Introduction to Data Structure

6-1Been-Chian Chien, Wei-Pang Yang, and Wen-Yang Li

nChapter 6 Graphs

Introduction to Data Structure

CHAPTER 6

GRAPHS

6.1 The Graph Abstract Data Type

6.2 Elementary Graph Operations

6.3 Minimum Cost Spanning Trees

6.4 Shortest Paths and Transitive Closure

6.5 Activity Networks


nChapter 6 Graphs

Chapter 1 Basic Concepts

Chapter 2 Arrays

Chapter 3 Stacks and Queues

Chapter 4 Linked Lists

Chapter 5 Trees

Chapter 6 Graph

Chapter 7 Sorting

Chapter 8 Hashing

Chapter 9 Heap Structures

Chapter 10 Search Structures

Contents


nChapter 6 Graphs

6.1 The Graph Abstract Data Type

structure Graph is objects: a nonempty set of vertices and a set of undirected edges, where

each edge is a pair of vertices functions: for all graph Graph, v, v1, and v2 Vertices

Graph Create() ::= return an empty graphGraph InsertVertex(graph, v) ::= return a graph with v inserted. Graph InsertEdge(graph, v1, v2) ::= return a graph with a new edge

(v1, v2).Graph DeleteVertex(graph, v) ::= return a graph with v and all its

incident edges removed.Graph DeleteEdge(graph, v1, v2) ::= return a graph with the edge

(v1, v2) removed.Boolean IsEmpty(graph) ::= if (graph==empty) return TRUE

else return FALSEList Adjacent(graph, v) ::= return a list of all vertices adjacent to v.

p. 263


nChapter 6 Graphs

6.1.1 Introduction 1736, Euler Koenigberg Bridge Problem:

Euler showed: there is a walk starting at any vertex, going through each edge exactly once and terminating at the start vertex iff the degree of each vertex is even

called Eulerian walk. the degree of a vertex: # of edges incident to a vertex

J. R. Newman: the world of Mathematics, 1956, p.573-580

C

A

B

D

g

c d

e

b

f

a

attaching


nChapter 6 Graphs

A Graph, G = (V, E), consists of two sets V and E V: finite non-empty set of vertices E: set of pairs of vertices, edges, e.g.(1,2) or 1,2

Note: V(G): set of vertices of graph G E(G): set of edges of graph G

Undirected Graph – the pair of vertices representing any edge is unordered. Thus, the pairs (V1,V2) and (V2,V1) represent the same edge.

Directed Graph – each edge is represented by a directed pairs V1, V2 Note: V1, V2 and V2, V1 represent two different edges.

6.1.2 Definitions

V1 V2

1

2

3

V(G) = {1, 2, 3}E(G) = {1,2 2,3 3,1}


nChapter 6 Graphs

Examples for Graph

0

1 2

3

0

1 2

3 4 5 6

G1 G2

0

1

2

G3

V(G1)={0,1,2,3} E(G1)={(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)}

V(G2)={0,1,2,3,4,5,6} E(G2)={(0,1),(0,2),(1,3),(1,4),(2,5),(2,6)}

V(G3)={0,1,2} E(G3)={0,1, 1,0, 1,2}

Examples for Graph

• Note: Graph G2 is also a tree, Tree is a special case of graph.


nChapter 6 Graphs

(a)

2

1

0

3

(b)

0 2

1

Graph with a self edgeMultigraph:multiple occurrencesof the same edge

Examples for Graphlike Structure

Not consider as a graph in this book


nChapter 6 Graphs

A Complete Graph is a graph that has the maximum number of edges

For undirected graph with n vertices, the maximum number of edges is n(n-1)/2

For directed graph with n vertices, the maximum number of edges is n(n-1)

Example: G1 is a complete graph

Complete Graph

0

1 2

3n(n-1)/2 = 6


nChapter 6 Graphs

If (v0, v1) is an edge in an undirected graph, Adjacent: v0 and v1 are adjacent Incident: The edge (v0, v1) is incident on vertices v0 and v1

