MELJUN CORTES ALGORITHM Divide-And-Conquer Algorithm Design Technique 1

Design and Analysis of Algorithm

TOPIC TITLE: Divide-and-Conquer Algorithm Design Technique Specific Objectives: At the end of the topic session, the students are expected to: Cognitive: 1. Define Divide-and-Conquer algorithm design technique. technique 2. Identify the importance of Divide-and-Conquer algorithm design technique in solving different problem types. 3. Explain how Divide-and-Conquer algorithm design technique is applied to sorting algorithms, specifically Merge and Quick sort. 4. Discuss the application of Divide-and-Conquer algorithm design technique in Binary Search and Binary Tree traversals. traversals Affective: 1. Listen to others with respect. 2. Participate in class discussions actively. 3. Share ideas to the class. MATERIALS/EQUIPMENT: o o

topic slides OHP

TOPIC PREPARATION: o

o o o

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Have the students research on the following: • Divide-and-Conquer algorithm and its application • Merge sort and Quick sort algorithm • Binary search algorithm and Binary tree traversals traversal The research work should be in a printed copy compiled in a folder with a title page. Provide sample problems that can be solved by using Divideand-Conquer algorithm design technique. It is imperative for the instructor to incorporate various kinds of teaching strategies while discussing the suggested topics. The instructor may use the suggested learning activities below to facilitate e a thorough and creative discussion of the topic. Prepare the slides to be presented in the class. Prepare seatwork for the students ts to apply the lessons learned in Divide-and-Conquer algorithm design technique.

Divide-and-ConquerAlgorithm ConquerAlgorithm Design Technique

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Design and Analysis of Algorithm

TOPIC PRESENTATION: The topic will cover the application of Divide-and-Conquer Conquer algorithm design technique technique. The following is the suggested flow of discussion for the course topic: 1. Identify the topics to be covered. 2. Ask the students to define Divide-and-Conquer algorithm design technique and cite some algorithms where it can be applied. applied 3. Discuss Divide-and-Conquer algorithm design technique and provide an example. 4. Enumerate the importance of Divide-and-Conquer Conquer algorithm design technique. 5. Ask the students to differentiate Merge Sort from Quick Sort. 6. Discuss merge sort and quick sort by explaining how DivideDivide and-Conquer Conquer algorithm design technique is applied to these types of sorting algorithm. 7. Ask the students to share their research about binary search and binary tree traversals. 8. Explain the application of Divide-and-Conquer Conquer algorithm design technique to binary search and binary tree traversals. Divide-and-Conquer Algorithm Design Technique Page 1 of 28

Divide Divide-and-Conquer Design Technique These are the topics to be discussed under Divide-and-Conquer Divide algorithm design ttechnique.

Design and Analysis of Algorithm

Divide-and-Conquer Algorithm Design Technique c Definition of Divide-and-Conquer algorithm design technique c Importance of Divide-and-Conquer algorithm design technique c Application of Divide-and-Conquer algorithm design technique in the following algorithms: q Merge sort q Quick sort q Binary search q Binary tree traversals

o o o

Definition of Divide-and-Conquer algorithm design technique t Importance of Divide-and-Conquer algorithm design technique Application of Divide-and-Conquer algorithm design esign technique in the following algorithms: o Merge sort o Quick sort o Binary search o Binary tree traversals

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Divide-and-ConquerAlgorithm ConquerAlgorithm Design Technique

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Design and Analysis of Algorithm

What is Divide-and-Conquer Algorithm Design Technique? Technique What is Divide-and-Conquer Algorithm Design Technique?

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Divide Divide-and-Conquer is a general neral algorithm design technique that solves a problem’s instance by dividing it into several smaller instances (ideally of equal size), solving each of them recursively, and then combining their solutions to get a solution to the original instance of the problem [LEV07].

