trees and graphs

Data Structure and Algorithm AnalysisSlide Set 6

Trees & Graph

13 CS- I & II

Engr. Maria Shaikh

[email protected]

Non-Linear Data Structure

These data structures do not have their elements in a sequence. Trees and

Graph is an example.

Trees are mainly used to represent data containing a hierarchical relationship

between elements, example : records, family trees and table of contents.

10/30/2014 Engr. Maria Shaikh 2

Non Linear Data Structures

Non Linear Data Structures are

Branched recursive data structures

Consisting of nodes

Each node can be connected to other nodes

Examples of tree-like structures

Trees: binary / other degree, balanced, etc.

Graphs: directed / undirected, weighted, etc.

Networks

3

4Project Manager

Team Leader

De-signer

QA Team Leader

Developer 1

Developer 2

Tester 1Developer

3 Tester 2

Tree

Graph

2

36

1

45

7

19

21

141

12 31

4

11

Network2

36

1

45

2

36

1

45

Basic Terminologies in Trees

Nodes: The boxes on the tree are called nodes.

Children: The nodes immediately below (to the left and right of) a given node

are called its children.

Parent: The node immediately above a given node is called its parent.

Root: The (unique) node without a parent is called the root.

Leaf: A node with no children is called a leaf

Siblings: Two children of the same parent are said to be siblings.

A node can be an ancestor (e.g.grandparent) or a descendant (e.g. great-

grandchild).


Basic Terminologies in Trees

Root: node with no parent.

A non-empty tree has exactly one root.

leaf: node with no children

siblings: two nodes with the same parent.


Basic Terminologies in Graph

Path: a sequence of nodes n1 , n2, ---------- nk such that ni is the

parent of ni+1

for 1 i < k,

In other words, a sequence of hops to get from one node to

Another.

The length of a path is the number of edges in the path, or 1 less

than the number of nodes in it.


Trees

Terminology

Root no parent

Leaf no child

Interior non-leaf

Height distance from root to leaf

Root node

Leaf nodes

Interior nodes Height

Trees Data Structures

Tree

Nodes

Each node can have 0 or more children

A node can have at most one parent

Binary tree

Tree with 02 children per node

Tree

Binary Tree

10

Size and depth

The size of a binary tree is the number of nodes

in it

This tree has size 12

The depth of a node is its distance from the root.

a is at depth zero

e is at depth 2

The depth of a binary tree is the depth of its

deepest node

This tree has depth 4

a

b c

d e f

g h i j k

l

Trees

Data structure terminology

Node, edge, root, child, children, siblings, parent, ancestor, descendant, predecessor, successor, internal node, leaf, depth,

height, subtree

Height = 2

Depth 0

Depth 1

Depth 2

17

15149

6 5 8

11

Binary Trees

A binary tree is a finite set of elements that are either empty or ispartitioned into three disjoint subsets. The first subset contains asingle element called the root of the tree. The other two subsets arethemselves binary trees called the left and right subtrees of theoriginal tree. A left or right subtree can be empty.

Each element of a binary tree is called a node of the tree.


Binary Trees

Binary trees: most widespread form

Each node has at most 2 children

10

17

159

6 5 8

Root

nodeLeft

subtree

Right

Subtree

Left child

13

Right child

Engr. Maria Shaikh

Complete Binary Tree

A complete binary tree is a binary tree in which every level,

except possibly the last, is completely filled, and all nodes are as

far left as possible.


Full and Complete Binary Tree

15

BINARY TREES: BASIC DEFINITIONS

leaves

left son right son

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Binary Search Trees

Binary search trees are ordered

For each node x in the tree

All the elements of the left subtree of x are < x

All the elements of the right subtree of x are x

17 Engr. Maria Shaikh

Binary Search Trees

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All items in the left subtree are less than the root.

All items in the right subtree are greater or equal to the root.

Each subtree is itself a binary search tree.

Binary search trees provide an excellent structure for searching a list

and at the same time for inserting and deleting data into the list.

