}\) as an ArrayStack, because we would like constant time access by position. Other options are also possible. Specifically, we could have implemented \(\mathtt{adj}\) as a DLList.

0	1	2	3	4	5	6	7	8	9	10	11
1	0	1	2	0	1	5	6	4	8	9	10
4	2	3	7	5	2	2	3	9	5	6	7
	6	6		8	6	7	11		10	11
	5				9	10
					4

Figure \(\PageIndex{1}\): A graph and its adjacency lists

The \(\mathtt{addEdge(i,j)}\) operation just appends the value \(\mathtt{j}\) to the list \(\mathtt{adj[i]}\):

    void addEdge(int i, int j) {
        adj[i].add(j);
    }

This takes constant time.

The \(\mathtt{removeEdge(i,j)}\) operation searches through the list \(\mathtt{adj[i]}\) until it finds \(\mathtt{j}\) and then removes it:

    void removeEdge(int i, int j) {
        Iterator<Integer> it = adj[i].iterator();
        while (it.hasNext()) {
            if (it.next() == j) {
                it.remove();
                return;
            }
        }    
    }

This takes \(O(\deg(\mathtt{i}))\) time, where \(\deg(\mathtt{i})\) (the degree of \(\mathtt{i}\)) counts the number of edges in \(E\) that have \(\mathtt{i}\) as their source.

The \(\mathtt{hasEdge(i,j)}\) operation is similar; it searches through the list \(\mathtt{adj[i]}\) until it finds \(\mathtt{j}\) (and returns true), or reaches the end of the list (and returns false):

    boolean hasEdge(int i, int j) {
        return adj[i].contains(j);
    }

This also takes \(O(\deg(\mathtt{i}))\) time.

The \(\mathtt{outEdges(i)}\) operation is very simple; it returns the list \(\mathtt{adj[i]}\) :

    List<Integer> outEdges(int i) {
        return adj[i];
    }

This clearly takes constant time.

The \(\mathtt{inEdges(i)}\) operation is much more work. It scans over every vertex \(j\) checking if the edge \(\mathtt{(i,j)}\) exists and, if so, adding \(\mathtt{j}\) to the output list:

    List<Integer> inEdges(int i) {
        List<Integer> edges = new ArrayStack<Integer>(Integer.class);
        for (int j = 0; j < n; j++)
            if (adj[j].contains(i))    edges.add(j);
        return edges;
    }

This operation is very slow. It scans the adjacency list of every vertex, so it takes \(O(\mathtt{n} + \mathtt{m})\) time.

The following theorem summarizes the performance of the above data structure:

Theorem \(\PageIndex{1}\)

The AdjacencyLists data structure implements the Graph interface. An AdjacencyLists supports the operations

• \(\mathtt{addEdge(i,j)}\) in constant time per operation;
• \(\mathtt{removeEdge(i,j)}\) and \(\mathtt{hasEdge(i,j)}\) in \(O(\deg(\mathtt{i}))\) time per operation;
• \(\mathtt{outEdges(i)}\) in constant time per operation; and
• \(\mathtt{inEdges(i)}\) in \(O(\mathtt{n}+\mathtt{m})\) time per operation.

The space used by a AdjacencyLists is \(O(\mathtt{n}+\mathtt{m})\).

As alluded to earlier, there are many different choices to be made when implementing a graph as an adjacency list. Some questions that come up include:

• What type of collection should be used to store each element of \(\mathtt{adj}\)? One could use an array-based list, a linked-list, or even a hashtable.
• Should there be a second adjacency list, \(\mathtt{inadj}\), that stores, for each \(\mathtt{i}\), the list of vertices, \(\mathtt{j}\), such that \(\mathtt{(j,i)}\in E\)? This can greatly reduce the running-time of the \(\mathtt{inEdges(i)}\) operation, but requires slightly more work when adding or removing edges.
• Should the entry for the edge \(\mathtt{(i,j)}\) in \(\mathtt{adj[i]}\) be linked by a reference to the corresponding entry in \(\mathtt{inadj[j]}\)?
• Should edges be first-class objects with their own associated data? In this way, \(\mathtt{adj}\) would contain lists of edges rather than lists of vertices (integers).

Most of these questions come down to a tradeoff between complexity (and space) of implementation and performance features of the implementation.