If <v0, v1> is an edge in a directed graph Adjacent: v0 is adjacent to v1, and v1 is adjacent from v0 Incident: The edge <v0, v1> is incident on v0 and v1

Adjacent and Incident

0

1

2

G3

0

1 2

3

The vertices adjacent to vertex 2: 0, 1 and 2

The edge incident on vertex 2: (0,2), (1,2) and (2,3)

The vertices adjacent to vertex 1: 0,

The edge incident on vertex 1: <0,1>, <1,0> and <1,2>


nChapter 6 Graphs

Subgraph Subgraph

A subgraph of G is a graph G’ →

• V(G’) V(G)

• E(G’) E(G) e.g. some of the subgraphs of G1:

00

1 2 3

1 2

0

1 2

3

(i) (ii) (iii) (iv)G1

0

1 2

3


nChapter 6 Graphs

0

0

1

0

1

2

0

1

2

(i) (ii) (iii) (iv)

0

1

2

G3

Subgraph (cont.)

e.g. some of the subgraphs of G3:

If G is connected with n vertices then its connected subgraph at least has n-1 edges.

• Why?


nChapter 6 Graphs

Path: from vp to vq in G is

a sequence of vertices vp, vi1, vi2,...,vin, vp (vp, vi1), (vi1, vi2),..., (vin, vq) E(G) if G is undirected

or vp, vi1, vi1, vi2,..., vin, vq E(G) if G is directed

eg.

Length: The length of a path is the number of edges on it eg. Length of path 1,2,4 is 2

1

2 3

4

- 1,2,4 is a path- 1,2,3 is not a path- 1,2,4,2 is a path

] E(G)(2,3) [

Path


nChapter 6 Graphs

Simple Path: is a path in which all vertices except possibly the first and last are distinct.

• e.g.1. 1,2,4 is a simple path, path: 1,2,4,2 is not a simple path.

• e.g.2.

1

2

3

1,2,3,1 also a simple path with length 3.1,2,3,1,2? path, length 4, not simple.1,2,3,1,3 not a path.

1

2 3

4

- 1,2,4 is a path- 1,2,3 is not a path- 1,2,4,2 is a path

Simple Path


nChapter 6 Graphs

Cycle Cycle: is a simple path, first and last vertices are same.

• eg. 1,2,3,1.

• Note: Directed cycle, Directed path.

Acyclic graph

1

2

30

1 2

3 4 5 6

G2

tree (acyclic graph)


nChapter 6 Graphs

Connected Connected

Two vertices V1 and V2 are Connected:

if in an undirected graph G,

a path in G from V1 to V2 (or from V2 to V1 undirected)∵

Strongly Connected V1, V2 are Strongly Connected:

if in a directed graph (digraph) G’, a path

in G from V1 to V2 and also from V2 to V1

Graph (Strongly) Connected graph connected (graph strongly connected)

if Vi, Vj V(G), a path from Vi to Vj in G

stronglyconnected

不是stronglyconnected

1

2 3

4

connected connected


nChapter 6 Graphs

Connected Component Connected Component (Strongly Connected Component):

is a maximal connected subgraph.

eg.

for graph for digraph

1

2

3

4

Two connected components of G1

1

2

1

2

3

Two strongly connected components of G2

1

2 3

4

3G2

G1


nChapter 6 Graphs

Degree Degree of Vertex

is the # of edges incident to that vertex eg.

In Direct Graph

If G has n vertices and e edge, di = degree of Vi, then

3 degree of V3 = 3

n

1ii2

1de

in-degreeout-degree

in-degree=1out-degree=2

degree=3


nChapter 6 Graphs

Network Network

A graph with weighted edges is called. eg.

台北

花蓮

高雄

台中

新竹

90 120

100

210

150 Diagraph G = (V,E)

Network N = (G,W) is a diagraph G together with a real valued for w: E R.