A Binary Search Tree is a binary tree with the following properties:

Engr. Maria Shaikh

Binary Search Trees

Balanced Binary Tree

A binary tree is balanced if every level above the lowest is full (contains 2n nodes)

In most applications, a reasonably balanced binary tree is desirable.

a

b c

d e f g

h i j

A balanced binary tree

a

b

c

d

e

f

g h

i jAn unbalanced binary tree

Engr. Maria Shaikh

(a), (b) - complete and balanced trees;

(d) nearly complete and balanced tree;

(c), (e) neither complete nor balanced trees

Engr. Maria Shaikh

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Invalid Binary Search Tree

Binary Search Tree Operations

We discuss four basic BST operations: traversal, search, insert, and delete; and

develop algorithms for searches, insertion, and deletion.

Traversals

Searches

Insertion

Deletion

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TRAVERSING BINARY TREES

One of the common operations of a binary tree is to traverse the tree. Traversinga tree is to pass through all of its nodes once. You may want to print thecontents of each node or to process the contents of the nodes. In either case eachnode of the tree is visited.

There are three main traversal methods where traversing a binary tree involvesvisiting the root and traversing its left and right subtrees. The only differenceamong these three methods is the order in which these three operations areperformed.

Binary Tree Traversal

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Engr. Maria Shaikh

TRAVERSING BINARY TREES

Traversing a binary tree in preorder (depth-first order)

1. Visit the root.

2. Traverse the left subtree in preorder.

3. Traverse the right subtree in preorder.

Traversing a binary tree in preorder

Preorder: ABDGCEHIF

Preorder Traversal

23 18 12 20 44 35 52

Engr. Maria Shaikh

Postorder Traversal

Traversing a binary tree in postorder

1. Traverse the left subtree in postorder.

2. Traverse the right subtree in postorder.

3. Visit the root.

Postorder Traversal

12 20 18 35 52 44 23

Engr. Maria Shaikh

Postorder Traversal

Postorder: GDBHIEFCA

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Inorder Traversing Binary Trees

Traversing a binary tree in inorder (or symmetric order)

1. Traverse the left subtree in inorder.

2. Visit the root.

3. Traverse the right subtree in inorder.

Inorder Traversal

12 18 20 23 35 44 52

Inorder traversal of a binary search tree produces a sequenced list

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Inorder Traversal

Inorder: DGBAHEICF

Right-Node-Left Traversal

52 44 35 23 20 18 12

Right-node-left traversal of a binary search tree produces a descending sequence

Engr. Maria Shaikh

Binary Expression Trees


*

+

b

-a

c

d

--

/

e + h

gf

[a + (b c)] * [(d e) / (f + g h)]

Preorder: * + a b c / - d e - + f g h

Postorder: a b c - + d e f g + h - / *

Tree Traversal Algorithms

Traversing a tree means to visit each of its nodes exactly one in particular order

Many traversal algorithms are known

Depth-First Search (DFS)

Visit node's successors first

Usually implemented by recursion

Breadth-First Search (BFS)

Nearest nodes visited first

Implemented by a queue38

Depth-First Search first visits all descendants of given node recursively, finally visits the node itself

DFS algorithm pseudo code


DFS(node)

{

for each child c of node

DFS(c);

print the current node;

}

7

1419

23 6

21

311 121 2 3

4 5 8

6 7

9

39

DFS in Action (Step 1)

Stack: 7

Output: (empty)

40

7

1419

23 6

21

311 12


Stack: 7, 19

Output: (empty)

41

7

1419

23 6

21

311 12


Stack: 7, 19, 1

Output: (empty)

