graphs 4/13/14 - parasol laboratoryamato/courses/221-prev/... · 2014-04-13 · graphs 4/13/14 1...

Graphs 4/13/14

1

Chapter 13: Graphs

Nancy Amato Parasol Lab, Dept. CSE, Texas A&M University

Acknowledgement: These slides are adapted from slides provided with Data Structures and Algorithms in C++, Goodrich, Tamassia and Mount (Wiley 2004)

http://parasol.tamu.edu

ORD

DFW

SFO

LAX

802

1743

1843

1233

337

Outline and Reading

•  Graphs (§13.1) • Definition • Applications • Terminology • Properties • Subgraphs, connectivity, spanning trees & forests • ADT

•  Data structures for graphs (§13.2) • Edge list structure • Adjacency list structure • Adjacency matrix structure

2

Graph

•  A graph is a pair (V, E), where •  V is a set of nodes, called vertices •  E is a collection of pairs of vertices, called edges •  Vertices and edges are positions and store elements

•  Example: •  A vertex represents an airport and stores the three-letter airport code •  An edge represents a flight route between two airports and stores the

mileage of the route

3

ORD PVD

MIA DFW

SFO

LAX

LGA

HNL

849

802

1387 1743

1843

1099 1120 1233

337 2555

142

Edge & Graph Types

•  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

•  Weighted edge •  Graph Types

•  Directed graph (Digraph) •  all the edges are directed •  e.g., route network

•  Undirected graph •  all the edges are undirected •  e.g., flight network

•  Weighted graph •  all the edges are weighted 4

ORD DFW flight

AA 1206

ORD DFW

u v (u,v)

flight route

802 miles

802 miles

Applications

•  Electronic circuits •  Printed circuit board •  Integrated circuit

•  Transportation networks • Highway network •  Flight network

•  Computer networks •  Local area network •  Internet • Web

•  Databases •  Entity-relationship diagram

5

John

DavidPaul

brown.edu

cox.net

cs.brown.edu

att.netqwest.net

math.brown.edu

cslab1bcslab1a

Terminology

•  End points (or end vertices) 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 (multiple) edges

•  h and i are parallel edges •  self-loop

•  j is a self-loop

6

X U

V

W

Z

Y

a

c

b

e

d

f

g

h

i

j

Graphs 4/13/14

2

Terminology (cont.)

•  outgoing edges of a vertex •  h and b are the outgoing

edges of X •  incoming edges of a

vertex •  e, g, and i are incoming

edges of X •  in-degree of a vertex

•  X has in-degree 3 •  out-degree of a vertex

•  X has out-degree 2

7

X

V

W

Z

Y

b

e

d

f

g

h

i

j

Terminology (cont.)

•  Path •  sequence of alternating vertices

and edges •  begins with a vertex •  ends with a vertex •  each edge is preceded and

followed by its endpoints •  Simple path

•  path such that all its vertices and edges are distinct

•  Examples •  P1=(V,b,X,h,Z) is a simple path •  P2=(U,c,W,e,X,g,Y,f,W,d,V) is a

path that is not simple

8

P1

X U

V

W

Z

Y

a

c

b

e

d

f

g

h P2

Terminology (cont.)

•  Cycle •  circular sequence of alternating

vertices and edges •  each edge is preceded and

followed by its endpoints •  Simple cycle

•  cycle such that all its vertices and edges are distinct

•  Examples •  C1=(V,b,X,g,Y,f,W,c,U,a,↵) is a

simple cycle •  C2=(U,c,W,e,X,g,Y,f,W,d,V,a,↵)

is a cycle that is not simple

9

C1

X U

V

W

Z

Y

a

c

b

e

d

f

g

h C2

Exercise on Terminology 1.  # of vertices? 2.  # of edges? 3.  What type of the graph is it? 4.  Show the end vertices of the edge with largest weight 5.  Show the vertices of smallest degree and largest degree 6.  Show the edges incident to the vertices in the above question 7.  Identify the shortest simple path from HNL to PVD 8.  Identify the simple cycle with the most edges

10

ORD PVD

MIA DFW

SFO

LAX

LGA

HNL

849

802

1387 1743

1843

1099 1120 1233

337 2555

142

11

Exercise: Properties of Undirected Graphs

Notation n number of vertices m number of edges deg(v) degree of vertex v

Property 1 – Total degree Σv deg(v) = ?