R is a real number w(u,v) is the weight of edge (arc) (u,v) E.


nChapter 6 Graphs

Method 1: Adjacency Matrix

Method 2: Adjacency Lists

Method 3: Adjacency Multilists

6.1.3 Graph Representations


nChapter 6 Graphs

Adjacency Matrix Let G = (V, E) with n vertices, n 1.

The adjacency matrix of G is a 2-dimensional n n matrix, A

A(i, j) = 1 iff (vi, vj) ( vi, vj for a diagraph) E(G),

A(i, j) = 0 otherwise.

6.1.3.1 Adjacency Matrix

eg.

1

2 3

4

101004

210013

100012

201101

degree =

di4321

- symmetric- space need n2 bits, but half can be saved

6 / 2 = 3 ... edges

1

0

]][[A n

j

jiA


nChapter 6 Graphs

Adjacency Matrix: Examples

0

1

2

111

00002

21011

10100

210- not be symmetric

out-degree =

in-degree

1

0

]][[A n

j

ji= dj

1

0

]][[A n

i

ji

Eg. 2.


nChapter 6 Graphs

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Examples for Adjacency Matrix

1

0

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3

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5

6

7G4

Adjacency Matrix: Examples


nChapter 6 Graphs

Questions:

• How many edges e are there in G?

• Determine the # of edges in G?

• Is G connected?

Answers:

• Need check entries of the matrix. check n2 – n time.

spend O(n2) time to determine e Note:

<< n2/2

n

1iid

2

1e

(n: Diagonal entries are zero, need not check it)

Adjacency Matrix: Determine Edges

0

1

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0

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0

1

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0

1

1

0

e = 7 << n2/2 = 32


nChapter 6 Graphs

Adjacency Lists

Adjacency matrices for G3

Adjacency list: edge e 2e nodes. Easy random access for any particular vertex

ListOrthogonal list

0

1

2

0 1 0

1 0 1

0 0 0

V0

V1

V2 0

1 0

0 2 0

V0

V1

V2 0

1 0

2 0 0

Head Nodes(sequential)

6.1.3.2 Adjacency Lists

G3


nChapter 6 Graphs

[0]

[1]

[2]

[3]

1 2 3

0 2 3

0 1 3

0 1 2

An undirected graph with n vertices and e edges

n head nodes and 2e list nodes

0

1 2

3

Adjacency Lists: Example G1


nChapter 6 Graphs

[0]

[1]

[2]

[3]

[4]

[5]

[6]

[7]

1 2

0 3

0 3

1 2

5

4 6

5 7

6

1

0

2

3

4

5

6

7G4

Adjacency Lists: Example G4


nChapter 6 Graphs

Number of nodes in a graph:

= the number of nodes in adjacency list

Number of edges in a graph

= determined in O(n+e)

Out-degree of a vertex in a directed graph

= the number of nodes in its adjacency list

In-degree of a vertex in a directed graph:

??? traverse the whole data structure

0

1

2

V0

V1

V2 0

1 0

2 0 0

G3

Adjacency Lists: Interesting Operations

n vertices# of edges

vs. O(n2) in adjacency matrix


nChapter 6 Graphs

[0]

[1]

[2]

1 NULL

0 NULL

1 NULL

determine in-degree of a vertex in a fast way.

0

1

2

Inverse Adjacency List

Adjacency Lists: Inverse Adjacency Lists


nChapter 6 Graphs

tail head column link for head row link for tail

Adjacency Lists: Orthogonal Representation

Orthogonal representation

0

1

0

1

0

0

0

1

0

Sparse Matrix ???

0 1 2

0

1

2

0 1

1 0 1 2

head node


nChapter 6 Graphs

6.1.3.3 Adjacency Multilists Adjacency Multilists ( 以 edge 為主 )

mark vertex1 vertex2 path1 path2

0

1 2

3

N1 N2N3

N4N5 N6

vertex 0: N1N2N3vertex 1: N1N4N5vertex 2: N2N4N6vertex 3: N3N5N6

[0]

[1]

[2]

[3]

0 1 N2 N4

0 2 N3 N4

0 3 N5

1 2 N5 N6

1 3 N6

2 3

N1

N2

N3

N4

N5

N6

edge(0,1)

edge(0,2)

edge(0,3)

edge(1,2)

edge(1,3)

edge(2,3)


nChapter 6 Graphs

Adjacency Multilists (cont.)