42

7

1419

23 6

21

311 12


Stack: 7, 19

Output: 1

43

7

1419

23 6

21

311 12


Stack: 7, 19, 12

Output: 1

44

7

1419

23 6

21

311 12


Stack: 7, 19

Output: 1, 12

45

7

1419

23 6

21

311 12


Stack: 7, 19, 31

Output: 1, 12

46

7

1419

23 6

21

311 12


Stack: 7, 19

Output: 1, 12, 31

47

7

1419

23 6

21

311 12


Stack: 7

Output: 1, 12, 31, 19

48

7

1419

23 6

21

311 12


Stack: 7, 21

Output: 1, 12, 31, 19

49

7

1419

23 6

21

311 12


Stack: 7

Output: 1, 12, 31, 19, 21

50

7

1419

23 6

21

311 12


Stack: 7, 14

Output: 1, 12, 31, 19, 21

51

7

1419

23 6

21

311 12


Stack: 7, 14, 23

Output: 1, 12, 31, 19, 21

52

7

1419

23 6

21

311 12


Stack: 7, 14

Output: 1, 12, 31, 19, 21, 23

53

7

1419

23 6

21

311 12


Stack: 7, 14, 6

Output: 1, 12, 31, 19, 21, 23

54

7

1419

23 6

21

311 12


Stack: 7, 14

Output: 1, 12, 31, 19, 21, 23, 6

55

7

1419

23 6

21

311 12


Stack: 7

Output: 1, 12, 31, 19, 21, 23, 6, 14

56

7

1419

23 6

21

311 12


Stack: (empty)

Output: 1, 12, 31, 19, 21, 23, 6, 14, 7

57

7

1419

23 6

21

311 12

Traversal finished

Breadth-First Search first visits the neighbor nodes, later their neighbors, etc.

BFS algorithm pseudo code

Breadth-First Search (BFS)

BFS(node)

{

queue node

while queue not empty

v queue

print v

for each child c of v

queue c

}

7

1419

23 6

21

311 125 6 7

2 3 4

8 9

1

58

BFS in Action (Step 1)

Queue: 7

Output: 7

59

7

1419

23 6

21

311 12


Queue: 7, 19

Output: 7

60

7

1419

23 6

21

311 12


Queue: 7, 19, 21

Output: 7

61

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14

Output: 7

62

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14

Output: 7, 19

63

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1

Output: 7, 19

64

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12

Output: 7, 19

65

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31

Output: 7, 19

66

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31

Output: 7, 19, 21

67

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31

Output: 7, 19, 21, 14

68

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31, 23

Output: 7, 19, 21, 14

69

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6

Output: 7, 19, 21, 14

70

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6

Output: 7, 19, 21, 14, 1

71

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6

Output: 7, 19, 21, 14, 1, 12

72

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6

Output: 7, 19, 21, 14, 1, 12, 31

73

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6

Output: 7, 19, 21, 14, 1, 12, 31, 23

74

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6

Output: 7, 19, 21, 14, 1, 12, 31, 23, 6

75

7

1419

23 6

21

311 12


Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6

Output: 7, 19, 21, 14, 1, 12, 31, 23, 6

76

7

1419

23 6

21

311 12

The queue is

empty stop

General Trees

A general tree ( sometimes called a tree) is defined to be a non empty finite set T of elements, called nodes, such that:

1) T contains a distinguished element R, called the root of T.

2) The remaining elements of T form an ordered collection of zero or more disjoint trees

T1, T2, -------------TM are called successors of R.

Engr. Maria Shaikh

General Trees


A

B

FE

HGK

D

J

NMGeneral Tree T with 13 Nodes

A, B, C, D, E, F, G, H, J, K, L, M, N

C

L

Presentation topic

GRAPH

Engr. Maria Shaikh

What is graph ?

A GRAPH is a mathematical structure consisting of a set of vertices and a set

of edges.

Node: Each element of a graph is called node of a graph.

Edge :Line joining two nodes is called an edge.

= Vertices

= Edeges

Engr. Maria Shaikh

Other Definitions of Graph

A data structure that consists of a set of nodes (vertices) and a set of

edges that relate the nodes to each other.

The set of edges describes relationships among the vertices.

Graph is a mathematical structure used to model pair wise

relations between objects from a certain collection.

10/30/2014 81Engr. Maria Shaikh

Formal definition of graphs

A graph G is defined as follows:

G = (V,E) is composed of:

V(G): a finite, nonempty set of vertices

E(G): a set of edges connecting the vertices in V

An edge e = (u,v) is a pair of vertices

Example: a b

c

d e

V= {a,b,c,d,e}

E= {(a,b),(a,c),(a,d),(b,e),(c,d),(c,e),

(d,e) }

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Types of graphs

There are two types of graph :

Directed graph

Undirected graph

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Directed graph

A graphs G is called directed graph or digraph if each edge has a direction.

