MELJUN CORTES ALGORITHM Transform-And-Conquer Algorithm Design Technique

8/21/2019 MELJUN CORTES ALGORITHM Transform-And-Conquer Algorithm Design Technique

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Transform-and-Conquer Algorithm Design Technique

Design and Analysis of Algorithm

* Property of STI

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Definition of Transform-and-Conquer algorithm

design technique

Application of Transform-and-Conquer algorithm

design technique in the following algorithms: Presorting algorithm

Gaussian Elimination

Balanced search tree

Problem reduction


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* Property of STI

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an algorithm design technique in solving a

problem by modifying instance to a simple one

applied in the same or different problem

involves three variations :

1. Instance simplification

2. Representation Change

3. Problem Reduction


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* Property of STI

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Transform-and-Conquer is applied in:

instance simplification through presorting and

Gaussian elimination

representation change through balanced search

trees, heaps and Horner’s rule

problem reduction through least common multiple

and optimization


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* Property of STI

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Presorting

data are sorted to make computation easier

solves instance of problem by transforming into

another simplified instance of the same problem

Many problems involving lists are easier to be solved

when list is sorted.

searching computing the median

checking if all elements are distinct (element

uniqueness)

sorting

Geometric algorithms


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* Property of STI

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If the array is sorted first, we can then apply the

binary search which requires only [log 2n]+1comparison in a worst case.

The total running time in the worst case will be:

Presorting algorithm is also applied in different

geometric algorithms engaging with sets of points.


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* Property of STI

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named after Carl Friedrich Gauss

The following equations illustrates the Gaussian

Elimination algorithm:


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* Property of STI

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The figure below is an illustration of the

application of Gaussian elimination application:

Solve the latter by substitutions starting with the last

equation and moving up to the first one.


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* Property of STI

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Matrix representation of the given equation:

Ax B A’x B’

where:


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The figure below shows the steps in elementary

operations:

Exchanging two equations of the system

Replacing an equation with its non-zero multiple

Replacing an equation with a sum or difference of

this equation and some multiple of another equation


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Apply Gaussian elimination using the equation below:

2x1 –

x2 + x3 = 14x1 + x2 – x3 = 5

x1 + x2 + x3 = 0

Solution:

We will start by creating the matrix. Subtract 2 *row 1. Afterwards, subtract ½ * row 1.


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* Property of STI

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Then subtract ½ * row 2:

Solution by backward substitutions:

X

3

= -2)/2 = -1, x

2

= 3- -3) x

3

)/3 =0,

and

x1 = 1-x

3

- -1)x

2

)/2 = 1


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* Property of STI

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Pseudo code:

// Gaussian Elimination (A(1..n, 1, …n], b[1..n])

// Uses Gaussian elimination to matrix A of a

system’s coefficients

//augmented with vector b of the system’s right-

hand side values

//input: Matrix A[i, n+ 1] and column vector b[1..n]

//output: An equivalent upper-triangular matrix inplace of A with the corresponding right-hand side

values in the (n+1) column


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* Property of STI

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// Gaussian Elimination (A(1..n, 1, …n], b[1..n])

// Uses Gaussian elimination to matrix A of a

system’s coefficients

//augmented with vector b of the system’s right-

hand side values

//input: Matrix A[i, n+ 1] and column vector b[1..n]

//output: An equivalent upper-triangular matrix in

place of A with the corresponding right-hand side

values in the (n+1) column


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* Property of STI

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Binary search tree is a type of binary tree that is

composed of nodes that contains elements of a set

of ordered items.

All elements of the left subtree should be less than

the root while all elements of the right subtree

should be greater than or equal to the root.


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* Property of STI

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AVL Tree

Adelson-Velskii and Landis Tree

the heights of the subtrees must differ by notmore than 1

The first example of a tree is a search tree where

each node is greater than its predecessor.


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* Property of STI

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The second example of AVL tree is illustrated

below.


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Four cases of AVL tree:

Left of left

Right of right

Left of right

Right of left


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* Property of STI

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Left of left - a subtree of a given tree that is left

high node has also become the left high node

Example: The figure below illustrates a sampleAVL tree before inserting 8.

Out of

balance

at 20

AFTER

INSERTING 8

BEFORE

INSERTING 8

LH

LH

LH


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* Property of STI

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Right of right - a subtree that is right high has

also become the right high

Example: The figure below illustrates the right of right case.

AFTER

INSERTING 38

BEFORE

INSERTING 38

RH

RH

RH


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Right of Left - a subtree that is left high has

become the right high

Example: The figure below illustrates the right of left case.

AFTER

INSERTING 15

BEFORE

INSERTING 15RH

LH


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Left of Right - a subtree that is right high has

become the left high

Example: The figure below illustrates the left of right case.

AFTER

INSERTING 18

BEFORE

INSERTING 18

LH

RH


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Types of rotations that can be applied to solve the

unbalancing of trees.

Single left rotation

Single right rotation

Double left-right rotation

Double right-left rotation


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Single left rotation - it solves the left of left case in

an unbalanced tree by rotating the edge

connecting the root and its left child in the binary

tree

Example: The figure below illustrates the singleleft rotation.


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Single right rotation - it the same with the single

left rotation but in opposite direction

Example: The figure below illustrates the singleright rotation.


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Double left-right rotation - solves the Right of Left

case of unbalanced tree

Example: The figure below illustrates the doubleleft-right rotation


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Double right-left rotation - solves the Left of Right

case of unbalanced tree

Example: The figure below illustrates the doubleright-left rotation


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Analysis of algorithm for Binary search using the

Binary Search tree

[log2n] ≤ h < 1.4405 log2 n + 2) – 3277

The inequalities immediately imply that the

operations of searching and insertion are Θ log n)

in the worst case.


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* Property of STI

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Problem reduction – process of transforming a

given problem to another problem using a known

algorithm

Illustration of Problem reduction strategy:


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Example for Problem Reduction

Problem: Spacing to Center Text Problem

Solution:• Given three numbers a, b, and c, where:

o a is the width of a normal space

o b is the width of an option-space

o c is the amount we want to indent

Find two more numbers

• x is the number of normal spaces to be

used

• y is the number of option-spaces to beused

So that ax + by is as close as possible to c.


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MELJUN CORTES ALGORITHM Transform-And-Conquer Algorithm Design Technique