Design and Analysis of Algorithm

What is Divide-andConquer Algorithm Design Technique? c Divide-and-Conquer q

solves a problem’s instance by dividing it into several smaller instances, solving each of them recursively, and then combining their solutions to get a solution to the original instance of the problem [LEV07].

c General Divide-and-Conquer Algorithm:

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It simply means that it is done by breaking down a specific problem into sub-problems problems of the same type repeatedly, until it become simple enough to be solved directly. Afterwards, solutions to the sub-problems sub are then combined to give a solution to the original problem. problem On the other hand, the capability to comprehend and design Divide-andConquer algorithms is a skill that should be practiced to become proficient. For instance, in proving a theorem by induction, inducti it is often needed to substitute the original problem by a more general or complex problem for us to get the recursion going, and there is no systematic method for finding the proper generalization. The correctness of a divide and conquer algorithm is usually proved by mathematical induction and its computational cost which is determined by solving recurrence relations relations. General Divide Divide-and-Conquer Algorithm Step 1: If the problem size is small, solve this problem directly; otherwise, split the original problem into 2 sub-problems problems with equal sizes. Step 2: Recursively solve these 2 sub-problems problems by applying this algorithm. Step 3: Merge the solutions of the 2 sub-problems problems into a solution of the original problem. A simple example of Divide-and-Conquer algorithm

What is Divide-and-Conquer Algorithm Design Technique? Page 3 of 28 Design and Analysis of Algorithm

What is Divide-andConquer Algorithm Design Technique? c Example: q

Let’s take a simple example of Divide Divide-and-Conquer Conquer algorithm for computing the sum of n numbers a0….an-1, If n>1, we can divide the problem into two instances of the same problem: to compute the sum of the first [n/2] numbers and to compute the sum of the remaining [n/2] numbers. Of course if n=1, we simply return a0 as the answer. One of these two sums is computed (by applying the same method recursively), we can add their values to get the sum question:

Computing the sum of n numbers

a0 + ...+ a n-1 = (a0 + …+ a[n/2]-1) + (a[n/2]+…+an-1)

a0 + ...+ an-1 = (a0 + …+ a[n/2]-1) + (a[n/2]+…+an-1) Maybe we can ask if this is the efficient way to compute the sum of n numbers. Not all divide-and-conquer conquer algorithms are necessary to more efficient than any algorithm even a brute force algorithm solution. But the spent on time executing the divide-and-conquer conquer algorithm turns out to be smaller and it yields some of the most important and efficient algorithms in computer science.

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The figure below illustrates an example of Divide Divide-and-Conquer Conquer algorithm in typical case:

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Design and Analysis of Algorithm

Figure 7.2 Example of Divide-and-Conquer Conquer algorithm application

The figure shows that the most typical case of divide-and-conquer divide algorithm is the instance of a problem which in this case the instance of size n is divided into two main instances of size n/2. Generally, the instance tance of size n can be divided into b instances of size n/b, with a of them needing to be solved. Here a and b are constants: a = q and b > 1, assuming that the size n is a power of b to simplify our analysis, we get the following recurrence for the runnin running time T(n):

T(n) = aT(n/b) + f(n) Where f(n) is a function that accounts for the time spent on dividing the problem into smaller ones and on combining their solutions. Obviously, the order of growth of its solution t(n) depends on the values of the constants a and b and the order of growth of the function f(n). In conclusion, the Divide and conquer technique leads to a kind of recurrence formula: T (n) ≤ 2T (n/2) + c n and nd time O(n log n). n) [What What is Divide-and-Conquer Conquer Algorithm Design Technique?, Technique Pages 2-3 3 of 28 28] Importance of Divide Divide-and-Conquer Algorithm Design esign Technique This approach allows more freedom in the choice of the sub-problem sub that is to be solved next, a feature that is important in some applications. applications 1. Solving difficult problems Divide and conquer algorithm design technique is an influential tool for solving theoretically tough problems which entails breaking problems into sub-problems problems and solving it in trivial cases. In reality, divide and conquer algorithm is considered con as a paradigm in providing simple solution to difficult problems.

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Design and Analysis of Algorithm Importance of Divide-andConquer Algorithm Page 4 of 28

2. Algorithm efficiency

Design and Analysis of Algorithm

Importance of Divide-andConquer Algorithm c Solving difficult problems c Algorithm efficiency c Parallelism c Memory Access

Divide and conquer often offers an ordinary way of creating efficient algorithms. For example, if the work of splitting the problem and combining the partial solutions is proportional to the problem's size n, there are a bounded number p of subsub problems of size ~ n/p at each stage, and the base cases require O(1) (constant-bounded) bounded) time, then the divide-anddivide conquer algorithm will have O(n log n) complexity. This is used for problems such as sorting and FFTs to reduce the complexity 2 from O(n ), although in general there may also be other approaches to designing efficient algorithms. 3. Parallelism

