Data Structures

KRISNA ADIYARTA

PASCA SARJANA (MAGISTER KOMPUTER)UNIVERSITAS BUDI LUHUR

JAKARTA

Data Structures and Algorithms in C++, Second Edition by Adam Drozdek, published by Brooks/Cole Thomson Learning, 2001, ISBN 0-534-37597-9

Definitions A data structure is a scheme for organizing data in

the memory of a computer. The way in which the data is organized affects the

performance of a program for different tasks. Computer programmers decide which data

structures to use based on the nature of the data and the processes that need to be performed on that data.

Some of the more commonly used data structures include : primitive data, arrays, lists, stacks, queues, trees, graphs

Program C++

#include #include #include main(){int keliling, panjang, lebar;cout panjang;cout lebar;keliling = (2 * panjang) + ( 2 * lebar)cout

Primitive (Atomic) Structures

Primitive (or Basic) Data Types User-Defined Ordinal Types Character String Types Pointer Types

Primitive Data Types

Number IntegerFloating-Point

Boolean CharacterASCII, ISO-8859-x, JIS, UNICODE

User-Defined Ordinal Type

Range of possible values mapped to positive integers

ExamplesEnumeration types (C, C++, Pascal, Ada)Subrange types (Pascal, Modula-2, Ada)

Character String Type

Design issues:Should strings be a special kind of character

array? Or a primitive type?Static or dynamic length?

Pointer Type

Range of values of memory address or null Provide explicit support for indirect referencing Available in: C, C++, Pascal, Ada

int *pi = new int;float *pf=new float;*pi =1024;*pf =3.14;

Record Structure

Possibly heterogeneous aggregation of named data elements

struct {char name[10];int age;int salary;} person;

strcpy(person.name, james);person.age=10;person.salary=3000;

Name SalaryAge

Person

Array Structure

A collection of pairs where index is an ordered set of integers and are values of some data type that is constant for the array.

not all languages require index to be continuous or contiguous or start at 0 or 1.

In C arrays are zero based and are contiguous from 0 to size-1 and can contain any simple or aggregate data type

Array Structure

Homogenous aggregation of data elements Constant-time access to elements Design issues:Subscript typesBounds checking?Subscript dimension, subscript orderSize static or dynamic? User-definable?

Single Linked List

A singly linked list is a concrete data structure consisting of a series of nodes

Each node storesData itemLink to the next node

next

Data item NODE

A B C D

HEAD CURRENT

Insertion

A B C

X

A B

X

C

1

2

3

A B CX

Deletion

C

A B D

2

1

A B D CX

A B D

Double Linked List A doubly linked list provides a natural

implementation of the List Special trailer and header nodes Nodes implement Position and store:element link to the previous node link to the next node

prev next

elem node

trailerheader nodes/positions

Insertion

A B X C

A B C

A B Cp

Xq

p q

Deletion

A B C D

A B C

Dp

A B C

Stack

Stores a set of elements in a particular order Stack principle: LAST IN FIRST OUT It means: the last element inserted is the first one

to be removed Basic Algorithm : Push & Pop

BA

DCBA

CBA

DCBA

EDCBAtop

toptop

toptop

A

Queue

Stores a set of elements in a particular order Queue principle: FIRST IN FIRST OUT It means: the first element inserted is the first

one to be removed Basic Algorithm : Enqueue & Dequeue

ABA

CBA

DCBA

DCBrear

front

rearfront

rear

front

rear

front

rear

front

Tree

ComputersRUs

Sales R&DManufacturing

Laptops DesktopsUS International

Europe Asia Canada

A tree is a finite nonempty set of elements.

It is an abstract model of a hierarchical structure.

consists of nodes with a parent-child relation.

Applications: Organization charts File systems Programming environments

Tree Terminology Root: node without parent Siblings: nodes share the same parent Internal node: node with at least one child External node (leaf ): node without children Ancestors of a node: parent, grandparent, grand-grandparent, etc.

subtree

A

B DC

G HE F

I J K

Descendant of a node: child, grandchild, grand-grandchild, etc.

Depth of a node: number of ancestors Height of a tree: maximum depth of any

node Degree of a node: the number of its

children Degree of a tree: the maximum number

of its node. Subtree: tree consisting of a node and

its descendants

Tree Property

A

B C

D

G

E F

IH

Property ValueNumber of nodesHeightRoot NodeLeavesInterior nodesAncestors of HDescendants of BSiblings of ERight subtree of ADegree of this tree

Intuitive Representation of Tree Node

List Representation ( A ( B ( E ( K, L ), F ), C ( G ), D ( H ( M ), I, J ) ) ) The root comes first, followed by a list of links to sub-

trees

Data Link 1 Link 2 Link n

How many link fields are needed in such a representation?

