12.2 AdjacencyLists: 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 $\ensuremath{\mathtt{(i,j)}}\in E$.

    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 12.3.) 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.

Figure 12.3: A graph and its adjacency lists

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

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 $O(\deg(\ensuremath{\mathtt{i}}))$ time, where $\deg(\ensuremath{\mathtt{i}})$ (the degree of $\ensuremath{\mathtt{i}}$) counts the number of edges in $E$ that have $\ensuremath{\mathtt{i}}$ 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 $O(\deg(\ensuremath{\mathtt{i}}))$ 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 $O(\ensuremath{\mathtt{n}} + \ensuremath{\mathtt{m}})$ time.

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

Theorem 12..2   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 $O(\deg(\ensuremath{\mathtt{i}}))$ time per operation;
• $\mathtt{outEdges(i)}$ in constant time per operation; and
• $\mathtt{inEdges(i)}$ in $O(\ensuremath{\mathtt{n}}+\ensuremath{\mathtt{m}})$ time per operation.

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

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 $\ensuremath{\mathtt{(j,i)}}\in E$? 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.