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Data Structures and Algorithms(CS210/ESO207/ESO211)

Lecture 34

Greedy strategy: a new algorithm paradigmPart III

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Correctness of a Greedy algorithm

the optimal solution of original instance of the problem

the optimal solution of ( smaller ) instance obtained by the greedy step

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A suitable Theorem

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Continuing Problem from last class

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Minimum spanning tree

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Problem Description

Input: an undirected graph G=(V,E) with : E ,

Aim: compute a spanning tree (V,E), E E such that ( )E is minimum .

Lemma 1 (proved in last class):

If E is the edge of least weight in G, then there is a MST containing .

Lemma 2:If E is the edge of least weight in G, then must be present in the MST.

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How to use Lemma 2 to designan algorithm for MST ?

Assume all edge weights are distinct

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How to compute a MST ?

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This is graph

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Let (u ,v) be the least weight edge in =( , ). Transform into as follows. Remove vertices u and v and add a new vertex w For each edge ( u ,x) , add edge ( w ,x) in . For each edge ( v,x) , add edge ( w ,x) in . In case of multiple edges between w and x, keep only the lighter weight edge.

Theorem1: W_MST ( ) = W_MST ( ) + w (u ,v)Proof: (by construction)

1. W_MST ( ) W_MST ( ) + w (u ,v)

2. W_MST ( ) W_MST ( ) - w (u ,v)

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Use Lemma 2

straightforward

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Using Theorem1 , we get an algorithm for MST with the following drawbacks :

Complicated to implement.

Inefficient.

less insightful (does not convey much about the structure of MST).

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A useful lesson for design of a graph algorithm

If you have a complicated algorithm for a graph problem,

search for some graph theoretic property

to design simpler and more efficient algorithm

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Two graph theoretic properties of MST

Cut property

Cycle property

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Every algorithm till date is based onone of these properties !

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Cut Property

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Cut Property

Definition: For any subset , such that , cut ( , ) is defined as:cut ( , ) = { ( , ) | and or and }

Cut-property : The least weight edge of a cut ( , ) must be in MST.

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cut ( , )

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Pursuing greedy strategyto minimize weight of MST,what can we say about the

edges of cut ( , ) ?

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Proof of cut-property

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Let be the MST, and ( u ,v) .

Question: What happens if we remove (x,y) from , and add (u ,v) to .

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Question: What happens if we remove (x,y) from , and add (u ,v) to .

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We get a spanning tree with weight < weight ( )

A contradiction !

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Cycle Property

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Cycle Property

Let be any cycle in .

Cycle-property :Maximum weight edge of any cycle can not be present in MST.

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Pursuing greedy strategy

to minimize weight of MST,what can we say about the

edges of cycle ?

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Proof of Cycle property


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Swapping ( u ,v) with ( f ,s), we get a spanning tree with weight < weight ( )

A contradiction !

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Algorithms based on cut Property

Spend some time on designing such an algorithmbefore going to the next slide

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How to use cut property to compute a MST ?

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Algorithm based on cycle Property

Spend some time on designing such an algorithmbefore going to the next slide

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An Algorithm based on cycle propertyDescription

Algorithm (Input: graph =( , ) with weights on edges)While ( ?? ) do

{ Compute any cycle ;Let (u ,v) be the maximum weight edge of the cycle ;

Remove (u ,v) from ; }

Return ;

Number of iterations of the While loop : ??

Time spent in one iteration of While loop: ??

Running time of the algorithm: O( )

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has any cycle

+

O ( )

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Homework

Ponder over the following questions before coming for the next class

Use cycle property and/or cut property to design a new algorithm for MST

Use some data structure to improve the running time of the algorithms discussed inthis class to O( )

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