Tree

Every tree node: object useful information children pointers to its children

Data

Data Data Data

Data Data Data

A Tree Representation

A node is represented by an object storing Element Parent node Sequence of

children nodes

B

A D F

C

E

B

DA

C E

F

Left Child, Right Sibling Representation

Data

Left Child

Right Sibling A

B C D

IHGFE

J K L

Tree Traversal

Two main methods: Preorder Postorder

Preorder visit the root traverse in preorder the children (subtrees)

Postorder: traverse in postorder the children (subtrees) visit the root

Preorder Traversal

Algorithm preOrder(v)visit(v)for each child w of v

preorder (w)

A traversal visits the nodes of a tree in a systematic manner

In a preorder traversal, a node is visited before its descendants

Application: print a structured document

Become Rich

1. Motivations 3. Success Stories2. Methods

2.1 Get a CS PhD

2.2 Start a Web Site

1.1 Enjoy Life

1.2 Help Poor Friends

2.3 Acquired by Google

1

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9

Postorder Traversal In a postorder traversal, a

node is visited after its descendants

Application: compute space used by files in a directory and its subdirectories

Algorithm postOrder(v)for each child w of v

postOrder (w)visit(v)

cs16/

homeworks/ todo.txt1Kprograms/

DDR.java10K

Stocks.java25K

h1c.doc3K

h1nc.doc2K

Robot.java20K

9

3

1

7

2 4 5 6

8

Decision Tree Binary tree associated with a decision process

internal nodes: questions with yes/no answer external nodes: decisions

Example: dining decision

Want a fast meal?

How about coffee? On expense account?

Starbucks Spikes Al Forno Caf Paragon

Yes No

Yes No Yes No

Binary Tree

A binary tree is a tree with the following properties: Each internal node has at most two

children (degree of two) The children of a node are an

ordered pair

We call the children of an internal node left child and right child

Alternative recursive definition: a binary tree is either a tree consisting of a single node,

OR a tree whose root has an ordered

pair of children, each of which is a binary tree

Applications: arithmetic expressions decision processes searching

A

B C

F GD E

H I

Examples of the Binary Tree

A

B C

GE

I

D

H

F

Complete Binary Tree

1

2

3

4

A

B

A

B

Skewed Binary Tree

E

C

D

5

Differences Between A Tree and A Binary Tree

The subtrees of a binary tree are ordered; those of a tree are not ordered.

A

B

A

B

Are different when viewed as binary trees. Are the same when viewed as trees.

Data Structure for Binary Trees A node is represented

by an object storing Element Parent node Left child node Right child node

B

DA

C E

B

A D

C E

Arithmetic Expression Tree

Binary tree associated with an arithmetic expression internal nodes: operators external nodes: operands

Example: arithmetic expression tree for the expression (2 (a 1) + (3 b))

+

2a 1

3 b

Maximum Number of Nodes in a Binary Tree

The maximum number of nodes on depth i of a binary tree is 2i, i>=0.

The maximum nubmer of nodes in a binary tree of height k is 2k+1-1, k>=0.

Prove by induction.

122 10

= += kki

i

Full Binary Tree A full binary tree of a given height k has 2k+11

nodes.

Height 3 full binary tree.

Labeling Nodes In A Full Binary Tree Label the nodes 1 through 2k+1 1. Label by levels from top to bottom. Within a level, label from left to right.

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

Node Number Properties

Parent of node i is node i / 2, unless i = 1. Node 1 is the root and has no parent.

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

Node Number Properties

Left child of node i is node 2i, unless 2i > n, where n is the number of nodes.

If 2i > n, node i has no left child.

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2 3

4 5 6 7

8 9 10 11 12 13 14 15

Node Number Properties

Right child of node i is node 2i+1, unless 2i+1 > n, where n is the number of nodes.

If 2i+1 > n, node i has no right child.

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

Complete Binary Trees A labeled binary tree containing the labels 1 to n with root 1,

branches leading to nodes labeled 2 and 3, branches from these leading to 4, 5 and 6, 7, respectively, and so on.

A binary tree with n nodes and level k is complete iff its nodes correspond to the nodes numbered from 1 to n in the full binary tree of level k.

1

2 3

75

11

4

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6

98 15141312

Full binary tree of depth 3

1

2 3

75

9

4

8

6

Complete binary tree

Binary Tree Traversals Let l, R, and r stand for moving left, visiting

the node, and moving right.

There are six possible combinations of traversal lRr, lrR, Rlr, Rrl, rRl, rlR

Adopt convention that we traverse left before right, only 3 traversals remain lRr, lrR, Rlr inorder, postorder, preorder

Inorder Traversal In an inorder traversal a

node is visited after its left subtree and before its right subtree

Algorithm inOrder(v)if isInternal (v)

inOrder (leftChild (v))visit(v)if isInternal (v)

inOrder (rightChild (v))

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Graph A graph, G=(V, E), consists of two sets:

a finite set of vertices(V), and a finite, possibly empty set of edges(E) V(G) and E(G) represent the sets of vertices and edges of G,

respectively Undirected graph

The pairs of vertices representing any edges is unordered e.g., (v0, v1) and (v1, v0) represent the same edge

