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Alexandro B. Ponteras

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DATA

STRUCTURES &ALGORITHMS

Graphs & Trees

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Data Structures:A data structure is a way to store and organize data in order

to facilitate access and modifications.

Algorithm

an algorithm is any well-defined computational procedure that takes

input and produces output.

A sequence of computational steps that transform the input into the

output.

A tool for solving a well-specified computational problem.

The algorithm describes a specific computational procedure for

achieving that input/output relationship.

e.g.) sorting problem sorting algorithm

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.. continued.

An algorithm is said to be correct if it produce correct

output for every input instance.

A correct algorithm solves the given problem.

An incorrect algorithm might not result at all or

it might result with an answer other than the desiredone.

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Outline

Graphs Definition

Applications

Terminology

Digraphs

Undirected Graph

Weighted graphs

Trees

Definition

Tree traversal

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Graph A graph is a pair(V, E), where

Vis a set of nodes, called vertices Eis a collection of pairs of vertices, called edges

Vertices and edges are positions and store elements

Example:

A vertex represents a town and stores the three-letter towncode

An edge represents a route between two towns and storesthe mileage of the route

SAGSC

BINHIN

BAC

BAG

KAB

PUL

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Edge Types Directed edge

ordered pair of vertices (u,v) first vertex u is the origin

second vertex v is thedestination

e.g., a trip

Undirected edge unordered pair of vertices

(u,v)

e.g., a trip route

Directed graph

all the edges are directed e.g., trip network

Undirected graph

all the edges are undirected

e.g., route network

BAC SCtrip

AA 1206

BAC SC849

miles

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John

DavidPaul

brown.edu

cox.net

cs.brown.edu

att.net

qwest.net

math.brown.edu

cslab1bcslab1a

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

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Terminology End vertices (or endpoints)

of an edge

U and V are theendpoints 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

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P1

Terminology(continued) Path

sequence of alternatingvertices and edges

begins with a vertex

ends with a vertex

each edge is preceded andfollowed by its endpoints

Simple path

path such that all its verticesand 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) isa path that is not simple

XU

V

W

Z

Y

a

c

b

e

d

f

g

hP2

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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

C1

XU

V

W

Z

Y

a

c

b

e

d

f

g

hC2

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Digraphs

A digraph is a graph

whose edges are all

directed

Short for directed graph

A

C

E

B

D

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Digraph Properties

A graph G=(V,E) such that

Each edge goes in one direction:

Edge (a,b) goes from a to b, but not b to a.

A

C

E

B

D

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Undirected Graph

Simply connects two vertices

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Weighted Graphs

In a weighted graph, each edge has an associatednumerical value, called the weight of the edge

Edge weights may represent, distances, costs, etc.

Example:

In a trip route graph, the weight of an edge represents thedistance in miles between the endpoint towns

SAGSC

BINHIN

BAC

BAG

KAB

PUL

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Shortest Path Problem

Given a weighted graph and two vertices u and v, we want to

find a path of minimum total weight between u and v.

Length of a path is the sum of the weights of its edges.

Example:

Shortest path between Pulupandan and San Carlos

SAGSC

BINHIN

BAC

BAG

KAB

PUL

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Tree

A tree is a non-empty set, one element of

which is designated the root of the tree while

the remaining elements are partitioned into

non-empty sets each of which is a sub tree ofthe root.

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Example of a Tree

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Tree Traversal

tree-traversal refers to the process of visiting

each node in a tree data structure, exactly

once, in a systematic way. Such traversals

are classified by the order in which the nodesare visited.

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Tree Traversal Order

preorder

inorder

postorder

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Preorder traversal

Visit the root.

Traverse the left subtree.

Traverse the right subtree.

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Preorder traversal sequence

F, B, A, D, C, E,G, I, H

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Inorder traversal


Visit the root.


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Inorder traversal sequence

A, B, C, D, E, F,G, H, I

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Postorder traversal



Visit the root.

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Postorder traversal sequence

A, C, E, D, B, H, I,G, F

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END

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