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Divide and conquer algorithm design technique is tailored for implementation in multi-processor processor machines, especially sharedshared memory systems where the communication of data between processors does not need to be planned in advance, because different sub-problems problems can be executed on different processors. 4. Memory Access Divide-and-conquer conquer algorithms design technique is expected to use memory caches efficiently. This is because once a subsub problem is small enough, it can be solved within the cache without accessing the slower main memory. In addition, it is also an algorithm designed to make use of the cache in this way that is referred to as the cache oblivious obliviou since it does not contain the cache size(s) as an explicit parameter. In fact, it can be applied to different algorithms algorithm such as sorting, matrix multiplication. [Importance Importance of Divide Divide-and-Conquer Conquer Algorithm, Page 4 of 28] 28

Divide-and-Conquer in Merge Sort Page 5 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Merge Sort

Merge sort is a type of sorting algorithm that combines two files ordered into one ordered file on the same given key. It is a perfect example of a successful application of the divide divide-and-conquer conquer technique. Pseudo code of Merge Sort algorithm

c Merge sort q

Application of Divide-and-Conquer Algorithm Design Technique in Merge Sort

combines two files ordered into one ordered file on the same given key

The figure below illustrates the Pseudo code for Merge sort algorithm:

c Pseudocode of Merge Sort

Merge-Sort A[1..n] 1 if n =1, done. 2 Recursively sort A[1..[n/2] ] and A[[n/2]+1.. n ] 3 Merge the two sorted lists Divide-and-Conquer Algorithm Design Technique

Figure 7.3 3 Pseudo code for Merge sort algorithm

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The figure shows that Merge sort algorithm is consists of three main steps, these are: 1. Divide the array into two sub-arrays arrays each with n/2 items. Divide-and-ConquerAlgorithm ConquerAlgorithm Design Technique

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Design and Analysis of Algorithm Divide-and-Conquer in Merge Sort Page 6 of 28

2. Conquer (solve) each sub-array array by sorting it. Unless the array is sufficiently small, use recursion to do this. 3. Combine the solutions to the sub-array array by merging them into a single sorted array.

Design and Analysis of Algorithm

Divide-and-Conquer in Merge Sort c Example of Merge Sort: q

Given the list below: 29

12

14

22

27

15

27

15

17

24

c Solution: 1.

22

17

24

Sort each sub-array: 12 14 22

3.

To understand Merge sort algorithm, let’s examine a sample problem. For instance, you have a one one-dimensional dimensional array that contains the following values:

Divide the array: 29 12 14

2.

Example of Merge Sort

29

15

17

24

27

Merge the sub-arrays:

29

12

14

22

15

17

24

1. Divide the array:

29 Divide-and-Conquer Algorithm Design Technique

27

Figure 7.4 One dimensional array list for Merge sort algorithm example

12

14

22

17

24

and

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27

15

Figure 7.5 5 The illustration of the one dimensional array after dividing the array

2. Sort each sub-array:

12

14

22

29

24

27

and

15

17

Figure 7.6 The illustration of the sorted one dimensional array after dividing the array

3. Merge the sub-arrays:

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14

15

17

22

24

27 2

29

Figure 7.7 The illustration of the sorted one dimensional array using Merge sort

The illustration below is the simulation of the Merge sort algorithm: Design and Analysis of Algorithm

Divide-and-Conquer in Merge Sort c Simulation of the Merge sort algorithm:

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Design and Analysis of Algorithm

Figure 7.8 The simulation of Merge sort algorithm

Divide-and-Conquer in Merge Sort Page 8 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Merge Sort c Analysis of algorithm of Merge sort using the Divide-and-Conquer approach

Analysis of algorithm for Merge sort Applying Divide-and-Conquer Divide algorithm The conquer step of merge merge-sort consists of merging two sorted sequences, each with n/2 elements and implemented by means of a doubly linked list, takes at most bn steps, for some constant b. Likewise, the basis case ((n < 2) will take at b most steps. Therefore, if we let T(n) denote enote the running time of merge sort we will come up with the formula below:

c Merge Sort Time complexity

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We can therefore analyze the running time of merge merge-sort sort by finding a closed form solution to the above equation. That is, a solution that has T(n)) only on the left-hand side.