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Un Directed graph

A graphs G is called directed graph if each edge has no direction.

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Graph terminologies

Adjacent nodes: Two nodes are adjacent if they are connected by an edge

Path: A sequence of vertices that connect two nodes in a graph

Complete graph: A graph in which every vertex is directly connected to every other vertex.

5 is adjacent to 7

7 is adjacent from 55

Engr. Maria Shaikh

What is the number of edges in a complete directed graph with N vertices?

N * (N-1)

Graph terminologies (cont.)

2( )O N

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What is the number of edges in a complete undirected graph with N vertices?

N * (N-1) / 2

Graph terminologies (cont.)

2( )O N

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Terminologies

subgraph: subset of vertices and edges forming a graph

connected component: maximal connected subgraph. E.g., the graph below has 3 connected components.

connected not connected

connected graph: any two vertices are connected by some path

Engr. Maria Shaikh

0 0

1 2 3

1 2 0

1 2

3(i) (ii) (iii) (iv)(a) Some of the subgraph of G1

0 0

1

0

1

2

0

1

2

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

(b) Some of the subgraph of G3

0

1 2

3

G1

0

1

2G3

Subgraphs Examples

Some different types of graph

1. Sub graph

2. Complete graph

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3. Weighted graph:

A graph in which each edge carries a value.

Some different types of graph

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Representation of graphs

Graphs can be represented in 2 ways :

Array representation

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Linked list representation

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Engr. Maria Shaikh

Real Life Example

Real Life Example

electronic circuits networks (roads, flights, communications)

CS16

LAX

JFK

LAX

DFW

STL

HNL

FTLEngr. Maria Shaikh

Paths

A path in a graph is a sequence of vertices such that from

each of its vertices there is an edge to the next vertex in the

sequence.

The length of a path is the number of edges on it. The length can be zero for the case of a single vertex.

Engr. Maria Shaikh

A path may be infinite.

A finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex.

Both of them are called terminal vertices of the path.

The other vertices in the path are internal vertices.

Path= A B D C E

C A- Start vertex

E- End vertex

AB

D

E

Paths

Engr. Maria Shaikh

Path

101

3

3 3

3

2

a b

c

d e

a b

c

d e

a b e d c b e d c

Engr. Maria Shaikh

A graph with no loops or multiple edges is called a simple graph.

A path with no repeated vertices is called a simple path.

The path from v1 to v4 is said to be simple path as

vertices is touched more than once.

The path from v1 to v4 is not simple as

v1 is touched twice or looped.

cycle: Simple path, except that the last vertex is the same as

the first vertex. Its also known as a circuit or circular path.

Path= A E C A

A B

C D

E

Simple Paths

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Paths

simple path

cycle:

a b

c

d e

b e c

a c d a

a b

c

d e

Engr. Maria Shaikh

Degree The degree of vertex in an undirected graph is the number of edges incident to

that vertex.

A vertex with degree one is called pendent vertex or end vertex.

A vertex with degree zero and hence has no incident edges is called an isolated vertex.

In the undirected graph vertex v3 has the degree 3 And vertex v2 has the degree 2

V1

B

A

Pendent vertexIsolated vertex

Engr. Maria Shaikh

Degree in directed graph

Degree of directed graph has two types

i. Indegree

No of edges with their head towards the vertex.

ii. Outdegree

No of edges with their tail towards the vertex.

Indegree of vertex v2 is 2 and

Outdegree of vertex v1 is 1.

Degree

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01 2

3 4 5 6

G

G2

3 2

3 3

1 1 1 1

Directed graph

in-degree

out-degree

0

1

2

G3

in:1, out: 1

in: 1, out: 2

in: 1, out: 0

0

1 2

3

33

3

Example:

Engr. Maria Shaikh

Adjacent and Incident

If (v0, v1) is an edge in an undirected graph,

v0 and v1 are adjacent

The edge (v0, v1) is incident on vertices v0 and v1

If is an edge in a directed graph

v0 is adjacent to v1, and v1 is adjacent from v0

The edge is incident on v0 and v1

Engr. Maria Shaikh

More

Tree - connected graph without cycles.

