Graphs Chapter 12. Chapter 12: Graphs2 Graphs A graph is a data structure that consists of a set of...

Graphs

Chapter 12

Chapter 12: Graphs 2

Graphs

• 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

ORD PVD

MIADFW

SFO

LAX

LGA

HNL

849

802

13871743

1843

10991120

1233337

2555

142


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


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)}


Edge Types

• 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

ORD PVDflight

AA 1206

ORD PVD849

miles


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


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


Terminology

• 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

XU

V

W

Z

Y

a

c

b

e

d

f

g

h

i

j


P1

Terminology (cont.)

• 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

XU

V

W

Z

Y

a

c

b

e

d

f

g

hP2


Terminology (cont.)

• 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

C1

XU

V

W

Z

Y

a

c

b

e

d

f

g

hC2


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


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


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


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


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


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


Edge Class


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


Adjacency List

• An adjacency list representation of a graph uses an array of lists

• One list for each vertex


Adjacency List

• 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

25

1

4 3

52

51 4

4

2 3 5

4 1 2

1

2

3

4

5


Adjacency List (continued)


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

25

1

4 3

0 1 0 0 1

1 0 0 1 1

0 0 0 1 0

0 1 1 0 1

1 1 0 1 0

1

2

3

4

5

1 2 3 4 5


Adjacency Matrix


Overview of the Graph Class Hierarchy


Class AbstractGraph


The ListGraph Class


Performance

• n vertices, m edges• no parallel edges• no self-loops• Bounds are “big-Oh”

AdjacencyList

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) 1 1

Graphs Chapter 12. Chapter 12: Graphs2 Graphs A graph is a data structure that consists of a set of...

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