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Design and Analysis of Algorithm

Merge sort time complexity:

Figure 7.9 Time complexity of Merge Sort algorithm

Therefore, refore, the Divide Divide-and-conquer algorithm is simplified to different steps which indicates that: • for the i’th th merging iteration, the complexity of the problem is O(n) • number of iterations is O(log n) • running time: O(n log n) [Divide Divide-and-Conquer in Merge Sort, Pages 5-8 of 28]] Divide-and-Conquer in Merge Sort Page 9 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Quick Sort

Application of Divide Divide-and-Conquer Algorithm Design Technique in Quick Sort Quick sort is an exchange sort developed by C.A.R. Hoare in 1962. It is a sorting algorithm that is quite similar with Merge sort ort algorithm where sorting is done by dividing the array into two partitions and then sorting sor each partition recursively recursively.

c Quick sort q

q

q

an exchange sort developed by C.A.R. Hoare in 1962 sorting is done by dividing the array into two partitions and then sorting each partition recursively a pivot key is placed in its correct position in the array while rearranging other elements widely dispersed across the list

In Quick sort, however, the arrays is partitioned by placing a pivot key in its correct position in the array while rearranging other elements widely dispersed across the list. Specifically, it rearranges elements of a given array ray A[0…n-1] to achieve its partition, a situation where all the elements before some position s are smaller than or equal to A[s] and all the elements after position s are greater to A[s]: ? ?? ?g ? ?? ? ? ? ? ??? ? ?? ? ? ?g ? ?? ? ? ?

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All are = A[s]

All are =A[s]

When tthe partition has been achieved, A[s] will be in its final position in the sorted array, and we can continue sorting the two sub-arrays sub of the elements preceding and following A[s] independently.

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Design and Analysis of Algorithm Divide-and-Conquer in Merge Sort Page 10 of 28

Pseudo code of Quick Sort algorithm The figure below illustrates the Pseudo code for Quick sort algorithm:

Design and Analysis of Algorithm

Divide-and-Conquer in Quick Sort c Pseudo code for Quick sort algorithm

1 2 3 4

5 6 q q

q q

QuickSort (Array[ ]) If length (array)>1 Choose bound; //partition array into subArray1 and subArray2 while there are elements left in array 1 include element either in subArray1 = {target = bound} 2 or in subArray2 = {target = bound}; QuickSort(subArray1); QuickSort(subArray2);

Select one element target from the input Partition the input into part containing elements not greater than target and part containing all bigger elements. Sort each part separately. Concatenate these sorted parts.

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Figure 7.10 Pseudo code for Quick sort algorithm

The figure shows that Quick sort algorithm is consists of different steps as follows follows: 1. Select one element target from the input 2. Partition the input into part containing elements not greater than target and part containing all bigger elements. 3. Sort each part separately. 4. Concatenate these sorted parts.

Design and Analysis of Algorithm

Divide-and-Conquer in Quick Sort

To understand Quick sort algorithm, let’s examine a sample problem. For instance, you have a one one-dimensional dimensional array that contains the following values:

c Example of Quick sort algorithm q

Given the original unsorted list:

q

Solution: 1. Partition the array :

Example of Quick Sort

16

23

14

28

13

11

21 2

26

Figure 7.11 One dimensional array list for Quick sort algorithm example

1. Partition the array so that all items smaller than the pivot item are to the left of it and the items larger are to the right right: Divide-and-Conquer Algorithm Design Technique

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Design and Analysis of Algorithm

Divide-and-Conquer in Quick Sort c Example of Quick sort algorithm q

Given the original unsorted list:

q

Solution: 2. Sort each sub-array:

Figure 7.12 The illustration of the one dimensional array after dividing the array

2. Sort each sub-array:

Figure 7.13 The illustration of the sorted one dimensional array after dividing the array

3. Merge the sub sub-arrays: Divide-and-Conquer Algorithm Design Technique

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Design and Analysis of Algorithm Divide-and-Conquer in Merge Sort Page 13 of 28 After Pass 2

11

13

14

16

21

23

26 2

28

Figure 7.13 The illustration of the sorted one dimensional array using Quick sort

Design and Analysis of Algorithm

Divide-and-Conquer in Quick Sort

The illustration below is the simulation of the Quick sort algorithm:

c Example of Quick sort algorithm q

Given the original unsorted list:

q

Solution: 3. Merge sub-arrays:

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Figure 7.14 The simulation of Quick e sort algorithm

