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FIT 1029 Algorithmic Problem Solving

Lecture 2 Fundamental Data Structures

Prepared by Julian Garca

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Lists

GraphsFirst Item.Second Item.

Third Item.

Tables

List

Simple List

First itemAnother itemYet another item

0.1.2.

ItemsIndices

NameStart numbering at 0.

Notation

0. stop1. pots2. tops

Anagrams

Anagrams[1] = pots

Anagrams = [stop, pots, tops]

Referring to elements in a list

L 1 2 3 4 5 6 70 1 2 3 4 5 6

L[0, 6] or L

L 1 2 3 4 5 6 70 1 2 3 4 5 6

L[0, 2]

L 1 2 3 4 5 6 70 1 2 3 4 5 6

L[2, 5]

L[0, n-1]A list L of size n

L 1 2 3 4 5 6 70 1 2 3 4 5 6

L[1] is 2

Using lists to solve a problem

Age of the kidsTwo men meet on the street A: All three of my sons celebrate their birthday today. Can you tell

me how old each one is? B: Yes, but you have to tell me something about them A: The product of their ages is 40. B: I need more info A: The sum of their ages is equal to the number of windows in the

building next to us B: I need more info A: My youngest son has blue eyes. B: That is sufficient!

Age of the kidsTwo men meet on the street A: All three of my sons celebrate their birthday today. Can you

tell me how old each one is? B: Yes, but you have to tell me something about them A: The product of their ages is 40. B: I need more info A: The sum of their ages is equal to the number of windows in the

building next to us B: I need more info A: My youngest son has blue eyes. B: That is sufficient!

2 Z+Y,M,E

1 Y M E

Y M E = 40

40

All three of my sons celebrate their birthday today. Can you tell me how old each one is?

The product of their ages is 40.

Possible solutions

1, 1, 400.1. 1, 2, 202. 1, 4, 10

1, 5, 83.2, 2, 104.2, 4, 55.

Have we got all possible solutions? check!


tell me how old each one is? B: Yes, but you have to tell me something about them A: The product of their ages is 40. B: I need more info A: The sum of their ages is equal to the number of windows in

the building next to us B: I need more info A: My youngest son has blue eyes. B: That is sufficient!

Simple List

1, 1, 400.1.2.

1, 2, 201, 4, 101, 5, 82, 2, 102, 4, 5

3.4.5.

422315141411

Simple List

1, 1, 400.1.2.

1, 2, 201, 4, 101, 5, 82, 2, 102, 4, 5

3.4.5.

422315141411

y = 1, m =5, e = 8

Age of the kidsConstruct a list of possible solutions

For each possible solution, find the sum of the ages

Find those solutions which have the same sum of ages

Find out of those solutions which ones have an youngest son

Start

End

Lists

GraphsFirst Item.Second Item.

Third Item.

Tables

Binomial[4, 2] = 6

0 1 2 3 4

0 1

1 1 1

2 1 2 1

3 1 3 3 1

4 1 4 6 4 1

Binomial:

Name of Table

Row Column

Column numbers

Row numbers

Tables

Distance[2,1] = 12

3 2 14 5

1 1 4 2

1 12 13 2

11 3 3 1

Distance:0 1 2 3

0

1

2

3

M[0..n-1 , 0.. m-1], table M with n rows and m columns.

Distance = [[3, 2, 14, 5], [1,1,4,2], [1,12,13,2], [11,3,3,1]]

Graphs

Source: http://www.facebook.com/notes/facebook-engineering/visualizing-friendships/469716398919

Visualizing friendship, Graph by Paul Butler.

Creating MazesDevelop a procedure for creating mazes.

Nine crossroads Andrea Gilbert (www.clickmazes.com)

First how do we represent a maze?

Simple Maze

Enter

Exit

l l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l lEnter

Exit

Mark the junctions

Make passage ways

l l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l


Exit

Representation of Maze

l l

l l l

l

l

l l

l l

l

l

l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l

l

Graphs

VertexVertex

Vertex

GraphsEdge

Edge

Edge

Graphs

Path

Start vertex

Endvertex

Graphs

Circuit

Circuit

No repetition of edges allowed

Undirected vs Directed

Non-simple vs Simple

Non-simple vs SimpleNo Multiple

edges

No Loops

Non-simple vs Simple

Disconnected vs Connected

Disconnected vs Connected

There is a path between any two vertices

Unweighted/Weighted2

16

12

10

1

0 2

43

5

24

Unlabelled/Labelled

Q

W

CBA

X

TE

P

Trees

Graphs which are: Simple

no loops or multiple edges

Connected No circuits.

Spanning Tree of a graph G

Spanning Tree of a graph G

Assume G is simple and connected.

A spanning tree of G is a tree which:

Contains all the vertices of G, and

The edges of the tree are edges of G.

How to get a spanning tree?

Prims algorithmChoose a vertex in the

Graph

Put the vertex in the Tree

Is there a vertex not in the Tree?

Choose an edge between the Tree and this vertex

Add the edge to the Tree.

Yes

Start

EndNo

OutputInput Is the output a tree? Contains all vertices in input?

Edges are edges of the input?

Can you envision any real world applications of Prims algorithm?

Start

End

Start

End

Iteration

Selection

Conditions

Fixed iteration vs Conditional Iteration

What about mazes?

Start with a Grid

l l

l l l

l

l

l l

l l

l

l

l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

Find a Spanning Tree

l l

l l l

l

l

l l

l l

l

l

l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

Add entrance and exit

l l

l l l

l

l

l l

l l

l

l

l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l

l

Enter

Exit

Put in walls

l l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l

l l l l l l l


Exit

Remove Tree

Enter

Exit

Summary

Lists Tables Graphs

Water Bottle ProblemSuppose you have a 5 litre bottle and a 3 litre bottle (without markings). How can you get

exactly 4 litres from the well?

3 5

Before Next LectureOn moodle: Watch the following videos:

What is an Algorithm? The components of a complex algorithm?

Read Describing Algorithms, by Robyn A. McNamara (again).

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