3223-05-1

CIS 3223. Data Structures and Algorithms

Graphs (1)

1. Basic concepts

Intuitively, a graph represents a network of relations among data items, in which each data item can be related to many other items. Graph is a very flexible data structure that can be used for various purposes.

Formally, a graph G consists of a set V whose members are the vertices (nodes) of G, together with a set E of pairs of vertices from V. These pairs are called edges (or arcs, links) of G.

The size of each graph is measured by |V| and |E|, where 0 ≤ |V|, 0 ≤ |E| ≤ |V²|.

Linear and tree data structures are special cases of graph.

For example, we can have G = < V, E >, where V = {1, 2, 3, 4, 5} and E = {(1, 2), (2, 3), (3, 4), (4, 5), (1, 5), (3, 5), (2, 4)}. The graph looks like:

Please note that the same graph can be drawn in many ways.

Based on the edges, we can further define the following notions:

adjacency: 2 vertices are adjacent if there is an edge between them, and they are neighbors to each other.
path: A sequence of vertices such that each is adjacent to the next.
simple path: In a path all vertices are distinct, except the first and the last may be the same.
cycle: A simple path where the last vertex is the same as the first.

If the pairs comprising an edge are unordered, i.e. (1, 2) or (2, 1) represent the same edge, the graph is called an undirected graph . If the pairs are ordered, the graph is a directed graph or digraph. To draw such a graph, it is necessary to mark the direction of the edges with arrows. Also, the above definitions are slightly different in directed graphs, since in a directed graph, we must follow the direction of the arrows to form a path. In a digraph, vertex A is adjacent to vertex B only if there is a directed edge from A to B (some books use the term in the opposite direction).

Then we can talk about connectivity:

connected undirected graph: There is a (undirected) path from each vertex to every other vertex.
strongly connected digraph: There is a (directed) path from each vertex to every other vertex.
weakly connected digraph: The graph is connected only when we suppress the direction of the edge.

Now we can define a tree as a weakly connected digraph with a unique vertex (the root) from which there is a unique path to each of the other vertices.

A weighted graph has a number attached to each edge, to represent some kind of "distance". In such a graph, the weight of a path can be defined as the sum of the weights on the path.

2. Storage structure

To represent a graph within a program, you can specify each vertex and the set of vertices adjacent to it.

Example: vertex set of adjacent vertices is the following:

0 {1, 2}
1 {2, 3}
2 {}
3 {0, 1, 2}

The above must be a digraph, because 0 is adjacent to 1 but 1 is not adjacent to 0. The graph looks like the following (with '+' for arrowhead):

       0----+1
       |+   /+
       | \ / |
       |  X  |
       | / \ |
       ++   \+
       2+----3

One way to represent a graph is as an array of linked lists:

0 --> 1 --> 2
1 --> 2 --> 3
2
3 --> 0 --> 1 --> 2

In general, the vertices in each list are not necessarily sorted.

Another way to represent a graph is to use an adjacency matrix of boolean values to represent a graph. If the graph contains an edge from 0 to 1, then row 0, column 1 should be True. For instance, the above graph can be represented in the following matrix:

\ 0 1 2 3

0 F T T F

1 F F T T

2 F F F F

3 T T T F

\	0	1	2	3
0	F	T	T	F
1	F	F	T	T
2	F	F	F	F
3	T	T	T	F

In both implementations, if the vertices are not sequentially numbered, but labeled in another way, some kind of mapping between the two will be needed.

Complexity in terms of |V| and |E|:

Space efficiency analysis: sparse and dense graphs.
Time efficiency analysis: search, insert, and delete, of vertex or edge.

A Java implementation of graph is in the textbook.