Design and Analysis of Algorithm

Divide-and-Conquer in Quick Sort

Analysis of algorithm for Quick sort applying pplying Divide-and-Conquer Divide algorithm

c Simulation of the Quick sort algorithm:

Generally, the quick sort efficiency is O(nlog2n). For the analysis of w worst-case time complexity,, we will consider the formula below:

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Divide-and-Conquer in Merge Sort Page 15 of 28 Design and Analysis of Algorithm

In the given formula, w we are using the notation T(n)) because we are presently determining the every every-case case complexity for the class of instances that are already sorted in non non-decreasing decreasing order. Since, T(0) = 0, we have the recurrence:

Divide-and-Conquer in Quick Sort c Analysis of algorithm of Quick sort using the Divide-and-Conquer technique q

quick sort efficiency: O(nlog2n)

q

Analysis of Worst-Case Time Complexity

Divide-and-Conquer Algorithm Design Technique

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Design and Analysis of Algorithm

Divide-and-Conquer in Binary Search Page 16 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Binary Search c Binary search q q

used to locate an element in an ordered array starts by testing the data in the element at the middle of the array to determine if the target is in the first or second half of the list

Application of Divide-and-Conquer Algorithm Design Technique in Binary search Binary search is a type of search algorithm that is used to locate an element in an ordered array. It is advisable to use a binary search whenever the list starts to become larger. Binary search starts by testing the data in the element at the middle of the array to determine if the target is in the first or second half of the list. If it is in the first half, we do not need to test the second half. Same goes with the second half. In other words, we eliminate half the list from further consideration. We repeat this process until we find the target to determine that it is not in the list. To find the middle of the list, we need three variables to identify the beginning beginning, the middle, and the end of the list. We analyze two cases here: the target that is in the list and the target that is not in the list.

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The divide divide-and-conquer steps of Binary search: The steps of Binary search starts by testing the target. If target is equal to the middle item, quit. Otherwise: 1. DIVIDE the array into two sub-arrays arrays about half as large. If target is smaller than the middle item, choose the left subsub array. If target is larger than the middle item, choose the right sub-array. 2. DIVIDE (Solve) the sub-array array by determining whether target is in that sub-array. Unless the sub-array array is sufficiently small, use recursion to do this. 3. OBTAIN the solution to the array from the solution to subsub array.

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Design and Analysis of Algorithm Divide-and-Conquer in Binary Search Page 17 of 28

Pseudo code for Binary search algorithm

BinarySearch (val list , val end , val target val locn ) 1 first = 0 2 last = end 3 loop ( first list[mid]) Look in upper half 1 first = mid + 1 3 else if (target < list[mid]) Look in lower half 1 Last = mid – 1 4 else Found equal: force exit 1 first = last + 1 5 end if 4 end loop 5 locn = mid 6 if (target equal list [mid]) 1 found = true 7 else 1 found = false 8 end if 9 return found end BinarySearch

Design and Analysis of Algorithm

Divide-and-Conquer in Binary Search c Pseudo code for Binary search algorithm

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Figure 7.15 Pseudo do code for Binary search algorithm Divide-and-Conquer in Binary Search Page 18 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Binary Search c Example:

q

Example of Binary search algorithm To understand Quick sort algorithm, let’s examine a sample problem. For instance, you have a one one-dimensional dimensional array that contains the following values:

5 8

9

11 15 22 23 37 63 78 7 82 92

Figure 7.16 One dimensional array list for Binary search algorithm example

Solution:

The figure below illustrated the process of searching the 23 as the target value in the given array list:

Divide-and-Conquer Algorithm Design Technique

Firs First

Mid

0

5

Last

Target: 23

11

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5 8

9

11 15 22 23 37 63 78 82 92

23 > 22 Divide-and-ConquerAlgorithm ConquerAlgorithm Design Technique

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Design and Analysis of Algorithm

5 8

5 8

9

9

First

Mid

6

8

Last

11

11 15 22 23 37 63 78 82 92 First

Mid

Last

6

6

7

23 < 63

11 15 22 23 37 63 78 82 92

23 = 23 Figure 7.17 Binary search algorithm example

In the he illustration illustration, the array length is 12 (0-11 indices). indices The binary search process starts by getting the middle index through adding the first and last index then dividing the sum by two. Hence, (0+11)/2 = 5. The middle index (5) has an element of 22 that will be compared to the target value which is 23. Since 23 is not equal to 22 then the search value is not yet found. Afterwards, a comparison will be done ifi the target value is greater than or less than the current value. If the target is greater than the value, we will proceed to the upper bound bound. Otherwise, Otherwise we will proceed to the lower bound of the given array. The process will be repeated until it finds the target or not. Divide-and-Conquer in Binary Search Page 19 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Binary Search c Analysis of algorithm of Binary search using the Divide-and-Conquer approach