Forest - collection of trees.

A forest is an undirected graph

without cycles

The connected components of a

forest are trees.

tree

forest

tree

tree

tree

Forest

Engr. Maria Shaikh

109

Cycles and trees

Graph with no cycles: acyclic

Directed Acyclic Graph: DAG

Undirected forest:

Acyclic undirected graph

Tree: undirected acyclic connected graph

one connected component

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Connectivity

Let n = #vertices, and m = #edges

A complete graph: one in which all pairs of vertices are adjacent

How many total edges in a complete graph?

Each of the n vertices is incident to n-1 edges, however, we would have counted each edge twice!

Therefore, intuitively, m = n(n -1)/2.

Therefore, if a graph is not complete, m < n(n -1)/2

n 5m (5

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More Connectivity

n = #vertices

m = #edges

For a tree m = n - 1

n 5m 4

n 5m 3

If m < n - 1, G is

not connected

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Graph Representations

Adjacency Matrix

Adjacency Lists

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113

Representing graphs

Adjacency matrix:

When graph is dense

|E| close to |V|2

Adjacency lists:

When graph is sparse

|E|

Adjacency Matrix

Let G=(V,E) be a graph with n vertices.

The adjacency matrix of G is a two-dimensional n by n array, say adj_mat

If the edge (vi, vj) is in E(G), adj_mat[i][j]=1

If there is no such edge in E(G), adj_mat[i][j]=0

The adjacency matrix for an undirected graph is symmetric; the adjacency

matrix for a digraph need not be symmetric

Engr. Maria Shaikh

115

Examples

1 2 3

1 0 0 1

2 0 1 0

3 1 1 0

1

3

2

32

1

4

1 2 3 4

1 0 1 1 0

2 1 0 0 0

3 1 0 0 0

4 0 0 0 0Engr. Maria Shaikh

Adjacency Matrix

|V| |V| matrix A.

Number vertices from 1 to |V| in some arbitrary manner.

A is then given by:

otherwise0

),( if1],[

EjiajiA ij

a

dc

b1 2

3 4

1 2 3 4

1 0 1 1 1

2 0 0 1 0

3 0 0 0 1

4 0 0 0 0

a

dc

b1 2

3 4

1 2 3 4

1 0 1 1 1

2 1 0 1 0

3 1 1 0 1

4 1 0 1 0

A = AT for undirected graphs.

Engr. Maria Shaikh

Examples for Adjacency Matrix

0

1

1

1

1

0

1

1

1

1

0

1

1

1

1

0 0

1

0

1

0

0

0

1

0

0

1

1

0

0

0

0

0

1

0

0

1

0

0

0

0

1

0

0

1

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

1

0

G1

G2

G4

0

1 2

3

0

1

2

1

0

2

3

4

5

6

7

symmetricundirected: n2/2

directed: n2

Engr. Maria Shaikh

Merits of Adjacency Matrix

From the adjacency matrix, to determine the connection of vertices is

easy

The degree of a vertex is

For a digraph (= directed graph), the row sum is the out_degree, while

the column sum is the in_degree

adj mat i jj

n

_ [ ][ ]

0

1

ind vi A j ij

n

( ) [ , ]

0

1

outd vi A i jj

n

( ) [ , ]

0

1

Engr. Maria Shaikh

01

2

3

0

1

2

0

1

2

3

4

5

6

7

1 2 3

0 2 3

0 1 3

0 1 2

G1

1

0 2

G3

1 2

0 3

0 3

1 2

5

4 6

5 7

6

G4

0

1 2

3

0

1

2

1

0

2

3

4

5

6

7

An undirected graph with n vertices and e edges ==> n head nodes and 2e list nodes

Adjacency Lists(data structures)

120

Traversing a Graph

A traversal of a graph is a systematic procedure for exploring a

graph by examining all of its vertices and edges.

Until finding a goal vertex or until no more vertices

Only for connected graphs

Example:

Web spider or crawler is the data collecting part of a search engine that

visits all the hypertext documents on the web. (The documents are

vertices and the hyperlinks are the edges).

