Decrease-and-Conquer Approach

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Decrease-and-Conquer Approach. Lecture 06. ITS033 – Programming &amp; Algorithms. Asst. Prof. Dr. Bunyarit Uyyanonvara IT Program, Image and Vision Computing Lab. School of Information, Computer and Communication Technology (ICT) Sirindhorn International Institute of Technology (SIIT)

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### Decrease-and-Conquer Approach

Lecture 06

ITS033 – Programming & Algorithms

Asst. Prof. Dr. Bunyarit Uyyanonvara

IT Program, Image and Vision Computing Lab.

School of Information, Computer and Communication Technology (ICT)

Sirindhorn International Institute of Technology (SIIT)

Thammasat University

http://www.siit.tu.ac.th/[email protected] 5013505 X 2005

ITS033

Topic 01-Problems & Algorithmic Problem Solving

Topic 02 – Algorithm Representation & Efficiency Analysis

Topic 03 - State Space of a problem

Topic 04 - Brute Force Algorithm

Topic 05 - Divide and Conquer

Topic 06-Decrease and Conquer

Topic 07 - Dynamics Programming

Topic 08-Transform and Conquer

Topic 09 - Graph Algorithms

Topic 10 - Minimum Spanning Tree

Topic 11 - Shortest Path Problem

Topic 12 - Coping with the Limitations of Algorithms Power

http://www.siit.tu.ac.th/bunyarit/its033.php

and http://www.vcharkarn.com/vlesson/showlesson.php?lessonid=7

This Week Overview
• Problem size reduction
• Insertion Sort
• Recursive programming
• Examples
• Factorial
• Tower of Hanoi

### Decrease & Conquer: Concept

ITS033 – Programming & Algorithms

Lecture 06.1

Asst. Prof. Dr. Bunyarit Uyyanonvara

IT Program, Image and Vision Computing Lab.

School of Information, Computer and Communication Technology (ICT)

Sirindhorn International Institute of Technology (SIIT)

Thammasat University

http://www.siit.tu.ac.th/[email protected] 5013505 X 2005

Introduction
• The decrease-and-conquer technique is based on exploiting the relationship between a solution to a given instance of a problem and a solution to a smaller instance of the same problem.
• Once such a relationship is established, it can be exploited either top down (recursively) or bottom up (without a recursion).
Introduction
• There are three major variations of decrease-and-conquer:

1. Decrease by a constant

2. Decrease by a constant factor

3. Variable size decrease

Decrease by a constant
• In the decrease-by-a-constant variation, the size of an instance is reduced by the same constant on each iteration of the algorithm.
• Typically, this constant is equal to 1
Decrease by a constant
• Consider, as an example, the exponentiation problem of computing an for positive integer exponents. The relationship between a solution to an instance of size n and an instance of size n - 1 is obtained by the

obvious formula: an= an-1 x a

• So the function f (n) = an can be computed either “top down” by using its recursive definition
• or “bottom up” by multiplying a by itself n - 1 times.
Decrease by a Constant Factor
• The decrease-by-a-constant-factor technique suggests reducing a problem’s instance by the same constant factor on each iteration of the algorithm.
• In most applications, this constant factor is equal to two.
Decrease by a Constant Factor
• If the instance of size n is to compute an, the instance of half its size will be to compute an/2, with the obvious relationship between the two: an= (an/2)2.
• But since we consider instances of the exponentiation problem with integer exponents only, the former works only for even n. If n is odd, we have to compute an-1 by using the rule for even-valued exponents and then multiply the result by
Variable Size Decrease
• the variable-size-decrease variety of decrease-and-conquer, a size reduction pattern varies from one iteration of an algorithm to another.
• Euclid’s algorithm for computing the greatest common divisor provides a good example of such a situation.

### Decrease & Conquer: Insertionsort

Lecture 06.2

ITS033 – Programming & Algorithms

Asst. Prof. Dr. Bunyarit Uyyanonvara

IT Program, Image and Vision Computing Lab.

School of Information, Computer and Communication Technology (ICT)

Sirindhorn International Institute of Technology (SIIT)

Thammasat University

http://www.siit.tu.ac.th/[email protected] 5013505 X 2005

Insertion Sort
• we consider an application of the decrease-by-one technique to sorting an array A[0..n - 1].
• Following the technique’s idea, we assume that the smaller problem of sorting the array A[0..n - 2] has already been solved to give us a sorted array of size n - 1: A[0]= . . . = A[n - 2].
• How can we take advantage of this solution to the smaller problem to get a solution to the original problem by taking into account the element A[n - 1]?
Insertion Sort
• we can scan the sorted subarray from right to left until the first element smaller than or equal to A[n - 1] is encountered and then insert A[n - 1] right after that element. =>straight insertion sort or simply insertion sort.
• Or we can use binary search to find an appropriate position for A[n - 1] in the sorted portion of the array. => binary insertion sort.

unsorted

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Analysis –Worst Case
• The basic operation of the algorithm is the key comparison A[j ]> v.
• The number of key comparisons in this algorithm obviously depends on the nature of the input.
• In the worst case, A[j ]> v is executed the largest number
• of times, i.e., for every j = i - 1, . . . , 0. Since v = A[i], it happens if and only if A[j ]>A[i] for j = i - 1, . . . , 0.
Analysis –Worst Case
• In other words, the worst-case input is an array of strictly decreasing values. The number of key comparisons for such an input is
Analysis – Best Case
• In the best case, the comparison A[j ]> v is executed only once on every iteration of the outer loop. It happens if and only if A[i - 1] = A[i] for every i =1, . . . , n-1, i.e., if the input array is already sorted in ascending order.
• Thus, for sorted arrays, the number of key comparisons is

### Decrease & Conquer: Recursive Programming

Lecture 06.3

ITS033 – Programming & Algorithms

Asst. Prof. Dr. Bunyarit Uyyanonvara

IT Program, Image and Vision Computing Lab.