Divide-and-Conquer Algorithm Design Technique

Analysis of algorithm for Binary search using the Brute Force The standard way to analyze the efficiency of binary search is to count the number of times the search key is compared with the element of the array. Moreover, for the sake of simplicity, we will count the so-called so three way comparisons. This assumes th that at after one comparison of K with A[m], the algorithm can determine whether K is smaller, equal to, or larger than A[m].

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Design and Analysis of Algorithm Divide-and-Conquer in Binary Tree Traversals Page 20 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Binary Tree Traversals

Application of Divide Divide-and-Conquer Conquer Algorithm Design Technique in Binary Tree Traversals Binary Tree is a type of tree data structure in which no node can have more than two subtrees. In other words, a node can have zero, one, or two subtrees. These subtrees are designated as the left subtree and right subtree.

c Binary Tree q

no node can have more than two subtrees.

The figure below illustrate a sample b binary tree:

c sample binary tree:

Left Subtree

Divide-and-Conquer Algorithm Design Technique

Right Subtree

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Divide-and-Conquer in Binary Tree Traversals c Balance factor q

height difference between the left subtree and the right subtree: B = H L – HR

A B

A C

B

C

B=0 B = -1

D

A B

D

C

B=1

Divide-and-Conquer Algorithm Design Technique

In the given figure, it shows that o one ne of the properties of o the Binary tree is its height. Given that we need to store N,, which refers to the number of nodes in a binary tree, the maximum height: Hmax = N. But a tree with maximum height is rare. It occurs when the entire tree is built in one direction. The minimum height of a tree Hmin = [log2N] + 1. 1

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The distance of a node from the root determines how efficiently it can be located located.. The children of any node in a tree can be accessed by following only one branch path. In addition, it is easier to locate a node in i a shorter tree. Balancing of tree is an important characteristic of a binary tree to determine whether the tree is balanced factor. A balance factor is the height difference between the left subtree and the right subtree. B = HL – HR. Example:

B HL = 1; HR = 1 B = H L – HR B=0

A

A

A C

B

C

D HL = 2; HR = 1 B = HL – HR B=1

B

C D

HL = 1; HR = 2 B = HL – HR B = -1

A binary tree is considered balanced if the height of its subtrees differs by no more than one. Its balance factor is -1, 1, 0, and 1 considering that each subtree is also balanced.

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Design and Analysis of Algorithm Divide-and-Conquer in Binary Tree Traversals Page 22 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Binary Tree Traversals c Algorithm for defining the height of a binary tree

Algorithm for d defining the height of a binary tree

Algorithm Height(T) //Compute recursively the height of a binary tree //Input: A binary tree T //Output: The height of T If T = 0 return -1 else return max[Height(TL), Height(TR) + 1 Figure 7.19 Pseudo code for determining the height of a binary tree

A(n(T)) = A(n(TL)) + A(n(TR) + 1 for n(T) > 0, A(0)=)

Divide-and-Conquer Algorithm Design Technique

We measure the problems instance size by the number of nodes n(T) in a given tree T. Obviously, the number of comparisons made to compute the maximum of two numbers and the number of addition A(n(T)) made by the algorithm are the same. We have the following relation relatio for A(n(T)):

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A(n(T)) = A(n(TL)) + A(n(TR) + 1 for n(T) > 0, A(0)=) Divide-and-Conquer in Binary Tree Traversals Page 23 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Binary Tree Traversals c Binary tree traversal q

requires that each node of the tree be processed once and only once in a predetermined sequence.

c Approaches to the traversal sequence: q

Depth-first traversal • the processing proceeds along a path from the root through one child to the most distant descendent of that first child before processing a second child

q

Breadth-first traversal • the processing proceeds horizontally from the root to all of its children, then to its children’s children, and so forth until all nodes have been processed

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Divide-and-Conquer in Binary Tree Traversals Page 24 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Binary Tree Traversals c Types of Depth-first search Binary Tree traversals : q