Engr. Maria Shaikh

1.121

Graph Search (traversal)

How do we search a graph?

At a particular vertices, where shall we go next?

Two common framework:

the depth-first search (DFS)

the breadth-first search (BFS) and

In DFS, go as far as possible along a single path until reach a dead

end (a vertex with no edge out or no neighbor unexplored) then

backtrack

In BFS, one explore a graph level by level away (explore all

neighbors first and then move on)

Engr. Maria Shaikh

122

Breadth-first search

Level-by-level traversal.

Search for all vertices that are directly reachable from the root (called

level 1 vertices)

After mark all these vertices, visit all vertices that are directly reachable

from any level 1 vertices (called level 2 vertices), and so on.

In general, level k vertices are directly reachable from a level k 1

vertices

Engr. Maria Shaikh

123

BFS in a binary tree

BFS: visit all siblings before their descendants

5

2

1 3

8

6 10

7 95 2 8 1 3 6 10 7 9

Engr. Maria Shaikh

124

BFS for General Graphs

This version assumes vertices have two children

left, right

This is trivial to fix

But still no good for general graphs

It does not handle cycles

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125

Start with A. Put in the queue (marked red)

A

B

G C

E

D

F

Queue: A

Example for Undirected Graph

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126

B and E are next

A

B

G C

E

D

F

Queue: A B E

Example for Undirected Graph (cont.)

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127

When we go to B, we put G and C in the queue

When we go to E, we put D and F in the queue

A

B

G C

E

D

F

Queue: A B E C G D F


Engr. Maria Shaikh

128

When we go to B, we put G and C in the queue

When we go to E, we put D and F in the queue

A

B

G C

E

D

F



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129

Suppose we now want to expand C. We put F in the queue again!

A

B

G C

E

D

F

Queue: A B E C G D F F


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Generalizing BFS

Cycles:

We need to save auxiliary information

Each node needs to be marked

Visited: No need to be put on queue

Not visited: Put on queue when found

What about assuming only two children vertices?

Need to put all adjacent vertices in queue

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The general BFS algorithm

Each vertex can be in one of three states:

Unmarked and not on queue

Marked and on queue

Marked and off queue

The algorithm moves vertices between these states

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Handling vertices

Unmarked and not on queue:

Not reached yet

Marked and on queue:

Known, but adjacent vertices not visited yet (possibly)

Marked and off queue:

Known, all adjacent vertices on queue or done with

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Start with A. Mark it.

A

B

G C

E

D

F

Queue: A

Example for BFS Undirected Graph Modified

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134

Expand As adjacent vertices.

Mark them and put them in queue.

A

B

G C

E

D

F

Queue: A B E


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135

Now take B off queue, and queue its neighbors.

A

B

G C

E

D

F

Queue: A B E C G


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Do same with E.

A

B

G C

E

D

F



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Visit C.

Its neighbor F is already marked, so not queued.

A

B

G C

E

D

F



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Visit G.

A

B

G C

E

D

F



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Visit D. F, E marked so not queued.

A

B

G C

E

D

F



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Visit F.

E, D, C marked, so not queued again.

A

B

G C

E

D

F


Example for Undirected Graph Modified

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141

Done. We have explored the graph in order:

A B E C G D F.

A

B

G C

E

D

F



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1.142

BFS for Directed GraphThe Color Scheme

White vertices have not been discovered All vertices start out white

Grey vertices are discovered but not fully explored They may be adjacent to white vertices

Black vertices are discovered and fully explored They are adjacent only to black and gray vertices

Explore vertices by scanning adjacency list of grey vertices

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1.143Data Structure and Algorithm

A

ONM

LKJ

E F G H

DCB

I

P

An Example


A

ONM

LKJ

E F G H

DCB

I

P

0


A

ONM

LKJ

E F G H

DCB

I

P

0

1 1

1


A

ONM

LKJ

E F G H

DCB

I

P

0

1 1

1

2

2


A

ONM

LKJ

E F G H

DCB

I

P

0

1 1

1

2

2

3 3

3

3

3


A

ONM

LKJ

E F G H

DCB

I

P

0

1 1

1

2

2

3 3

3

3

3

4 4

4


A

ONM

LKJ

E F G H

DCB

I

P

0

1 1

1

2

2

3 3

3

3

3

4 4

4

55

152

Interesting features of BFS

Complexity: O(|V| + |E|)

All vertices put on queue exactly once

For each vertex on queue, we expand its edges

In other words, we traverse all edges once

BFS finds shortest path from s to each vertex

Shortest in terms of number of edges

Why does this work?