0

1 2

3

N1 N2N3

N4N5 N6

vertex 0: N1N2N3vertex 1: N1N4N5vertex 2: N2N4N6vertex 3: N3N5N6

mark vertex1 vertex2 Link 1 for V1 Link 2 for V2

Node Structures

[0]

[1]

[2]

[3]

0 1 N2 N4

0 2 N3 N4

0 3 N5

1 2 N5 N6

1 3 N6

2 3

N1

N2

N3

N4

N5

N6

edge(0,1)

edge(0,2)

edge(0,3)

edge(1,2)

edge(1,3)

edge(2,3)


nChapter 6 Graphs

Adjacency MultiLists vs. Adjacency List Adjacency Multilists ( 以 edge 為主 )

e=6 6 nodes

Adjacency List

0

1 2

3

e=6 12 nodes

0

1 2

3

N1 N2N3

N4N5 N6


nChapter 6 Graphs

TraversalGiven G = (V, E) and vertex v, find or visit all w V, such that w connects v.

Method 1: Depth First Search (DFS) preorder tree traversal

Method 2: Breadth First Search (BFS) level order tree traversal

Connected Components (Application 1 of Graph traversal) Spanning Trees (Application 2 of Graph traversal) Minimum Cost Spanning Tree (Application 3 of Graph

traversal) …

6.2 Elementary Graph Operations

1

0

2

3

4

5

6

7G4


nChapter 6 Graphs

Graph TraversalGiven: an undirected graph G = (V, E), a vertex v V(G)Interested in: visiting all vertices connected to v. (reachable)Approach: 1. Depth First Search (DFS). 2. Breadth First Search (BFS).

1

2 3

4 5 6

DFS: 1,2,4,5,3,6BFS: 1,2,3,4,5,6

4

2

1

stack

1 2 3

Queue

Graph Operations: Graph Traversal


nChapter 6 Graphs

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1 2

3 4 5 6

7

[0]

[1]

[2]

[3]

[4]

[5]

[6]

[7]

1 2

0 3

0 5

1 7

2 7

3

4

6

1 7

2 7

4 5 6

Example

6.2.1 Depth First Search

Adjacency Lists


nChapter 6 Graphs

Example 6.1: Depth first search v0, v1, v3, v7, v4, v5, v2, v6

Program 6.1, p.273

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3 4 5 6

7

[0]

[1]

[2]

[3]

[4]

[5]

[6]

[7]

1 2

0 3

0 5

1 7

2 7

3

4

6

1 7

2 7

4 5 6

Depth First Search: Example

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1

3

7

stack

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1

3

7

5

2 4


nChapter 6 Graphs

Depth First SearchProcedure DFS(v) (* Array VISITED(n) 0 initially and VISITED(i) 1, if has visited*) VISITED(v) 1 print(v) for each vertex w adjacent to v do if VISITED(w) = 0 then call DFS(w) end end DFS

Depth First Search: Algorithm

VISITED

1 2 3 4 5 6 7 8

1

2 3

4 5 6 7

8

Output DFS: 1,2,4,8,5,6,3,7

V1

V2

V3

V2

V4

V5

V4

V8

V8

V5

V6

V7

V5 V6

V3

V3

V7

V7


nChapter 6 Graphs

Time complexity: Adjacency list

Time complexity: Adjacency matrix1 2 8

1

2

3

8

1

2

3

4

5

6

7

8

2 3 0

1 4 5 0

e edges 2e nodes

1 6 7 0

Depth First Search: Time Complexity

Adjacency matrix: O(n2) n: # of vertex

Adjacency list: O(e) e: # of edges


nChapter 6 Graphs

Example 6.2: Breadth first search v0, v1, v2, v3, v4, v5, v6, v7

Program 6:2 , p.275

0

1 2

3 4 5 6

7

[0]

