Kruskal_s2009

8/2/2019 Kruskal_s2009

http://slidepdf.com/reader/full/kruskals2009 1/41

1

NETWORK

PN NORIZAH BT AHMAD

8/2/2019 Kruskal_s2009


2

Network

• A graph in which each edge (orarc) is given a value called itsweight.

• The weight on an edge mayrepresent a distance, a time ora cost.

• A graph also called asweighted graph.

8/2/2019 Kruskal_s2009


3

Example

• If we add distances to our route map theresulting network would look like this.

8/2/2019 Kruskal_s2009


4

MINIMAL SPANNING TREE (MST) (or

MINIMUM CONNECTOR)

• To make a selection of the available arcs

• So, any node can be reached from anyother node such that the total length of the chosen arcs or edges is as small aspossible

• A connected set of arcs with no loops iscalled a tree

• The set which solves the minimalconnector problem is called minimalspanning tree for the network.

8/2/2019 Kruskal_s2009


5

APPLICATION OF MINIMAL SPANNING

TREE

(1) Design of network – example: telephone networks

- To determine the least costly paths with no cycles inthis network, therefore connecting everyone at a

minimum cost

(2) Airline routes- vertices represent cities and edgesrepresent routes between the cities

- MST can be applied to optimize airlines routes byfinding the least costly paths with no cycles

(3) cable TV companies – to connect each town in some wayto the base station to received the signal but usingminimum cable.

8/2/2019 Kruskal_s2009


6

The Minimum-ConnectorProblem

8/2/2019 Kruskal_s2009


7


Example

If A, B and F are linked as shown inthe figure below, B is already

connected to F via A so it isunnecessary to choose arc BF.

A

B

F

8/2/2019 Kruskal_s2009


8

There are two ideas

• Kruskal’s and Prim’s Algorithmsto find the minimal spanningtree (The Minimum-Connector

Problem) of a weighted graph.• Dijkstra’s Algorithm to find the

shortest or least value path

between two nodes on aweighted graph.

8/2/2019 Kruskal_s2009


9


•Kruskal’s Algorithm

•Prim’s Algorithm

8/2/2019 Kruskal_s2009


• KRUSKAL’S ALGORITHM

10

8/2/2019 Kruskal_s2009


11

Kruskal’s algorithmn

• choose the smallest availableedges.

• not worrying about any connection

to edges that have already beenchosen

• except that it is careful not to form

a cycle

8/2/2019 Kruskal_s2009


INTRODUCTION

• Kruskal's algorithm is an algorithm ingraph theory to find a minimum spanning tree for

a connected weighted graph.

• This means it finds a subset of the edges that

forms a tree that includes every vertex, wherethe total weight of all the edges in the tree is

minimized.

http://en.wikipedia.org/wiki/Algorithm

http://en.wikipedia.org/wiki/Graph_theory

http://en.wikipedia.org/wiki/Minimum_spanning_tree

http://en.wikipedia.org/wiki/Edge_%28graph_theory%29

http://en.wikipedia.org/wiki/Vertex_%28graph_theory%29

http://en.wikipedia.org/wiki/Vertex_%28graph_theory%29

http://en.wikipedia.org/wiki/Edge_%28graph_theory%29

http://en.wikipedia.org/wiki/Minimum_spanning_tree

http://en.wikipedia.org/wiki/Graph_theory

http://en.wikipedia.org/wiki/Algorithm

8/2/2019 Kruskal_s2009


INTRODUCTION

• If the graph is not connected, then it finds aminimum spanning forest (a minimum spanning

tree for each connected component).

• Kruskal's algorithm is an example of a greedy

algorithm.

8/2/2019 Kruskal_s2009


14

The algorithm can be stated likethis :

• Step 1 – Sort the edges in ascending orderof length / weight.

• Step 2 – Select the shortest edges in thenetwork.

• Step 3 – Select, from the edges which arenot in the solution, the shortest edgeswhich does not form a cycle. (Where twoedges have the same weight, select at

random)

• Step 4 – Repeat step 3 until all the verticesare in the solution or form a spanning tree.

8/2/2019 Kruskal_s2009


STEPS IN FINDING MST

STEP 1

This is our original

graph. The numbersnear the arcsindicate their weight.None of the arcs arehighlighted.

8/2/2019 Kruskal_s2009


STEP 2

AD and CE are the

shortest arcs, withlength 5, and AD has been arbitrarilychosen, so it ishighlighted.

