Randomized Algorithms CS648

Lecture 26

A few memorable problems

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RANDOMIZED INCREMENTAL CONSTRUCTION

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Convex Hull algorithm

Theorem:

There is an O(𝒏 𝐥𝐨𝐠 𝒏 ) time Las Vegas algorithm for computing convex hull.

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Assignment 4: Concentration of running time.

Randomized QuickSort

Aim: To show P(𝑿 > 8𝑛 l𝑜𝑔4/3 𝑛) < 𝑛−7

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𝑒𝑖 Elements of A arranged in Increasing order of values

𝑒𝑗

Right perspective

Randomized QuickSort

Aim: To show P(𝑿 > 8𝑛 l𝑜𝑔4/3 𝑛) < 𝑛−7

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𝑒𝑖 Elements of A arranged in Increasing order of values

Right perspective

Randomized QuickSort from perspective of 𝑒𝑖

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Elements of A arranged in Increasing order of values

𝑒𝑖

𝑒𝑖 leaves the algorithm

Randomized QuickSort from perspective of 𝑒𝑖

𝐘𝑖: no. of recursive calls in which 𝑒𝑖 participates before being selected as a pivot.

Question: Is there any relation between 𝐗 and 𝐘𝑖 ?

Answer: 𝐗 = 𝐘𝑖𝑛𝑖=1

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CONVEX HULL ALGORITHM

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Randomized algorithm for convex hull

Convex-hull-algorithm(𝑷) { Let <𝒑𝟏,𝒑𝟐,…,𝒑𝒏 > be a uniformly random permutation of 𝑷;

𝑯 triangle(𝒑𝟏,𝒑𝟐, 𝒑𝟑);

For 𝒊 = 𝟒 to 𝒏 do

insert 𝒑𝒊 and update 𝑯

return 𝑯;

}

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Conflict graph : a powerful data structure

𝒄𝟒

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

cones 𝒏 − 𝒊 points 𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟏

Before entering the for loop

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒏 − 𝟑 points cones

INSERTING 𝒊TH POINT

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𝒊th iteration

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𝒄𝟏

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒊th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 + 𝟏 points cones 𝒄𝟏

𝒊th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 + 𝟏 points cones 𝒄𝟏

𝒊th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 + 𝟏 points cones 𝒄𝟏

𝒊th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 points cones 𝒄𝟏

𝒊th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 points cones 𝒄𝟏

𝒊th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 points cones 𝒄𝟏

𝒊th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 points cones 𝒄𝟏

𝒊th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 points cones 𝒄𝟏

𝒄′

𝒄"

𝒊th iteration

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𝒄𝟏

𝒄′

𝒄"

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄′

𝒄"

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 points cones 𝒄𝟏

Running time of 𝒊th iteration

Running time of 𝒊th iteration is of the order of

• Number of edges of convex hull that are destroyed

• Number of new edges of convex hull that are created

• Number of points in the two adjacent cones that get created

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Total time for 𝒏 iterations = O(𝒏)

𝑿𝒊

Key idea: No. of cones to which a point belongs during the algorithm

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LOAD BALANCE

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Balls into Bins

Ball-bin Experiment: There are 𝑚 balls and 𝑛 bins.

Each ball selects its bin randomly uniformly and independent of other balls

and falls into it.

Applications: • Hashing

• Load balancing in distributed environment 32

1 2 3 … 𝑖 … 𝑛

1 2 3 4 5 … 𝑚 − 1 𝑚

Balls into Bins

Question : What is the load of most heavily loaded bin ?

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1 2 3 … 𝑖 … 𝑛

1 2 3 4 5 … 𝑚 − 1 𝑚

O(log 𝒏 / log log 𝒏)

Balls into Bins

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1 2 3 … 𝑖 … 𝑛

𝒊

Balls into Bins

Question : What is the load of most heavily loaded bin ?

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1 2 3 … 𝑖 … 𝑛

O(log log 𝒏)

𝒊

A NOVEL SOLUTION

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Balls into Bins

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1 2 3 … 𝑖 … 𝑛

Generalization

What if each ball chooses 𝒅 bin randomly uniformly ?

What if there is asymmetry in breaking ties ?

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O(𝐥𝐨𝐠 𝐥𝐨𝐠 𝒏

𝐥𝐨𝐠 𝒅)

Balls into Bins

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1 2 3 … 𝑛

2+ 1 … 𝑛

𝒊

O(𝐥𝐨𝐠 𝐥𝐨𝐠 𝒏

𝒅)

Berthold Vöcking: How asymmetry helps load balancing. J. ACM 50(4): 568-589 (2003)

Interested students should ponder over it. The following paper gives an elegant proof for this claim. This is the most beautiful proof I ever read.

This is the end of the course.

• If you liked the course, it is because of the radiating beauty of “randomized algorithms”.

• If you dislikes the course, it might be because I did not do a good job.

• I look forward to your feedback/evaluation to improve.

• Thanks a lot and best wishes for your wishes.

• I am truly honored to teach the brightest students of my country

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