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1.153


The basic idea behind this algorithm is that it traverses the graph using

recursion

Go as far as possible until you reach a dead end

Backtrack to the previous path and try the next branch

The graph below, started at node a, would be visited in the following

order: a, b, c, g, h, i, e, d, f, j

da b

h i

c

g

e

j

f

154

Start with A. Mark it.

A

B

G C

E

D

F

Current vertex: A

Example for DFS Undirected Graph Modified

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155

Expand As adjacent vertices. Pick one (B).

Mark it and re-visit.

A

B

G C

E

D

F

Current: B


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Now expand B, and visit its neighbor, C.

A

B

G C

E

D

F

Current: C


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Visit F.

Pick one of its neighbors, E.

A

B

G C

E

D

F

Current: F


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Es adjacent vertices are A, D and F.

A and F are marked, so pick D.

A

B

G C

E

D

F

Current: E


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Visit D. No new vertices available. Backtrack to E. Backtrack to F.

Backtrack to C. Backtrack to B

A

B

G C

E

D

F

Current: D


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Visit G. No new vertices from here. Backtrack to B. Backtrack to A. E

already marked so no new.

A

B

G C

E

D

F

Current: G


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Done. We have explored the graph in order:

A B C F E D G

A

B

G C

E

D

F

Current:1

2

3

4

6

7

5


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1.162

DFS: Color Scheme

Vertices initially colored white

Then colored gray when discovered

Then black when finished

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1.163

DFS: Time Stamps

Discover time d[u]: when u is first discovered

Finish time f[u]: when backtrack from u

d[u] < f[u]

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DFS Example

source

vertex

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DFS Example

1 | | |

| | 3 |

2 | |

source

vertexd f

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DFS Example

1 | | |

| | 3 | 4

2 | |

source

vertexd f

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DFS Example

1 | | |

| 5 | 3 | 4

2 | |

source

vertexd f

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DFS Example

1 | | |

| 5 | 63 | 4

2 | |

source

vertexd f

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DFS Example

1 | | |

| 5 | 63 | 4

2 | 7 |

source

vertexd f

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DFS Example

1 | 8 | |

| 5 | 63 | 4

2 | 7 |

source

vertexd f

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DFS Example

1 | 8 | |

| 5 | 63 | 4

2 | 7 9 |

source

vertexd f

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DFS Example

1 | 8 | |

| 5 | 63 | 4

2 | 7 9 |10

source

vertexd f

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DFS Example

1 | 8 |11 |

| 5 | 63 | 4

2 | 7 9 |10

source

vertexd f

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DFS Example

1 |12 8 |11 |

| 5 | 63 | 4

2 | 7 9 |10

source

vertexd f

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DFS Example

1 |12 8 |11 13|

| 5 | 63 | 4

2 | 7 9 |10

source

vertexd f

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DFS Example

1 |12 8 |11 13|

14| 5 | 63 | 4

2 | 7 9 |10

source

vertexd f

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DFS Example

1 |12 8 |11 13|

14|155 | 63 | 4

2 | 7 9 |10

source

vertexd f

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DFS Example

1 |12 8 |11 13|16

14|155 | 63 | 4

2 | 7 9 |10

source

vertexd f

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Interesting features of DFS

Complexity: O(|V| + |E|)

All vertices visited once, then marked

For each vertex on queue, we examine all edges

In other words, we traverse all edges once

DFS does not necessarily find shortest path

Why?

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Graph Traversal

Problem: Search for a certain node or traverse all nodes in the

graph

Depth First Search

Once a possible path is found, continue the search until the end of the

path

Breadth First Search

Start several paths at a time, and advance in each one step at a time

Engr. Maria Shaikh

END OF SLIDE SET 6


trees and graphs

Documents