Binary Tree Traversal A binary tree traversal requires that each node of the tree be processed once and only once in a predetermined sequence. The two general approaches to the trave traversal rsal sequence are depth first and breadth first. Dept Depth-first traversal is where the processing proceeds along a path from the root through one child to the most distant descendent of that first child before processing a second child. In other words, in the depth-first depth search traversal, all of the descendents of a child are processed before going on to the next child. Breadth Breadth-first traversal is where the processing proceeds horizontally from the root to all of its children, then to its children’s children, and so forth until all nodes have been processed. In other words, in the breadth breadth-first traversal, each level is completely processed before the next level is started. Types of Depth Depth-first search Binary Tree traversals There are th three types of depth-first first search binary tree traversal, the preorder, inorder and postorder traversals. In the preorder traversal,, the root is visited first, followed by the left subtree and then the right subtree. The figure below illustrates the preorder traversal:

Preorder traversal

Root

• root – left subtree – right subtree

Right Subtree

Left Subtree

//Pre-condition: Root is the entry node of a tree or subtree //Post-condition: Each node has been processed in order 1 if (root is not equal to null) 1 Process (root) 2 preOrder (root -> leftSubtree) 3 preorder (root -> rightSubtree) 2 end if 3 return Divide-and-Conquer Algorithm Design Technique

Figure 7.20 Preorder traversal * Property of STI Page 24 of 28

Divide-and-ConquerAlgorithm ConquerAlgorithm Design Technique

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Design and Analysis of Algorithm

Algorithm for preorder traversal: //Pre-condition: condition: Root is the entry node of a tree or subtree //Post-condition: condition: Each node has been processed in order 1. if (root is not equal to null) 1. Process (root) 2. preOrder (root -> leftSubtree) 3. preorder (root -> rightSubtree) 2. end if 3. return

Divide-and-Conquer in Binary Tree Traversals Page 25 of 28 Design and Analysis of Algorithm

In the inorder traversal,, the left subtree is visited first, followed by the root and then the right subtree. The figure below illustrates the inorder traversal:

Divide-and-Conquer in Binary Tree Traversals c Types of Depth-first search Binary Tree traversals : q

Inorder traversal • left subtree – root – right subtree

//Pre-condition: Root is the entry node of a tree or subtree //Post-condition: Each node has been processed in order 1 if (root is not equal to null) 1inOrder (root -> leftSubtree) 2 process (root) 3 inOrder(root -> rightSubtree) 2 end if return Divide-and-Conquer Algorithm Design Technique

Figure 7.21 Inorder traversal * Property of STI Page 25 of 28

Divide-and-Conquer in Binary Tree Traversals Page 26 of 28 Design and Analysis of Algorithm

Divide-and-Conquer in Binary Tree Traversals c Types of Depth-first search Binary Tree traversals : q

Algorithm for inorder traversal: //Pre-condition: condition: Root is the entry node of a tree or subtree //Post-condition: condition: Each node has been processed in order 1. if (root is not equal to null) 1. inOrder (root -> leftSubtree) 2. process (root) 3. inOrder(root -> rightSubtree) 2. end if return

In the postorder traversal,, the left subtree is visited first, followed by the right subtree and then the root. The figure below illustrates the postorder traversal:

Postorder traversal • left subtree – right subtree – root

//Pre-condition: Root is the entry node of a tree or subtree //Post-condition: Each node has been processed in order 1 if (root is not equal to null) 1 postOrder (root -> leftSubtree) 2 postOrder(root -> rightSubtree) 3 process (root) 2 end if return Divide-and-Conquer Algorithm Design Technique

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Divide-and-ConquerAlgorithm ConquerAlgorithm Design Technique

Figure 7.22 Postorder traversal

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Design and Analysis of Algorithm

Algorithm for postorder traversal //Pre-condition: condition: Root is the entry node of a tree or subtree //Post-condition: condition: Each node has been processed in order 1. if (root is not equal to null) 1. postOrder (root -> leftSubtree) 2. postOrder(root -> rightSubtree) 3. process (root) 4. 2. end if return Divide-and-Conquer in Binary Tree Traversals Page 27 of 28

Sample application of Binary Tree traversals Consider the given binary tree below:

Design and Analysis of Algorithm

Divide-and-Conquer in Binary Tree Traversals c Sample application of Binary Tree Traversals: q

