8.4: Graph Representations

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Adjacency Matrix Representation

An adjacency matrix is one of the two common ways to represent a graph. The adjacency matrix shows which nodes are adjacent to one another. Two nodes are adjacent if there is an edge connecting them. In the case of a directed graph, if node \(j\) is adjacent to node \(i\), there is an edge from \(i\) to \(j\). In other words, if \(j\) is adjacent to \(i\), you can get from \(i\) to \(j\) by traversing one edge. For a given graph with \(n\) nodes, the adjacency matrix will have dimensions of \(n\times n\). For an unweighted graph, the adjacency matrix will be populated with boolean values.

For any given node \(i\), you can determine its adjacent nodes by looking at row \(\left(i,\left[1..n\right]\right)\) of the adjacency matrix. A value of true at \(\left(i,j\right)\) indicates that there is an edge from node \(i\) to node \(j\), and false indicating no edge. In an undirected graph, the values of \(\left(i,j\right)\) and \(\left(j,i\right)\) will be equal. In a weighted graph, the boolean values will be replaced by the weight of the edge connecting the two nodes, with a special value that indicates the absence of an edge.

The memory use of an adjacency matrix is \(O(n^{2})\).

Adjacency List Representation

The adjacency list is another common representation of a graph. There are many ways to implement this adjacency representation. One way is to have the graph maintain a list of lists, in which the first list is a list of indices corresponding to each node in the graph. Each of these refer to another list that stores the index of each adjacent node to this one. It might also be useful to associate the weight of each link with the adjacent node in this list.

Example: An undirected graph contains four nodes 1, 2, 3 and 4. 1 is linked to 2 and 3. 2 is linked to 3. 3 is linked to 4.

1 - [2, 3]

2 - [1, 3]

3 - [1, 2, 4]

4 - [3]

It might be useful to store the list of all the nodes in the graph in a hash table. The keys then would correspond to the indices of each node and the value would be a reference to the list of adjacent node indices.

Another implementation might require that each node keep a list of its adjacent nodes.