Directed graph Each edge as a directed pair of vertices e.g. represents an edge, v0 is the tail and v1 is the

head

Graph0

1 2

3

0

1

2

0

1 2

3 4 5 6G1

G2 G3V(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)={,,}

complete undirected graph: n(n-1)/2 edgescomplete directed graph: n(n-1) edges

complete graph incomplete graph

Complete Graph

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

Adjacent and Incident

If (v0, v1) is an edge in an undirected graph, v0 and v1 are adjacentThe edge (v0, v1) is incident on vertices v0 and v1

If is an edge in a directed graphv0 is adjacent to v1, and v1 is adjacent from v0The edge is incident on v0 and v1

Subgraph and Path A subgraph of G is a graph G such that V(G)

is a subset of V(G) and E(G) is a subset of E(G) A path from vertex vp to vertex vq in a graph G,

is a sequence of vertices, vp, vi1, vi2, ..., vin, vq, such that (vp, vi1), (vi1, vi2), ..., (vin, vq) are edges in an undirected graph

The length of a path is the number of edges on it

Subgraph and Path

0 0

1 2 3

1 2 0

1 2

3

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

0

1 2

3

G1

Simple Path and Style

A simple path is a path in which all vertices, except possibly the first and the last, are distinct

A cycle is a simple path in which the first and the last vertices are the same

In an undirected graph G, two vertices, v0 and v1, are connected if there is a path in G from v0 to v1

An undirected graph is connected if, for every pair of distinct vertices vi, vj, there is a path from vi to vj

Simple Path and Style

0

1 2

3

0

1 2

3 4 5 6G1

G2

connected

tree (acyclic graph)

Connected Component

A connected component of an undirected graph is a maximal connected subgraph.

A tree is a graph that is connected and acyclic. A directed graph is strongly connected if there

is a directed path from vi to vj and also from vj to vi.

A strongly connected component is a maximal subgraph that is strongly connected.

Degree The degree of a vertex is the number of edges

incident to that vertex For directed graph, the in-degree of a vertex v is the number of

edgesthat have v as the head

the out-degree of a vertex v is the number of edgesthat have v as the tail

if di is the degree of a vertex i in a graph G with n vertices and e edges, the number of edges is

e d in

=( ) /0

1

2

Degree

undirected graphdegree

G1

30

1 2

333

3

directed graphin-degreeout-degree

0

1

2

in:1, out: 1

in: 1, out: 2

in: 1, out: 0

G2

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

Adjacency Matrix

0111

1011

1101

1110

010

100

010

01100000

10010000

10010000

01100000

00000100

00001010

00000101

00000010

G1 G2

0

1 2

3

0

1

2

1

0

2

3

4

5

6

7

symmetricundirected: n2/2

directed: n2

G3

Adjacency lists N linked list

1

2 0

3 1 2

2 3 0

1 3 0

2102

1

0

G1

0

1 2

3

G2

Graph Operations Traversal

Given G=(V,E) and vertex v, find all wV, such that w connects v.Depth First Search (DFS)

preorder tree traversalBreadth First Search (BFS)

level order tree traversal Spanning Trees

Graph Operations

depth first search: v0, v1, v3, v7, v4, v5, v2, v6breadth first search: v0, v1, v2, v3, v4, v5, v6, v7

Weighted Graph In Many applications,each edge of a graph has an

associated numerical value, called a weight Usually, the edge weights are non negative

integers Weight graphs may either directed or undirected

Spanning Trees A spanning tree is a minimal subgraph G, such

that V(G)=V(G) and G is connected Weight and MST Either dfs or bfs can be used to create a

spanning treeWhen dfs is used, the resulting spanning tree is

known as a depth first spanning treeWhen bfs is used, the resulting spanning tree is

known as a breadth first spanning tree

Minimum-Cost Spanning Trees A minimum-cost spanning tree is a spanning

tree of least cost Design strategy greedy methodKruskals algorithm

Edge by edgePrims algorithm

Span out from one vertexSollins algorithm

Hint: Construct from connected components Leave as a homework

01

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5 6

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5 6

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1014

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1 6

1 2

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4 5

0 1

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Examples for Kruskals Algorithm

Examples for Kruskals Algorithm

0

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5 6

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Examples for Kruskals Algorithm0 5

2 3

1 6

1 2

3 6

3 4

4 6

4 5

0 1

10

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28

0

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22

0

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5 6

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14 16

22

25

cycle cost = 10 +25+22+12+16+14

Examples for Prims Algorithm

0

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5 6

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25 25

22

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1014

Examples for Sollins Algorithm

0

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1614

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25

vertex edge0 0 -- 10 --> 5, 0 -- 28 --> 11 1 -- 14 --> 6, 1-- 16 --> 2, 1 -- 28 --> 02 2 -- 12 --> 3, 2 -- 16 --> 13 3 -- 12 --> 2, 3 -- 18 --> 6, 3 -- 22 --> 44 4 -- 22 --> 3, 4 -- 24 --> 6, 5 -- 25 --> 55 5 -- 10 --> 0, 5 -- 25 --> 46 6 -- 14 --> 1, 6 -- 18 --> 3, 6 -- 24 --> 4