Property 2 – Total number of edges

In an undirected graph with no self-loops and no multiple edges

m ≤ Upper bound? Lower bound? ≤ m

Example   n = ?   m = ?   deg(v) = ?

A graph with given number of vertices (4) and maximum

number of edges

Exercise: Properties of Undirected Graphs


Property 1 – Total degree Σv deg(v) = 2m Proof: each edge is counted

twice Property 2 – Total number

of edges In an undirected graph with

no self-loops and no multiple edges

m ≤ n (n - 1)/2

Proof: each vertex has degree at most (n - 1)

12

Example   n = 4   m = 6   deg(v) = 3


number of edges

Graphs 4/13/14

3

Exercise: Properties of Directed Graphs


Property 1 – Total in-degree and out-degree Σv in-deg(v) = ?

Σv out-deg(v) = ? Property 2 – Total number

of edges In an directed graph with no

self-loops and no multiple edges

m ≤ Upper bound? Lower bound? ≤ m

13

Example   n = ?   m = ?   deg(v) = ?


number of edges

Exercise: Properties of Directed Graphs


Property 1 – Total in-degree and out-degree Σv in-deg(v) = m

Σv out-deg(v) = m Property 2 – Total

number of edges In an directed graph with

no self-loops and no multiple edges

m ≤ n (n - 1)

14

Example   n = 4   m = 12   deg(v) = 6


number of edges

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

15

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

16

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 This definition of tree is

different from the one of a rooted tree

•  A forest is an undirected graph without cycles

•  The connected components of a forest are trees

17

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

•  Spanning trees have applications to the design of communication networks

•  A spanning forest of a graph is a spanning subgraph that is a forest

18

Graph

Spanning tree

Graphs 4/13/14

4

Main Methods of the Graph ADT

•  Vertices and edges •  are positions •  store elements

•  Accessor methods •  incidentEdges(v) •  adjacentVertices(v) •  degree(v) •  endVertices(e) •  opposite(v, e) •  areAdjacent(v, w)

•  isDirected(e) •  origin(e) •  destination(e)

•  Update methods •  insertVertex(o) •  insertEdge(v, w, o) •  insertDirectedEdge(v, w, o) •  removeVertex(v) •  removeEdge(e)

•  Generic methods •  numVertices() •  numEdges() •  vertices() •  edges()

19

Specific to directed edges

Exercise on ADT 1.  incidentEdges(ORD) 2.  adjacentVertices(ORD) 3.  degree(ORD) 4.  endVertices((LGA,MIA)) 5.  opposite(DFW, (DFW,LGA)) 6.  areAdjacent(DFW, SFO)

7.  insertVertex(IAH) 8.  insertEdge(MIA, PVD, 1200) 9.  removeVertex(ORD) 10. removeEdge((DFW,ORD)) 11. isDirected((DFW,LGA)) 12. origin ((DFW,LGA)) 13. destination((DFW,LGA)))

20

ORD PVD

MIA DFW

SFO

LAX

LGA

HNL

849

802

1387 1743

1843

1099 1120 1233

337 2555

142

Edge List Structure

•  An edge list can be stored in a sequence, a vector, a list or a dictionary such as a hash table

21

ORD PVD

MIA DFW

LGA

849

802

1387 1099 1120

142

(ORD, PVD)

(ORD, DFW)

(LGA, PVD)

(LGA, MIA)

(DFW, LGA)

849

802

142

1099

1387

(DFW, MIA) 1120

Edge List

ORD

LGA

PVD

DFW

MIA

Vertex Sequence

Exercise: Edge List Structure

Construct the edge list for the following graph

22

u x

y

v z

a

Asymptotic Performance •  Vertices and edges

•  are positions •  store elements

•  Accessor methods •  Accessing vertex sequence

•  degree(v) •  Accessing edge list

•  endVertices(e) •  opposite(v, e)

•  isDirected(e) •  origin(e) •  destination(e)

•  Generic methods •  numVertices() •  numEdges() •  vertices() •  edges()

23

(ORD, PVD)

(ORD, DFW)

(LGA, PVD)

(LGA, MIA)

(DFW, LGA)

