DecreaseandConquer Approach

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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/bunyaritbunya
rit_at_siit.tu.ac.th02 5013505 X 2005
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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.ph
  p?lessonid7

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This Week Overview

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

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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/bunyaritbunya
rit_at_siit.tu.ac.th02 5013505 X 2005
5
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).

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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

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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

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Decrease by a Constant
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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.

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Decrease by a Constant Factor

• The decrease-by-a-constant-factor technique
  suggests reducing a problems instance by the
  same constant factor on each iteration of the
  algorithm.
• In most applications, this constant factor is
  equal to two.

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Decrease by a Constant Factor
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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

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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.
• Euclids algorithm for computing the greatest
  common divisor provides a good example of such a
  situation.

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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/bunyaritbunya
rit_at_siit.tu.ac.th02 5013505 X 2005
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Insertion Sort

• we consider an application of the decrease-by-one
  technique to sorting an array A0..n - 1.
• Following the techniques idea, we assume that
  the smaller problem of sorting the array A0..n -
  2 has already been solved to give us a sorted
  array of size n - 1 A0 . . . An - 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 An -
  1?

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Insertion Sort

• we can scan the sorted subarray from right to
  left until the first element smaller than or
  equal to An - 1 is encountered and then insert
  An - 1 right after that element. gtstraight
  insertion sort or simply insertion sort.
• Or we can use binary search to find an
  appropriate position for An - 1 in the sorted
  portion of the array. gt binary insertion sort.

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Insertion Sort
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Insertion Sort
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort Demo
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Insertion Sort
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Analysis Worst Case

• The basic operation of the algorithm is the key
  comparison Aj gt v.
• The number of key comparisons in this algorithm
  obviously depends on the nature of the input.
• In the worst case, Aj gt v is executed the
  largest number
• of times, i.e., for every j i - 1, . . . , 0.
  Since v Ai, it happens if and only if Aj
  gtAi for j i - 1, . . . , 0.

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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

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Analysis Best Case

• In the best case, the comparison Aj gt v is
  executed only once on every iteration of the
  outer loop. It happens if and only if Ai - 1
  Ai 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

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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/bunyaritbunya
rit_at_siit.tu.ac.th02 5013505 X 2005
39
Concept of Recursion

• A recursive definition is one which uses the word
  or concept being defined in the definition itself

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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)

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Recursive Calculation of Factorials
n! n (n-1)!

• In order for this to work, we need a stop case
  (the simplest case)
• Here 0! 1

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This is Iterative Problem Solving
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but, this is Recursion
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Iterative Solution
n! n ? (n-1) ? (n-2) ? ? 1, if n gt 0
long Factorial(int n) long fact 1
for (int i2 i lt n i) fact fact
i return fact
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Programming with Recursion
Recursive definition - definition in which
something isdefined in terms of a smaller
version of itself, e.g.
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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.
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Recursion Solution
long MyFact (int n) if (n 0) return 1
return (n MyFact (n 1))
Recursive call - a call made to the function from
within the function itself
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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
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Example 1
Iterative programming
include ltvcl.hgt include ltstdio.hgt include
ltconio.hgt void iforgot_A(int n) for (int
i1 iltn i) printf("d, I will
remember to do my homework.\n",i)
printf("Maybe NOT!") void main()
iforgot_A(5) getch()
gtgt 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!
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Example 2
Recursive programming
include ltvcl.hgt include ltstdio.hgt include
ltconio.hgt void iforgot_B(int n) if (ngt0)
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()
gtgt 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!
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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.

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Concept 2 - Recursive Thinking

• Divide or decrease problem
• One step makes the problem smaller (but of the
  same type)
• Stopping case (solution is trivial)

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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

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Stopping the recursion

• The recursion must always STOP
• Stopping condition is important
• Recursive solutions to problems

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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

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Implementation of Hanoi

• See the implementation of Tower of Hanoi in the
  lecture

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Advantages of Recursion

• Advantages
• Some problems have complicated iterative
  solutions, conceptually simple recursive ones
• Good for dealing with dynamic data structures
  (size determined at run time).

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Disadvantages of Recursion

• Disadvantages
• Extra method calls use memory space other
  resources
• Thinking up recursive solution is hard at first
• Believing that a recursive solution will work

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Why Program Recursively?
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This Weeks Practice

• Write a recursive function to calculate Fibonacci
  numbers

• What is the result of f(6) ?

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Your recursive function
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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
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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/bunyaritbunya
rit_at_siit.tu.ac.th02 5013505 X 2005
64
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 ?

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End of Chapter 6

• Thank you!