Graph representations
Mathematical
Mathematically, a graph G is defined by a set V of vertices and a set E of edges. We write G=(V,E).
- V can be a set of anything (e.g. V={ my dog Tika, my cat Mittens, Mr. Ed the talking horse }).
- E must be a subset of the set { {u,v} : u,v ∈ V } (the unordered pairs of vertices in V).
If the graph is directed, then the edges are ordered pairs of vertices.
That is edge (u,v) is different than edge (v,u).
For example, the set of out-neighbors of a vertex w in a directed graph G=(V,E)
can be expressed as {u ∈ V : (w,u) ∈ E } .
Textual (e.g. in a file)
The digraph G=(V,E) where V={1,2,3,4} and E={(1,2), (1,3), (2,4), (3,4)} can be
represented in a file as:
1: 2 3
2: 4
3: 4
4:
Or possibly as
1
2
3
4
1 2
1 3
2 4
3 4
Data structures
Adjacency list representation:
- An array of vertices, where the ith vertex is stored at the ith position of the array. With the ith vertex we also store a doubly linked list of the neighbors of the vertex.
- Takes O(n+m) space, operations involving a vertex w typically take O(degree(w)) time.
Adjacency matrix representation:
- An n by n matrix M of boolean values, where M[i,j] is true if there is an edge from the ith vertex to the jth vertex.
- Takes O(n2) space. Finding all neighbors of a vertex takes Θ(n) time. Checking existence of an edge takes O(1) time.
Hash table of Hash tables:
- A hash table of vertices, each vertex is stored by ID (an integer counter) or some other hashable key (such as a string). With each vertex W, the neighbors of W are stored in another hash table. (So the data structure is a hash table of hash tables.)
- Takes O(n+m) space, most operations are constant time or take time proportional to the degree of the vertex involved.
For a concrete example of a Graph and Digraph class in C++,
see program assignment 2 at ClassS04CS141/Prog2 and ClassS04CS141/Prog2Soln .
References