849

802

142

1099

1387

(DFW, MIA) 1120

Edge List

ORD

LGA

PVD

DFW

MIA

Vertex Sequence

2

3

2

3

2

Degree

False

False

False

False

False

False

Weight Directed


Asymptotic Performance •  Vertices and edges

•  are positions •  store elements

•  Accessor methods •  Accessing vertex sequence

•  degree(v) O(1) •  Accessing edge list

•  endVertices(e) O(1) •  opposite(v, e) O(1)

•  isDirected(e) O(1) •  origin(e) O(1) •  destination(e) O(1)

•  Generic methods •  numVertices() O(1) •  numEdges() O(1) •  vertices() O(n) •  edges() O(m)

24

(ORD, PVD)

(ORD, DFW)

(LGA, PVD)

(LGA, MIA)

(DFW, LGA)

849

802

142

1099

1387

(DFW, MIA) 1120

Edge List

ORD

LGA

PVD

DFW

MIA

Vertex Sequence

2

3

2

3

2

Degree

False

False

False

False

False

False

Weight Directed


Graphs 4/13/14

5

Asymptotic Performance of Edge List Structure "   n vertices, m edges "   no parallel edges "   no self-loops "   Bounds are “big-Oh”

Edge List

Space n + m incidentEdges(v) adjacentVertices(v) m

areAdjacent (v, w) m insertVertex(o) 1 insertEdge(v, w, o) 1 removeVertex(v) m removeEdge(e) 1

25

(ORD, PVD)

(ORD, DFW)

(LGA, PVD)

(LGA, MIA)

(DFW, LGA)

849

802

142

1099

1387

(DFW, MIA) 1120

Edge List

ORD

LGA

PVD

DFW

MIA

Vertex Sequence

2

3

2

3

2

False

False

False

False

False

False

Weight Directed Degree

Edge List Structure

•  Vertex object •  element •  reference to position in

vertex sequence •  Edge object

•  element •  origin vertex object •  destination vertex object •  reference to position in

edge sequence •  Vertex sequence

•  sequence of vertex objects

•  Edge sequence •  sequence of edge objects

26

v

u

w

a c

b

a

z d

u v w z

b c d

Adjacency List Structure

27

ORD PVD

MIA DFW

LGA

849

802

1387 1099 1120

142

ORD

LGA

PVD

DFW

MIA

(ORD, PVD)

Adjacency List

(ORD, DFW)

(LGA, PVD) (LGA, MIA)

(PVD, ORD) (PVD, LGA)

(LGA, DFW)

(DFW, ORD) (DFW, LGA) (DFW, MIA)

(MIA, LGA) (MIA, DFW)

Exercise: Adjacency List Structure

Construct the adjacency list for the following graph

28

u x

y

v z

a

Asymptotic Performance of Adjacency List Structure

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

Adjacency List

Space n + m incidentEdges(v) adjacentVertices(v) deg(v)

areAdjacent (v, w) min(deg(v), deg(w)) insertVertex(o) 1 insertEdge(v, w, o) 1 removeVertex(v) deg(v)* removeEdge(e) 1*

29

ORD

LGA

PVD

DFW

MIA

(ORD, PVD)

Adjacency List

(ORD, DFW)

(LGA, PVD) (LGA, MIA)

(PVD, ORD) (PVD, LGA)

(LGA, DFW)

(DFW, ORD) (DFW, LGA) (DFW, MIA)

(MIA, LGA) (MIA, DFW)

Adjacency List Structure

•  Edge list structure •  Incidence sequence

for each vertex •  sequence of

references to edge objects of incident edges

•  Augmented edge objects •  references to

associated positions in incidence sequences of end vertices

30

u

v

w

a b

a

u v w

b

Graphs 4/13/14

6

Adjacency Matrix Structure

0 1 2 3 4

0 0 0 1 1 0

1 0 0 1 1 1

2 1 1 0 0 0

3 1 1 0 0 1

4 0 1 0 1 0

31

0 2

4 3

1

Exercise: Adjacency Matrix Structure

Construct the adjacency matrix for the following graph

32

u x

y

v z

a

Adjacency Matrix Structure: Weighted Graph

0 ORD

1 LGA

2 PVD

3 DFW

4 MIA

0 ORD 0 0 849 802 0

1 LGA 0 0 142 1387 1099

2 PVD 849 142 0 0 0

3 DFW 802 138 0 0 1120

4 MIA 0 109

9 0 1120 0

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0:ORD 2:PVD

4:MIA 3:DFW

1:LGA

849

802

1387 1099 1120

142

Exercise: Adjacency Matrix Structure: Weighted Digraph

0 ORD

1 LGA

2 PVD

3 DFW

4 MIA

0 ORD

1 LGA

2 PVD

3 DFW

4 MIA

34

0:ORD 2:PVD

4:MIA 3:DFW

1:LGA

849

802

1387 1099 1120

142

Asymptotic Performance of Adjacency Matrix Structure •  n vertices, m edges •  no parallel edges •  no self-loops •  Bounds are “big-Oh”

