Lecture 16

Randomized AlgorithmsCS648

Lecture 16Randomized Incremental Construction

(Backward analysis)

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PROBLEM 1FIND-MIN PROBLEM

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Find-Min algorithm

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

: no. of times is updated.

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??

𝒏−𝟏1 2 …

A

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

Forward analysis for

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

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

First elements

Forward analysis for

: a subset of elements. = “Given that first elements of A are some permutation of , what is prob. that = ?”

For this event to happen, A[] must be smaller than every element of . depends upon .

For example, if the smallest element of has rank in A, then = ??

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𝒌−𝟏𝒏−𝒊+𝟏

Dependency on makes it hard to calculate

Backward analysis for

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

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

First elements

Backward analysis for

: a subset of elements. = “Given that first elements of A are some permutation of , what is prob. that = ?”

For this event to happen, the smallest element of must appear at A[].

=

Every element of is equally likely to appear at place A[].

= ??

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“the smallest element of appear at A[]”

𝟏𝒊

Fact: A is permuted randomly uniformly

Same for each

Backward analysis for

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

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

First elements

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.

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

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th iteration

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𝜹𝒊−𝟏

𝜹𝒊−𝟏

Grid structure for first points

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 ?

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O(1) + ∙

Forward analysis for

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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Forward analysis for

: a subset of points from .:first points of are some permutation of

= ??

Calculating :

Let be the set of all subsets of of size .

=

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𝜹𝒊−𝟏

𝜹𝒊−𝟏

Grid structure for first points

Depends upon

Backward analysis for

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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Backward analysis for

: a subset of points from .: first points of are some permutation of

= ??

Calculating :

Let be the set of all subsets of of size .

= = =

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𝜹𝒊

𝜹𝒊

𝟐𝒊

= Grid structure for first points

running time of th iteration

: running time of th iteration

E[] = O(1) + ∙ = O(1) + ∙ = O(1)

Expected running time of the algorithm :

Theorem:There exists a linear time Las Vegas algorithm to compute closest pair of points in plane.

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¿∑𝒊≤𝒏

O(1) =

RANDOMIZED INCREMENTAL CONSTRUCTION

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Randomized Incremental Construction

• Permute the elements of input randomly uniformly.

• Build the structure incrementally.

• Keep some data structure to perform th iteration efficiently.

• Use Backward analysis to analyze the expected running time.

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Randomized Incremental Construction

• Convex Hull of a set of points

• Trapezoidal decomposition of a set of segments.

• Convex polytope of a set of half-planes

• Smallest sphere enclosing a set of points.

• Linear programming in finite dimensions.

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PROBLEM 3CONVEX HULL OF POINTS

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Convex hull of Points

Problem definition: Given points in a plane, compute a convex polygon of smallest area that encloses all the points.

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Convex hull of Points

Deterministic algorithm:• O() time algorithm.• Many algorithms exist:

Grahams Scan, Jarvis’s march, divide and conquer,…

Randomized algorithm:• O() time algorithm.• Based on Randomized Incremental Construction.• Generalizable to higher dimensions.

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Randomized algorithm for convex hull

Convex-hull-algorithm(){ Let <,,…, > be a uniformly random permutation of ; triangle(,, ); For to do insert and update return ;}

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A simple exercise from geometry

Exercise: Given a line L and two points p and q, determine whether the points lie on the same/different sides of L.

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L p

q

q

𝒚=𝒎𝒙+𝒄

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Conflict graph : a powerful data structure

𝒄𝟒

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟓𝒄𝟔

𝒄𝟕

𝒄𝟖

conespoints𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟏

Before entering the for loop

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Before entering the for loop

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟏

𝒄𝟐

𝒄𝟑

points cones

INSERTING TH POINT

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th iteration

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𝒄𝟏

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒𝒄𝟓𝒄𝟔

𝒄𝟕

𝒄𝟖

points cones𝒄𝟏

th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒𝒄𝟓𝒄𝟔

𝒄𝟕

𝒄𝟖

points cones𝒄𝟏

th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒𝒄𝟓𝒄𝟔

𝒄𝟕

𝒄𝟖

points cones𝒄𝟏

th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒𝒄𝟓𝒄𝟔

𝒄𝟕

𝒄𝟖

points cones𝒄𝟏

th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒𝒄𝟓𝒄𝟔

𝒄𝟕

𝒄𝟖

points cones𝒄𝟏

th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒𝒄𝟓𝒄𝟔

𝒄𝟕

𝒄𝟖

points cones𝒄𝟏

th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒𝒄𝟓𝒄𝟔

𝒄𝟕

𝒄𝟖

points cones𝒄𝟏

th iteration

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𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒𝒄𝟓𝒄𝟔

𝒄𝟕

𝒄𝟖

points cones𝒄𝟏

th iteration

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𝒄𝟏

𝒄 ′

𝒄

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄 ′

𝒄

𝒄𝟔

𝒄𝟕

𝒄𝟖

points cones𝒄𝟏

Running time of th iterationRunning time of th iteration is of the order of

• Number of edges destroyed

• Number of new edges created

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

Question: What is the max. number of new edges created in an iteration ?Answer: 2 Number of edges created during the algorithm = O() Since every edge destroyed was once created, so Total number of edges destroyed Total number of edges decreated

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Total time for iterations = O()

𝑿 𝒊

Backward analysis of th iteration

: a subset of points from .: first points of are some permutation of

= ??

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Backward analysis of th iteration

: a subset of points from .: first points of are some permutation of

= ??

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𝒄𝟏𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖 𝒑𝟏

𝒑𝟐

𝒑𝟑

𝒑𝟒

𝒑𝟓

𝒑𝟔

𝒑𝟕

𝒑𝟖

Backward analysis of th iteration

: a subset of points from .: first points of are some permutation of

= ??

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𝒄𝟏𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖 𝒑𝟏

𝒑𝟐

𝒑𝟑

𝒑𝟒

𝒑𝟓

𝒑𝟔

𝒑𝟕

𝒑𝟖

Backward analysis of th iteration

: a subset of points from .: first points of are some permutation of

= ??

Calculating :

Let be the set of all subsets of of size .

= =

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𝒄𝟏𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖 𝒑𝟏

𝒑𝟐

𝒑𝟑

𝒑𝟒

𝒑𝟓

𝒑𝟔

𝒑𝟕

𝒑𝟖 𝟐(𝒏− 𝒊)𝒊

=

Running time of the algorithm

Expected running time of ith iteration = + O(1) = O()

Expected running time of the algorithm = O()

Theorem: There is an O() time Las Vegas algorithm for computing convex hull.

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