Consider the given binary tree below:

q

Preorder: A, B, C, D, E, F, G Inorder: C, B, D, A, F, E, G Postorder: C, D, B, F, G, E, A

q q

Divide-and-Conquer Algorithm Design Technique

Figure 7.22 Sample binary tree to use depth-first traversals

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The values of preorder, inorder and postorder ostorder traversals applied in the given example are the following: Preo Preorder: A, B, C, D, E, F, G Inorder: C, B, D, A, F, E, G Postorder: C, D, B, F, G, E, A Analysis of algorithm for Depth-First traversal algorithm using DivideDivide and-Conquer Conquer The most important divide divide-and-conquer algorithms for binary trees are the three classic traversals: preorder, inorder and postorder ostorder traversals. All three traversals visit nodes of a binary tree recursively. [Divide [Divide-and-Conquer Conquer in Binary Tree Traversals, Pages 20-27 20 of 28] SEATWORK: Draw a binary tree consists of ten nodes based on the given inorder traversal: ABCEDF ABCEDFJGIH and preorder traversal: JCBADEFIGH. Answer: This problem is solved by setting the root of the tree to the first node in the preorder traversal traversal, that is J, and then from the inorder traversal, identifying its left subtrees as A B C E D F and the right subtree as G I H. The process is repeated with each subtree until the complete tree is developed.

Divide-and-ConquerAlgorithm ConquerAlgorithm Design Technique

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Design and Analysis of Algorithm SEATWORK Page 28 of 28 Design and Analysis of Algorithm

SEATWORK

c Draw a binary tree consists of ten nodes based on the given: q q

inorder traversal: ABCEDFJGIH preorder traversal: JCBADEFIGH

Divide-and-Conquer Algorithm Design Technique

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[SEATWORK, SEATWORK, Pages 28 of 28 28] GENERALIZATION: o

o

o

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o o

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o o o o

Divide-and-Conquer is a general algorithm design technique that solves a problem’s instance by dividing it into several smaller instances (ideally of equal size), solving each of them recursively, and then combining their solutions to get a solution. Divide-and-Conquer Conquer algorithm design technique is applied in Merge sort, Quick sort, Binary search and the depth-first depth traversals. Merge sort is a type of sorting algorithm that combines two files ordered into one ordered file on the same given key. It is a perfect ect example of a successful application of the divide-anddivide conquer technique. Quick sort is an exchange sort developed by C.A.R. Hoare in 1962. It is more similar to Merge Sort in that sort is accomplished by dividing the array into two partitions and then sorting each partition recursively. Binary Tree is a type of tree data structure in which no node can have more than two subtrees. In other words, a node can have zero, one, or two subtrees. Binary tree traversal requires that each node of the tree be processed once and only once in a predetermined sequence. Dept-first first traversal is where the processing proceeds along a path from the root through one child to the most distant descendent of that at first child before processing a second child. Breadth-first first traversal is where the processing proceeds horizontally from the root to all of its children, then to its children’s children, and so forth until all nodes have been processed. The three depth-first traversals are preorder, inorder and Postorder traversals. In the preorder traversal, the root is visited first, followed by the left subtree and then the right subtree. In the inorder traversal, the left subtree is visited first, followed by the root and then the right subtree In the postorder traversal, the left subtree is visited first, followed by the right subtree and then the root.

Divide-and-ConquerAlgorithm ConquerAlgorithm Design Technique

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Design and Analysis of Algorithm

REFERENCES: o www.csc.liv.ac.uk/~darek/COMP523/lecture4.ppt pt o www.csie.nuk.edu.tw/~cychen/96-1/AA_The%20 1/AA_The%20Divide-andConquer%20Algorithms.ppt o www.cs.unc.edu/Courses/comp790090f08/Lecturenotes/Lecture 12.ppt o http://www.cs.uml.edu/~bliu/404-f08/notes/divide f08/notes/divide-andconquer.pdf o http://fs.mis.kuas.edu.tw/~jwding/Algorithms/PPT/Ch2_Dividehttp://fs.mis.kuas.edu.tw/~jwding/Algorithms/PPT/Ch2_Divide and-Conquer.pdf o http://www.bambooweb.com/articles/d/i/Divide-and andconquer_algorithm.html o Anany Levitin,(2007), The design and analysis of algorithm (2nd ed.), Pearson Education Inc.

Divide-and-ConquerAlgorithm ConquerAlgorithm Design Technique

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