Adjacency Matrix

Space n2

incidentEdges(v) adjacentVertices(v) n

areAdjacent (v, w) 1 insertVertex(o) n2

insertEdge(v, w, o) 1 removeVertex(v) n2 removeEdge(e) 1

35

0 1 2 3 4

0 0 0 1 1 0

1 0 0 1 1 1

2 1 1 0 0 0

3 1 1 0 0 1

4 0 1 0 1 0

Adjacency Matrix Structure

•  Edge list structure •  Augmented vertex

objects •  Integer key (index)

associated with vertex •  2D-array adjacency

array •  Reference to edge

object for adjacent vertices

•  Null for non nonadjacent vertices

•  The “old fashioned” version just has 0 for no edge and 1 for edge

36

u

v

w

a b

0 1 2

0 ∅ ∅

1 ∅

2 ∅ ∅ a

u v w 0 1 2

b

Graphs 4/13/14

7

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

Edge List

Adjacency List

Adjacency Matrix

Space n + m n + m n2

incidentEdges(v) adjacentVertices(v) m deg(v) n

areAdjacent (v, w) m min(deg(v), deg(w)) 1 insertVertex(o) 1 1 n2

insertEdge(v, w, o) 1 1 1 removeVertex(v) m deg(v) n2 removeEdge(e) 1 1 1

37

Depth-First Search

38

D B

A

C

E

Outline and Reading

•  Depth-first search (§13.3.1) •  Algorithm •  Example •  Properties •  Analysis

•  Applications of DFS •  Path finding • Cycle finding

39

Depth-First Search

•  Depth-first search (DFS) is a general technique for traversing a graph

•  A DFS traversal of a graph G

•  Visits all the vertices and edges of G

• Determines whether G is connected

• Computes the connected components of G

• Computes a spanning forest of G

•  DFS on a graph with n vertices and m edges takes O(n + m ) time

•  DFS can be further extended to solve other graph problems

•  Find and report a path between two given vertices

•  Find a cycle in the graph •  Depth-first search is to

graphs what Euler tour is to binary trees

40

Example

41

D B

A

C

E

D B

A

C

E

D B

A

C

E

discovery edge back edge

A visited vertex A unexplored vertex

unexplored edge

A(A) = {B, C, D, E}

A(B) = {A, C, F} A(B) = {A, C, F}

A(C) = {A, B, D, E}

F

F

F G

G

G

A(C) = {A, B, D, E} A(C) = {A, B, D, E}

Example (cont.)

42

D B

A

C

E

D B

A

C

E

D B

A

C

E

D B

A

C

E

A(D) = {A, C} A(E) = {A, C}

F F

F F

A(C) = {A, B, D, E} A(C) = {A, B, D, E}

A(D) = {A, C} A(D) = {A, C}

A(E) = {A, C} A(E) = {A, C}

G

G

G

G

Graphs 4/13/14

8

Example (cont.)

43

D B

A

C

E

F

A(C) = {A, B, D, E} A(B) = {A, C, F}

G

D B

A

C

E

F G

A(G) = Φ

D B

A

C

E

F

A(F) = {B}

A(B) = {A, C, F} A(A) = {A, B, C, D}

G

DFS and Maze Traversal

•  The DFS algorithm is similar to a classic strategy for exploring a maze

•  We mark each intersection, corner and dead end (vertex) visited

•  We mark each corridor (edge ) traversed

•  We keep track of the path back to the entrance (start vertex) by means of a rope (recursion stack)

44

DFS Algorithm

•  The algorithm uses a mechanism for setting and getting “labels” of vertices and edges

