Randomized Algorithms Randomized Algorithms CS648 Lecture 15 Randomized Incremental Construction (building the background) Lecture 15 Randomized Incremental.

Randomized AlgorithmsCS648

Lecture 15Randomized Incremental Construction

(building the background)

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Partition Theorem

A set of events ,…, defined over a probability space (,P) is said to induce a partition of if• = • =∅ for all

Partition Theorem:

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P(B) = )

ΩB

) ∙ ) This theorem solves

many difficult problems magically. But one needs some

experience in order to apply it effectively. You will realize it

soon.

PROBLEM 1FIND-MIN PROBLEM

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Find-Min algorithm

Find-Min(A[1..]){ A[1]; For to do { if ( ?? ) ?? } return ; }

Question: If elements of A are permuted randomly uniformly, what is the expected number of times variable is updated ?

: no. of times is updated.

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8 5 16 11 32 4 57 6 19 82 7 42 2 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14

A

A[] A[] ;

??

Find-Min algorithm

= Probability that A[] is smaller than {A[],…, A[]}

Notations:• : set of all subsets of A of size .• For any , : first elements of A are (some permutation of) .

Using Partition Theorem, =

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𝒊 𝒏−𝟏1 2 …

A

𝐏 (𝑿¿¿ 𝒊=𝟏∨𝓔𝒂)¿ 𝐏 (𝓔𝒂)

First elements

Though this equation is perfectly correct, you won’t be able to proceed from this

point onwards to find . Can you find the reason behind its

uselessness ?

PROBLEM 2CLOSEST PAIR OF POINTS

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Closest Pair of Points

Problem Definition:Given a set of points in plane, compute the pair of points with minimum Euclidean distance.

Deterministic algorithms:• O() : Trivial algorithm• O() : Divide and Conquer based algorithm

Randomized algorithm:• O() : Randomized Incremental Construction based algorithm

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Notations and assumptions

Notations: • : set of points in plane.

• Coordinates of each point are positive integers.

• distance(,) : Euclidean distance between and .

Assumption:

• Distance between each pair of points is distinct.

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A discrete math exercise

Exercise: What is the maximum number of points that can be placed in a unit square such that the minimum distance is at least 1 ?Answer: 4.

This exercise is used is deterministic algorithm as well the randomized algorithm that we shall discuss now.

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1

1

√2

If there are more than 4 points, at least one

of the four small squares will have more

than 1 points.

Overview of the randomized algorithm

• Incremental algorithm: starts with a set of 2 points, computes their distance; inserts 3rd point and updates the closest pair distance; inserts 4th point and updates the closest pair distance; … • Uses an efficient data structure, called Grid, to facilitate efficient

processing during ith step: - To determine if ith point is going to change the closest pair distance.

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Grid()

A data structure with operations:• Locate_cell(, ): Locates the cell to which belongs.

• Report_points(, ): Report all points belonging to cell .

• Insert_point(, ): Insert point in grid .

• Build_Grid(): Build grid for with parameter .

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𝜹

𝜹

: set of points

: distance between closest pair of points in .

Grid()

Height Balanced Binary search tree

Dynamic hashing

Locate_cell(, ) (1)

Report_points(, ) (1)

Insert_point(, ) (1) expected

Build_Grid() ) expected

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The following time bounds are possible.

Excluding Insert_point() operation, show as a homework, that we can achieve all other bounds using static hashing discussed in this course.

Closest Pair of Points

Closest-pair-algorithm() Let <,,…, > be a uniformly random permutation of ; distance(,); Build_Grid(, ); For to do { Step 1: locate the cell of the grid containing ; Step 2: find the point closest to ; let = distance(, ); Step 3: If ; Insert(, ); Else ; Build_Grid(, ); } return ;

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th iteration

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𝜹𝒊−𝟏

𝜹𝒊−𝟏

:𝒑𝒊

We just need to

insert

th iteration

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𝜹𝒊−𝟏

𝜹𝒊−𝟏

𝜹𝒊

:𝒑𝒊

We need to rebuild the grid

th iteration

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𝜹𝒊

𝜹𝒊

:𝒑𝒊

Analysis of th iteration

Closest-pair-algorithm() Let <,,…, > be a uniformly random permutation of ; distance(,); Build_Grid(, ); For to do { Step 1: locate the cell of the grid containing ; Step 2: find the point closest to ; let = distance(, ); Step 3: If ; Insert(, ); Else ; Build_Grid(, ); } return ;

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O(1)

O(1)

O(1)

for constant

running time of th iteration

: running time of th iteration

E[] = ??

Question: What is ?

depends upon which depends upon the first points.

So we need to use Partition theorem to calculate .

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O(1) + ∙

Calculating

Notations:

• : set of all subsets of of size .

• For any , : first points are (some permutation of) .

=

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𝐏 (𝜹¿¿ 𝒊<𝜹𝒊−𝟏∨𝓔𝒂)¿ 𝐏 (𝓔𝒂)

Though this equation is perfectly correct, you won’t be able to proceed from this

point onwards to find Can you find the reason behind its

uselessness ?

Homework Exercise

• Investigate the cause of problem in our forward analysis for each of the two problems.

• Try to find alternate approach for analysis. (Backward analysis )

• Provide efficient implementation of Grid data structure.

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I am hopeful that at least one of

you will reinvent this technique on

his/her own before next class.