1. Graphs
Chapter 12

2. Graphs PVD ORD SFO LGA HNL LAX DFW MIA
A graph is a data structure that consists of a set of vertices and a set of edges between pairs of vertices
Edges represent paths or connections between the vertices
The set of vertices and the set of edges must both be finite and neither one be empty
Example:
A vertex represents an airport
An edge represents a flight route between two airports and stores the mileage of the route
849
PVD
1843
ORD
142
SFO
802
LGA
1743
337
1387
HNL
2555
1099
LAX
1233
DFW
1120
MIA
Chapter 12: Graphs

3. Visual Representation of Graphs
Vertices are represented as points or labeled circles and edges are represented as lines joining the vertices
The physical layout of the vertices and their labeling are not relevant
Chapter 12: Graphs

4. Graphs A graph is a pair (V, E), where V is a set of vertices
E is a set of pairs of vertices, called edges
Example:
V = {A, B, C, D, E}
E = {(A,B), (A,D), (C,E), (D, E)}
Chapter 12: Graphs

5. Edge Types flight AA 1206 ORD PVD 849 miles ORD PVD Directed edge
ordered pair of vertices (u,v)
first vertex u is the origin
second vertex v is the destination
e.g., a flight
Undirected edge
unordered pair of vertices (u,v)
e.g., a flight route
flight
AA 1206
ORD
PVD
849
miles
ORD
PVD
Chapter 12: Graphs

6. Directed and Undirected Graphs
A graph with directed edges is called a directed graph or digraph
Example: flight network
A graph with undirected edges is an undirected graph or simply a graph
Example: route network
Chapter 12: Graphs

7. Weighted Graphs
The edges in a graph may have values associated with them known as their weights
A graph with weighted edges is known as a weighted graph
Chapter 12: Graphs

8. Terminology X U V W Z Y a c b e d f g h i j
End vertices (or endpoints) of an edge
U and V are the endpoints of a
Edges incident on a vertex
a, d, and b are incident on V
Adjacent vertices
U and V are adjacent
Degree of a vertex
X has degree 5
Parallel edges
h and i are parallel edges
Self-loop
j is a self-loop X U V W Z Y a c b e d f g h i j
Chapter 12: Graphs

9. Terminology (cont.) V a b P1 d U X Z P2 h c e W g f Y
A path is a sequence of vertices in which each successive vertex is adjacent to its predecessor
In a simple path, the vertices and edges are distinct except that the first and last vertex may be the same
Examples
P1=(V,X,Z) is a simple path
P2=(U,W,X,Y,W,V) is a path that is not simple V a b P1 d U X Z P2 h c e W g f Y
Chapter 12: Graphs

10. Terminology (cont.) V a b d U X Z C2 h e C1 c W g f Y
A cycle is a simple path in which only the first and final vertices are the same
Simple cycle
cycle such that all its vertices and edges are distinct
Examples
C1=(V,X,Y,W,U,V) is a simple cycle
C2=(U,W,X,Y,W,V,U) is a cycle that is not simple V a b d U X Z C2 h e C1 c W g f Y
Chapter 12: Graphs

11. Subgraphs A subgraph S of a graph G is a graph such that
The vertices of S are a subset of the vertices of G
The edges of S are a subset of the edges of G
A spanning subgraph of G is a subgraph that contains all the vertices of G
Subgraph
Spanning subgraph
Chapter 12: Graphs

12. Non connected graph with two connected components
Connectivity
A graph is connected if there is a path between every pair of vertices
A connected component of a graph G is a maximal connected subgraph of G
Connected graph
Non connected graph with two connected components
Chapter 12: Graphs

13. Trees and Forests A (free) tree is an undirected graph T such that
T is connected
T has no cycles
A forest is an undirected graph without cycles
The connected components of a forest are trees
Tree
Forest
Chapter 12: Graphs

14. Spanning Trees and Forests
A spanning tree of a connected graph is a spanning subgraph that is a tree
A spanning tree is not unique unless the graph is a tree
A spanning forest of a graph is a spanning subgraph that is a forest
Graph
Spanning tree
Chapter 12: Graphs

15. The Graph ADT and Edge Class
Java does not provide a Graph ADT
In making our own, we need to be able to do the following
Create a graph with the specified number of vertices
Iterate through all of the vertices in the graph
Iterate through the vertices that are adjacent to a specified vertex
Determine whether an edge exists between two vertices
Determine the weight of an edge between two vertices
Insert an edge into the graph
Chapter 12: Graphs

16. Vertices and edges
Represent the vertices by integers from 0 up to |V|-1, where |V| represents the cardinality of V.
Define an Edge class containing
Source vertex
Destination vertex
Weight
An Edge object represents a directed edge
For undirected edges, use two Edge objects, one in each direction
Chapter 12: Graphs

17. Edge Class
Chapter 12: Graphs

18. Implementing the Graph ADT
Because graph algorithms have been studied and implemented throughout the history of computer science, many of the original publications of graph algorithms and their implementations did not use an object-oriented approach and did not even use abstract data types
Two representations of graphs are most common
Edges are represented by an array of lists called adjacency lists, where each list stores the vertices adjacent to a particular vertex
Edges are represented by a two dimensional array, called an adjacency matrix
Chapter 12: Graphs

19. Adjacency List
An adjacency list representation of a graph uses an array of lists
One list for each vertex
Chapter 12: Graphs

20. Adjacency List 1 2 3 4 5 2 5 1 1 5 4 5 2 4 2 3 5 4 3 4 1 2
An array (Adj) of lists, one list per vertex
For each vertex u in V,
Adj[u] contains all edges incident on u
Space required Θ(n+m), where n denotes the number of vertices, and m denotes the number of edges
Chapter 12: Graphs

21. Adjacency List (continued)
Chapter 12: Graphs

22. Adjacency Matrix
Uses a two-dimensional array to represent a graph
For an unweighted graph, the entries can be boolean values
Or use 1 for the presence of an edge
For a weighted graph, the matrix would contain the weights 1 1 2 3 4 5 1 5 2 4 3
Chapter 12: Graphs

23. Adjacency Matrix
Chapter 12: Graphs

24. Overview of the Graph Class Hierarchy
Chapter 12: Graphs

25. Class AbstractGraph
Chapter 12: Graphs

26. The ListGraph Class
Chapter 12: Graphs

27. Performance Adjacency List Adjacency Matrix Space n + m n2
n vertices, m edges
no parallel edges
no self-loops
Bounds are “big-Oh”
Adjacency List
Adjacency Matrix
Space
n + m n2
Iterate over incident edges of v
deg(v) n
isEdge (v, w)
min(deg(v), deg(w)) 1
insert(v, w, o)
Chapter 12: Graphs