8/2/2019 Kruskal_s2009


STEP 3

However, CE isnow the shortestarc that does not

form a loop, withlength 5, so it ishighlighted as thesecond arc.

http://en.wikipedia.org/wiki/Image:Kruskal_Algorithm_2.svg

8/2/2019 Kruskal_s2009


STEP 4

The next arc,DF with length6, is highlightedusing much thesame method

8/2/2019 Kruskal_s2009


STEP 5

The next-shortest arcsare AB and BE, bothwith length 7. AB ischosen arbitrarily, andis highlighted. The arcBD has beenhighlighted in red,because it would forma loop ABD if it werechosen.

8/2/2019 Kruskal_s2009


STEP 6

The process continues to highlightthe next-smallest arc, BE withlength 7. Many more arcs arehighlighted in red at this stage: BC because it would form the loopBCE, DE because it would form

the loop DEBA, and FE because itwould form FEBAD.

8/2/2019 Kruskal_s2009


STEP 7

Finally, the

processfinishes withthe arc EG of length 9, andthe minimum

spanning treeis found.

8/2/2019 Kruskal_s2009


22

Example 1

• Use Kruskal’s algorithm to obtain a minimumspanning tree for this network.

A

8

B

D

C

E5

6

7

4

1

3

8/2/2019 Kruskal_s2009


23

STEP 1 – Sort the edges inascending order of weights

Weight Edge

1 EC

3 EB

4 BC

5 AD

6 ED

7 CD

8 AB

8/2/2019 Kruskal_s2009


24

STEP 2 – Select the edge of leastweight

• The edge of least weight is edge EC.

A

8

B

D

C

E5

7

4

1

3

8/2/2019 Kruskal_s2009


25

Step 3 – select the edge of leastweight that does not form a cyclewith the edges already included

• The edge of smallest weight available is edgeEB.

A

8

B

D

C

E5

6

7

4

1

3

8/2/2019 Kruskal_s2009


26

• The edge of smallest weight available is edgeBC. However, this would form a cycle. So it is

not chosen.

• Choose edge AD.

A8

B

D

C

E5

67

4

1

3

8/2/2019 Kruskal_s2009


27

• The smallest edge is now DE.

A

B

D

C

E5

6

1

3

8/2/2019 Kruskal_s2009


28

• All vertices are now in the solution and form aminimum spanning tree.

• A total weight = 6 + 5 + 3 + 1 = 15

A

B

D

C

E5

6

1

3

8/2/2019 Kruskal_s2009


29

Example 2

• Find the minimal Spanning Tree

8/2/2019 Kruskal_s2009


30

Solution

• Arcs (edges) ranked in order of increasinglength.

Length Arc/Edge

________________________________________ 2 AB, DE

3 DF, CF

4 EF5 AF, CD

6 AE, BF

8 BC

8/2/2019 Kruskal_s2009


31

Solution

• We can start by selecting AB or DE,so let us arbitrarily select AB.

• Select DE

B

2

2

2

8/2/2019 Kruskal_s2009


32

Solution

• We can select CF or DF, so let usarbitrarily select DF.

2

3

2

8/2/2019 Kruskal_s2009


33

Solution

• Select CF

2 3

3

2

8/2/2019 Kruskal_s2009


34

Solution

• The next shortest arc is EF but E and F arealready connected via D, so we do notselect EF.

• The next shortest arcs are CD and AF but

C and D are already connected via F, sowe choose AF

2

5

3

3

2

8/2/2019 Kruskal_s2009


35

Solution

• The minimal spanning trees is2 + 2 + 3 + 3 + 5 = 15

8/2/2019 Kruskal_s2009


36

Example 3

Use Kruskal’s algorithm to find the leastamount of cable needed to solve theproblem below.

8/2/2019 Kruskal_s2009


37

Solution

• First, rank the edges in order of length.Halifax to Huddersfield 8

Halifax to Bradford 10

Bradford to Leeds 12Bradford to Huddersfield 12

Leeds to Wakefield 13

Huddersfield to Wakefield 14Bradford to Wakefield 15

Leeds to Harrogate 20

8/2/2019 Kruskal_s2009


38

Solution

• Now we begin to select edges,starting with the smallest.

• We have now connected all the

vertices into the spanning tree.• The length of our minimum

spanning tree is 8 + 10 + 12 + 13+ 19 = 62 miles

8/2/2019 Kruskal_s2009


39

Solution

• The solution can be shown in the diagram below.

8/2/2019 Kruskal_s2009


40

ISL

• Reading - Decision Mathematics D1 pg 60 – 65

• Self – assessment exercise 3A pg 66 No 1 & 2

Th k f

8/2/2019 Kruskal_s2009


41

Thank you for yourattention

Can we proceed to anothersubtopic ?

Prim’s Algorithm to find another

MST.

Kruskal_s2009

Documents