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Data Structures and Algorithms

(CS210/ESO207/ESO211)

Lecture 32

Greedy strategy: a new algorithm paradigm

PartI

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Overview of this lecture

In addition to solving a job scheduling problem, this lecture will give you an

outline of how any algorithmic problem is solved. In short, designing an

efficient and provably correct algorithm for a problem involves

Developing familiarity with the problem through some examples.

Examples help you develop some intuition/insight into the problem.

The intuitions guide us towards some strategy to solve the problem.

But intuitions may be sometimes misleading (due to lack of full understanding

about the problem).

So many times you fail, but each failure adds to your understanding of the problem

and further guides you towards the solution of the problem.

If one has positive attitude and lot of perseverance, he/she arrives at the

algorithm through a sequence of milestones of failures ...

I hope the reader will get a glimpse of the abovejourneyin this lecture.

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Two problems

1. Job scheduling problems

- a set of jobs

- a set of servers

- some optimization criteria

- constraintsAim:to select an optimal subset of jobs to be executed on servers

such that all constraints are satisfied.

2. Graph problems

- Minimum spanning tree (to be explained in detail in the the next class)

- Shortest paths

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Problem 1

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JOB Scheduling

Largest subset of non-overlapping job

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A job scheduling problemDescription

INPUT:

Aset of jobs {,,, } where jobis specified by two real numbers

s(): start time of job

f(): finish time of job

A single server

Constraint: Server can execute at most one job at any moment of time.

Aim:To select the largest subset of non-overlapping jobs which can be executed by

the server.

(job is said to be non-overlapping with job if [s(),f()] [s(),f()]= )

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A job scheduling problemDeveloping familiarity with the problem through an example

INPUT: (,),(. ,. ), (. ,. ), (. ,), (,), (. ,. ), (. ,. ), (. ,. )

It makes sense to work with pictures than these numbers

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Try to find solution for the above

example.

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A job scheduling problemDeveloping familiarity with the problem through an example

INPUT: (,),(. ,. ), (. ,. ), (. ,), (,), (. ,. ), (. ,. ), (. ,. )

Seeing the solution (red bars) of this example, what strategy comes to your mind tosolve this problem ?

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Designing algorithm for a problem

1. Choose a strategy based on some intuition

2. Transform the strategy into an algorithm.

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Try to prove

correctness

of the

algorithm

cTry to design a

conterexample

c

Stop as soon as either of

these goals is reached

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Designing algorithm for the problemstrategy 1: Select the earliest start timejob

Intuition:

It might be better to assign jobs as early as possible so as to make optimum use of server.

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When one fails to prove the correctness of

this strategy, one should search for a

counterexample. Can you transform the

above example into a counterexample ?

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Designing algorithm for the problemstrategy 1: Select the earliest start timejob

Intuition:

It might be better to assign jobs as early as possible so as to make optimum use of server.

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Designing algorithm for the problemstrategy 1: Select the earliest start timejob

Intuition:

It might be better to assign jobs as early as possible so as to make optimum use of server.

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This is indeed a counterexample. Instead of

getting disappointed, try to realize that this

counterexample points towards some other

strategy which might work.

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Designing algorithm for the problemstrategy 2: Select the job with smallest duration

Intuition:

Such a job will make least use of the server and so might lead to larger number of jobs to be

executed

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When one fails to prove the correctness of

this strategy, one should search for a

counterexample. Can you transform the

above example into a counterexample ?

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Designing algorithm for the problemstrategy 2: Select the job with smallest duration

Intuition:

Such a job will make least use of the server and so might lead to larger number of jobs to be

executed

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Designing algorithm for the problemstrategy 2: Select the job with smallest duration

Intuition:

Such a job will make least use of the server and so might lead to larger number of jobs to be

executed

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This is indeed a counterexample. Instead of

getting disappointed, try to realize that this

counterexample points towards some other

strategy which might work.

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Designing algorithm for the problemstrategy 3: Select the job with smallest number of overlaps

Intuition:

Selecting such a job will avoid least number of other jobs to be discarded.

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When one fails to prove the correctness of

this strategy, one should search for a

counterexample. Can you transform the

above example into a counterexample ?

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Designing algorithm for the problemstrategy 3: Select the job with smallest number of overlaps

Intuition:

Selecting such a job will avoid least number of other jobs to be discarded.

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Counterexample again

!Let us not give upour perseverance. After all, these strategies

and counterexamples have increased our

understanding about the problem. Think of

some other strategy after studying these

counterexamples.

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Designing algorithm for the problemstrategy 4: Select the job with earliest finish time

Intuition:

Selecting such a job will free the server earliest and hence more no. of jobs will get scheduled.

Let us apply this strategy on some reasonably tough example.

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Designing algorithm for the problemstrategy 4: Select the job with earliest finish time

Intuition:

Selecting such a job will free the server earliest and hence more no. of jobs will get scheduled.

Let us apply this strategy on some reasonably tough example.

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Designing algorithm for the problemstrategy 4: Select the job with earliest finish time

Intuition:

Selecting such a job will free the server earliest and hence more no. of jobs will get scheduled.

Let us apply this strategy on some reasonably tough example.

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Designing algorithm for the problemstrategy 4: Select the job with earliest finish time

Intuition:

Selecting such a job will free the server earliest and hence more no. of jobs will get scheduled.

Let us apply this strategy on some reasonably tough example.

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Designing algorithm for the problemstrategy 4: Select the job with earliest finish time

Intuition:

Selecting such a job will free the server earliest and hence more no. of jobs will get scheduled.

Let us apply this strategy on some reasonably tough example.

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It is indeed a correct solution forthis example. We should try to

prove the correctness of the

algorithm.

But first we give a full description

of the algorithm based on this

strategy.

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Algorithmearliest finish timeDescription

Algorithm (Input : set of jobs.)

1. Define;

2. While do

{ Let be the job from with earliest finish time;

U{};Remove and all jobs that overlap with from set;

}

3. Return;

Running time for a trivial implementation of the above algorithm: O()

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Algorithmearliest finish timeCorrectness

Let

be the job with earliest finish time.

Lemma1: There exists an optimal solution for in which is present.

Proof: Consider any optimal solution for. Let us suppose .

Let be the job from with earliest finish time.

Let be the set obtained by replacing in by .

We claim thatis also an optimal solution.

(Suitable arguments were given in the class to show that consists of non-overlapping intervals.The fact that has earliest finish time played a key role for this. Try to reconstruct these arguments.)

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Homework exercises

Spend 30 minutes today on the following problems.

1. Use Lemma1to complete the proof of correctness of the algorithm.

2. Design an O(log ) implementation of the algorithm.

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