12.2 $ \mathtt{AdjacencyLists}$: A Graph as a Collection of Lists

Adjacency list representations takes a more vertex-centric approach. There are many different 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 *adj;
(An example is shown in Figure 12.3.) In this particular implementation, we represent each list in $ \mathtt{adj}$ as a subclass of $ \mathtt{ArrayStack}$, because we would like constant time access by position. Other options are also possible. Specifically, we could have implemented $ \mathtt{adj}$ as a $ \mathtt{DLList}$.

Figure 12.3: A graph and its adjacency lists
\includegraphics{figs/graph}



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) {
    for (int k = 0; k < adj[i].size(); k++) {
      if (adj[i].get(k) == j) {
        adj[i].remove(k);
        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):

  bool 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 simply copies the values in $ \mathtt{adj[i]}$ into the output list:

  void outEdges(int i, LisT &edges) {
    for (int k = 0; k < adj[i].size(); k++)
      edges.add(adj[i].get(k));
  }
This clearly takes $ O(\deg(\ensuremath{\mathtt{i}}))$ 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:

  void inEdges(int i, LisT &edges) {
    for (int j = 0; j < n; j++)
      if (adj[j].contains(i))  edges.add(j);
  }
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 $ \mathtt{AdjacencyLists}$ data structure implements the $ \mathtt{Graph}$ interface. An $ \mathtt{AdjacencyLists}$ supports the operations The space used by a $ \mathtt{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:

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

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