Graphs and Sets - PowerPoint Presentation

Slide1

Graphs and Sets

Dr. Andrew Wallace PhD

BEng

(hons)

EurIng

andrew.wallace@cs.umu.seSlide2

Overview

Sets

Implementation

ComplexityGraphsConstructing GraphsGraph examplesSlide3

Sets

Collection of items

No specified ordered

Unique valuesImplementation of mathematical concept of finite setStatic of dynamicSlide4

Sets

Items in a set are

members

of the set No sets of sets but …

Subsets

Union of sets

Intersections

 Slide5

Sets

Examples:

int

, float, charArraysFunctionsObjects (struct)Slide6

Set operations

Create

Insert

RemoveIs member ofIs emptySelectSize ofEnumerateSlide7

Implementation

Simple

Array

ListEfficientTreesRadix treesHash tablesSlide8

Implementation

Insert

Check for duplicates

UnionIf list has duplicatesCheck when doingEqualRemoveIntersectionDifferenceSlide9

Implementation

Bit vector

0 or 1

Set the bit if the item is in the queueFaster operationsBit operations in hardware (bitwise AND or OR)Masking0010110101 AND 0010000000Minimise memory

 Slide10

Implementation

Bit field

struct

bField{ int

: 6

;

int

m_nVal1 : 3; int m_nVal2 : 4; int m_nVal3 : 6;}Slide11

Implementation

Limitations

Can’t

use bit field variables in an arrayCan’t take the memory address of a bit field variableCan’t overlap integer boundariesSlide12

Implementation

Priority Queues

Linux kernel

Caching algorithms / memory pagesSlide13

Complexity

Depends on implementation

Improve set operations such as union or intersection

Improve insert, search, removeO(n) or O(logn

)

Some set operations can take O(

m

*

n)Slide14

Graphs

Set of nodes or vertices + pairs of nodes

G

= (V, A)Directed or undirectedUndirected a to b is the same as b to aNode - undirectedVertices - directed

Edge

Arcs (directed)

Connection between nodes

Weighted or

unweightedacbdSlide15

Graphs

V = {a, b, c, d}

A = {(a, b), (a, c), (b, d), (c, b)}

Adjacency2 edges are adjacent if the share a common vertex(a, c) and (a, b)2 vertices are adjacent if they share a common edgea and cJoin

Incident

An edge and vertex on the edge

a

c

bdSlide16

Graphs

struct

Node

{ int nNodeID;

Node*

pOut

;

int nOut;};Slide17

Graphs

s

truct

Edge{ int nEdgeID;

int

nStart

; int nEnd;}; Slide18

Graphs

Trivial graph

One vertex

Edgeless graphVertices and no edgesNull graphEmpty setSlide19

Graphs

Paths

Sequence of nodes

{a, b, d}Simple pathNo repetition of nodesCyclic pathAround and around in circles!

Walk

Open walk = path

Closed walk = cyclic path

Trail = walk with unique edges

acbdSlide20

Graphs

Connected graph

All nodes have a path to all other nodes

Sub graphsVertices are a sub set of GAdjacency relationship are a subset of G’s and restricted to the subgraphComplete graph

All nodes connected to all other nodes

Undirected graph A = n(n-1)/2

If A < n-1, then the graph is not connected

a

cbd

c

b

aSlide21

Graphs

Weighted graphs

Maps

Places as nodeRoads as arcsDistances as weights on the edgesSlide22

Graphs

Weights can represent “cost”

Some algorithms require restrictions on weights

Rational numbers or integersAll positive integersWeight of a pathSum of the weights for a given path

a

c

b

d

10

20

23

13Slide23

Constructing Graphs

Adjacency lists

An array of arrays of adjacent vertices

ac

b

d

a

b

d

c

c

b

d

bSlide24

Constructing Graphs

int

**

pNodeArray;pNodeArray = (int**)

malloc

(

sizeof

(

int) * 10);for(i=0;i<10;i++){ pNodeArray[i

] = malloc

(sizeof

(int) *

NumNodesOut[

i

]);

}

pNodeArray

[

i

][j] =

nNodeOut

;Slide25

Constructing Graphs

Adjacency matrix

Node x node matrix (

n x n)a

c

b

d

1

1

1

1Slide26

Constructing Graphs

Undirected graph

symmetrical

ac

b

d

1

1

1

1

1

1

1

1Slide27

Constructing Graphs

int

**

pNodeArray;pNodeArray = (int**)

malloc

(

sizeof

(

int) * 10);for(i=0;i<10;i++){ pNodeArray[i] = malloc

(sizeof

(int)

* 10);

}

pNodeArray

[

i

][j] =

1;Slide28

Graph examples

Robot navigation

AGV (automatic Guided Vehicles)

Free space pathsPick up and drop off pointsMap as a graphSlide29

Graph exampleSlide30

Graph example

Computer NetworksSlide31

Questions?