45

Algorithm DFS(G, v) Input graph G and a start vertex v of G Output labeling of the edges of G in the connected component of v as discovery edges and back edges setLabel(v, VISITED) for all e ∈ G.incidentEdges(v)

if getLabel(e) = UNEXPLORED w ← opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) DFS(G, w) else setLabel(e, BACK)

Algorithm DFS(G) Input graph G Output labeling of the edges of G as discovery edges and back edges

for all u ∈ G.vertices() setLabel(u, UNEXPLORED)

for all e ∈ G.edges() setLabel(e, UNEXPLORED)

for all v ∈ G.vertices() if getLabel(v) = UNEXPLORED DFS(G, v)

Exercise: DFS Algorithm

•  Perform DFS of the following graph, start from vertex A •  Assume adjacent edges are processed in alphabetical order • Number vertices in the order they are visited •  Label edges as discovery or back edges

46

C B

A

E

D

F

Properties of DFS

Property 1 DFS(G, v) visits all the vertices and edges in the connected component of v

Property 2 The discovery edges labeled by DFS(G, v) form a spanning tree of the connected component of v

47

D B

A

C

E

F G

v1

v2

Analysis of DFS •  Setting/getting a vertex/edge label takes O(1) time •  Each vertex is labeled twice

•  once as UNEXPLORED •  once as VISITED

•  Each edge is labeled twice •  once as UNEXPLORED •  once as DISCOVERY or BACK

•  Function DFS(G, v) and the method incidentEdges are called once for each vertex

48

D B

A

C

E

F G

Graphs 4/13/14

9

Analysis of DFS •  DFS runs in O(n + m) time

provided the graph is represented by the adjacency list structure

•  Recall that ∑v deg(v) = 2m

49

Algorithm DFS(G, v) Input graph G and a start vertex v of G Output labeling of the edges of G in the connected component of v as discovery edges and back edges setLabel(v, VISITED) for all e ∈ G.incidentEdges(v)

if getLabel(e) = UNEXPLORED w ← opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) DFS(G, w) else setLabel(e, BACK)

Algorithm DFS(G) Input graph G Output labeling of the edges of G as discovery edges and back edges

for all u ∈ G.vertices() setLabel(u, UNEXPLORED)


for all v ∈ G.vertices() if getLabel(v) = UNEXPLORED DFS(G, v)

O(n)

O(m)

O(n +m)

O(deg(v))

Path Finding

•  We can specialize the DFS algorithm to find a path between two given vertices u and z using the template method pattern

•  We call DFS(G, u) with u as the start vertex

•  We use a stack S to keep track of the path between the start vertex and the current vertex

•  As soon as destination vertex z is encountered, we return the path as the contents of the stack

50

Algorithm pathDFS(G, v, z) setLabel(v, VISITED) S.push(v) if v = z

return S.elements() for all e ∈ G.incidentEdges(v)

if getLabel(e) = UNEXPLORED w ← opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) S.push(e) pathDFS(G, w, z) S.pop(e) else setLabel(e, BACK)

S.pop(v)

Cycle Finding

•  We can specialize the DFS algorithm to find a simple cycle using the template method pattern

•  We use a stack S to keep track of the path between the start vertex and the current vertex

•  As soon as a back edge (v, w) is encountered, we return the cycle as the portion of the stack from the top to vertex w

51

Algorithm cycleDFS(G, v, z) setLabel(v, VISITED) S.push(v) for all e ∈ G.incidentEdges(v)

if getLabel(e) = UNEXPLORED w ← opposite(v,e) S.push(e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) pathDFS(G, w, z) S.pop(e) else T ← new empty stack repeat o ← S.pop() T.push(o) until o = w return T.elements()

S.pop(v)

Breadth-First Search

52

C B

A

E

D

L0

L1

F L2

Outline and Reading

•  Breadth-first search (Sect. 13.3.5) • Algorithm • Example • Properties • Analysis • Applications

•  DFS vs. BFS • Comparison of applications • Comparison of edge labels

53

Breadth-First Search

•  Breadth-first search (BFS) is a general technique for traversing a graph

•  A BFS traversal of a graph G •  Visits all the vertices and

edges of G • Determines whether G is

connected • Computes the connected

components of G • Computes a spanning

forest of G

•  BFS on a graph with n vertices and m edges takes O(n + m ) time

•  BFS can be further extended to solve other graph problems •  Find and report a path

with the minimum number of edges between two given vertices •  Find a simple cycle, if

there is one

54

Graphs 4/13/14

10

Example

55

C B

A

E

D

discovery edge cross edge

A visited vertex A unexplored vertex

unexplored edge

L0

L1

F

C B

A

E

D

L0

L1

F

C B

A

E

D

L0

L1

F

Example (cont.)