[1]

[2]

[3]

[4]

[5]

[6]

[7]

1 2

0 3

0 5

1 7

2 7

3

4

6

1 7

2 7

4 5 6

6.2.2 Breadth First Search


nChapter 6 Graphs

Breadth First Search

procedure BFS(v) VISITED (v) 1 print (v) initialize Q with v while Q not = do call DELETE Q(v,Q) for all vertices w adjacent to v do if VISITED (w)=0 then [ call ADDQ(w,Q); VISITED(w) 1, print(w)] end end end BFS 1 2 3 4 5 6 7 8

1

visited

V …

Breadth First Search: Algorithm

Q


nChapter 6 Graphs

eg.

V1

V2V3

V4 V5V6 V7

V8

Q

V1 V2 V3 V4 V5 V6 V7 V8

Print: V1 , V2 , V3 , V4 , V5 , V6 , V7 , V8 .

Breadth First Search: Example


nChapter 6 Graphs

Time complexity: by using Adjacency Matrix

1 2 3 4 5 6 7 8

1 0 1 1 0 0 0 0 0

2 1 0 0 1 1 0 0 0

3 0

4 0

5 0

6 0

7 0

8 0

while

for n

n

Breadth First Search: Time Complexity

O(n2)


nChapter 6 Graphs

Time complexity: by using Adjacency List:

Total: d1 + d2 +...+ dn = O(e) where di = degree (vi)

while

for di: degree of (vi)di

n

Breadth First Search: Time Complexity

O(e)


nChapter 6 Graphs

• Application 1. Finding components of a graph

• Application 2. Finding a spanning tree of a connected

graph

• Application 3. Minimum cost spanning tree

Applications of Graph Traversal


nChapter 6 Graphs

Procedure COMP(G)(* determine the connected components of G *) for i 1 to n do VISITED(i) 0 for i 1 to n do if VISITED(i) = 0 then call DFS(i); print(“||”) end end COMP

Determine connected graph Invoke DFS or BFS

Application 1. Finding Components of a Graph

6.2.3 Connected Components

1

2 3

4

5

6 7

8

DFS: 1 2 4 3 || 5 6 7 8 || BFS: 1 2 3 4 || 5 6 7 8 ||

...

8 3 4 ...21

VISITED

1 1 1 1


nChapter 6 Graphs

Application 2. Finding a Spanning Tree of a Connected Graph

Def. Application 2.1– obtaining circuit equations for an electrical network Application 2.2 – minimum cost spanning tree

Def. Spanning Tree

Any tree consisting only edges in G and including all vertices in G is called.

• i.e. minimal subgraph G’ of G, such that V(G’) =V(G) and G’ is connected.

6.2.4 Spanning Trees


nChapter 6 Graphs

eg.

How to find it?

• Modify BFS Procedure BFS(v)

If VISITTED(w)=0 then If VISITTED(w)=0 then

1.call ADDQ(w,Q) 1.

2.VISITTED(w)←1 2.

3.print(w) 3.

4.T = T {(∪ v, w)}

...(How many?)

Ans ： 16 (EX.)

Spanning Trees (cont.)


nChapter 6 Graphs

Modify Procedure DFS：同 modify Procedure BFS e.g.

Application 1– obtaining circuit equations for an electrical network

1

2 3

4

1

2 3

4

BFS: 1,2,3,4

1

2 3

4

DFS: 1,2,4,3

BF spanning tree DF spanning tree

Spanning Trees (cont.)


nChapter 6 Graphs

BFS Spanning

DFS Spanning Tree vs. BFS Spanning Tree

0

1 2

3 4 5 6

7DFS Spanning

0

1 2

3 4 5 6

7

0

1 2

3 4 5 6

7

Introduction to Data Structure

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