School of Information, Computer and Communication Technology (ICT)

Sirindhorn International Institute of Technology (SIIT)

Thammasat University

http://www.siit.tu.ac.th/[email protected] 5013505 X 2005

Concept of Recursion

A recursive definition is one which uses the wordor concept being defined in the definition itself

= (n-1)!

Factorials
• How is this recursive?

n! = n × (n-1) × (n-2) × … × 3 × 2 × 1

• So: n! = n × (n-1) !
• The factorial function is defined in terms of itself (i.e. recursively)
Recursive Calculation of Factorials
• In order for this to work, we need a stop case (the simplest case)
• Here: 0! = 1

n! = n × (n-1)!

Iterative Solution

n! = n  (n-1)  (n-2)  …  1, if n > 0

long Factorial(intn)

{

long fact = 1;

for (int i=2; i <= n; i++)

fact = fact * i;

return fact;

}

1, if n = 0

n! =

n  (n-1)!, if n > 0

Programming with Recursion

Recursive definition - definition in which something isdefined in terms of a smaller version of itself, e.g.

1, if n = 0

n! =

n  (n-1)!, if n > 0

Programming with Recursion

Stopping Condition in a recursive definition is the case for which the solution can be stated nonrecursively

General (recursive) case is the case for which the solution is expressed in terms of a smaller version of itself.

Stopping cond.

recursive case

Recursion Solution

longMyFact(intn)

{

if (n == 0) return 1;

return (n * MyFact(n – 1));

}

Recursive call- a call made to the function from within the function itself;

6

MyFact(3)

2

3*MyFact(2)

1

2*MyFact(1)

1

1*MyFact(0)

How does this work?

int x = MyFact(3);

int MyFact (int n)

{ if (n == 0) // The stop case

return 1;

else

return n * MyFact(n-1);

} // factorial

Example #1

Iterative programming

#include <vcl.h>

#include <stdio.h>

#include <conio.h>

void iforgot_A(int n)

{

for (int i=1; i<=n; i++)

{

printf("%d, I will remember to do my homework.\n",i);

}

printf("Maybe NOT!");

}

void main()

{

iforgot_A(5);

getch();

}

>> iforgot_A(5)

1, I will remember to do my homework.

2, I will remember to do my homework.

3, I will remember to do my homework.

4, I will remember to do my homework.

5, I will remember to do my homework.

Maybe NOT!

Example #2

Recursive programming

#include <vcl.h>

#include <stdio.h>

#include <conio.h>

void iforgot_B(int n)

{

if (n>0)

{

printf("%d, I will remember to do my homework.\n",n);

iforgot_B(n-1);

}

else

printf("Maybe NOT!");

}

void main()

{

iforgot_B(5);

getch();

}

>> iforgot_B(5)

5, I will remember to do my homework.

4, I will remember to do my homework.

3, I will remember to do my homework.

2, I will remember to do my homework.

1, I will remember to do my homework.

Maybe NOT!

Writing Recursive Functions
• Get an exact definition of the problem to be solved.
• Determine the size of the input of the problem.
• Identify and solve the stopping condition(s) in which the problem can be expressed non-recursively.
• Identify and solve the general case(s) correctly in terms of a smaller case of the same problem.

Concept 2 - Recursive Thinking

• Divide or decrease problem
• One “step” makes the problem smaller (but of the same type)
• Stopping case (solution is trivial)
Recursion as problem solving technique
• Recursive methods defined in terms of themselves
• In code - will see a call to the method itself
• Can have more than one “activation” of a method going at the same time
• Each activation
• has own values of parameters
• Returns to where it was called from
• System keeps track of this
Stopping the recursion
• The recursion must always STOP
• Stopping condition is important
• Recursive solutions to problems
Stopping the recursion
• General pattern is
• test for stopping condition
• if not at stopping condition:
• either
• do one step towards solution
• call the method again to solve the rest
• or
• call the method again to solve most of the problem
• do the final step
Implementation of Hanoi
• See the implementation of Tower of Hanoi in the lecture
• Some problems have complicated iterative solutions, conceptually simple recursive ones
• Good for dealing with dynamic data structures (size determined at run time).
• Extra method calls use memory space & other resources
• Thinking up recursive solution is hard at first
• Believing that a recursive solution will work
This Week’s Practice
• Write a recursive function to calculate Fibonacci numbers
• What is the result of f(6) ?
Fibonacci Recursive Tree

fibo(6)

fibo(5) + fibo(4)

fibo(4) + fibo(3)

fibo(3) + fibo(2)

1

fibo(3) + fibo(2)

fibo(2) + fibo(1)

fibo(2) + fibo(1)

fibo(2) + fibo(1)

1

1

1

1

1

1

1

### Decrease & Conquer: Homework

ITS033 – Programming & Algorithms

Asst. Prof. Dr. Bunyarit Uyyanonvara

IT Program, Image and Vision Computing Lab.

School of Information, Computer and Communication Technology (ICT)

Sirindhorn International Institute of Technology (SIIT)

Thammasat University

http://www.siit.tu.ac.th/[email protected] 5013505 X 2005

Homework: Fake-Coin Problem

Design an algorithm using Decrease and Conquer approach to solve Fake Coin Problem

• Among n identically looking coins, one is fake (lighter than genuine).
• Using balance scale to find that fake coin.
• How many time do you use the balance to find a fake coin from n coins ?
• Is it optimum ?

Thank you!