56

C B

A

E

D

L0

L1

F

C B

A

E

D

L0

L1

F L2

C B

A

E

D

L0

L1

F L2

C B

A

E

D

L0

L1

F L2


visited vertex A A unexplored vertex

unexplored edge

Example (cont.)

57

C B

A

E

D

L0

L1

F L2

C B

A

E

D

L0

L1

F L2

C B

A

E

D

L0

L1

F L2


visited vertex A A unexplored vertex

unexplored edge BFS Algorithm


58

Algorithm BFS(G, s) L0 ← new empty sequence L0.insertLast(s) setLabel(s, VISITED) i ← 0 while ¬Li.isEmpty()

Li +1 ← new empty sequence for all v ∈ Li.elements() for all e ∈ G.incidentEdges(v) if getLabel(e) = UNEXPLORED w ← opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) setLabel(w, VISITED) Li +1.insertLast(w) else setLabel(e, CROSS) i ← i +1

Algorithm BFS(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u ∈ G.vertices() setLabel(u, UNEXPLORED)


for all v ∈ G.vertices() if getLabel(v) = UNEXPLORED BFS(G, v)

Exercise: BFS Algorithm

•  Perform BFS of the following graph, start from vertex A •  Assume adjacent edges are processed in alphabetical order • Number vertices in the order they are visited and note the level

they are in •  Label edges as discovery or cross edges

59

C B

A

E

D

F

Properties

Notation Gs: connected component of s

Property 1 BFS(G, s) visits all the vertices and edges of Gs

Property 2 The discovery edges labeled by BFS(G, s) form a spanning tree Ts of Gs

Property 3 For each vertex v in Li

•  The path of Ts from s to v has i edges

•  Every path from s to v in Gs has at least i edges

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C B

A

E

D

L0

L1

F L2

C B

A

E

D

F

Graphs 4/13/14

11

Analysis •  Setting/getting a vertex/edge label takes O(1) time •  Each vertex is labeled twice

•  once as UNEXPLORED •  once as VISITED

•  Each edge is labeled twice •  once as UNEXPLORED •  once as DISCOVERY or CROSS

•  Each vertex is inserted once into a sequence Li •  Method incidentEdges() is called once for each vertex •  BFS runs in O(n + m) time provided the graph is represented by

the adjacency list structure •  Recall that Σv deg(v) = 2m

61

Analysis of BFS


62

Algorithm BFS(G, s) L0 ← new empty sequence L0.insertLast(s) setLabel(s, VISITED) i ← 0 while ¬Li.isEmpty()

Li +1 ← new empty sequence for all v ∈ Li.elements() for all e ∈ G.incidentEdges(v) if getLabel(e) = UNEXPLORED w ← opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) setLabel(w, VISITED) Li +1.insertLast(w) else setLabel(e, CROSS) i ← i +1

Algorithm BFS(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u ∈ G.vertices() setLabel(u, UNEXPLORED)


for all v ∈ G.vertices() if getLabel(v) = UNEXPLORED BFS(G, v)

O(n)

O(m)

O(n +m)

O(deg(v))

Applications

•  Using the template method pattern, we can specialize the BFS traversal of a graph G to solve the following problems in O(n + m) time • Compute the connected components of G • Compute a spanning forest of G • Find a simple cycle in G, or report that G is a forest • Given two vertices of G, find a path in G between

them with the minimum number of edges, or report that no such path exists

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DFS vs. BFS

64

C B

A

E

D

L0

L1

F L2

C B

A

E

D

F

DFS BFS

Applications DFS BFS Spanning forest, connected components, paths, cycles

√ √

Shortest paths √

Biconnected components √

DFS vs. BFS (cont.)

Back edge (v,w) •  w is an ancestor of v in the

tree of discovery edges

Cross edge (v,w) •  w is in the same level as v

or in the next level in the tree of discovery edges

65

C B

A

E

D

L0

L1

F L2

C B

A

E

D

F

DFS BFS

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