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

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

1/28

Design and Analysis of Algorithm

D iv i de -and -Conque r

A l go r i t h m D esi gn Techn i q ue

c Definition of Divide-and-Conquer algorithm designtechnique

c Importance of Divide-and-Conquer algorithm

design techniquec Application of Divide-and-Conquer algorithm

design technique in the following algorithms:

q Merge sort

q Quick sort

Divide-and-Conquer Algorithm Design Technique * Property of STI

Page 1 of 28

q

q Binary tree traversals


2/28


Wh at i s D i v i d e - an d -

Co n q u er A l go r i t h m D esi gnTechn ique?

c Divide-and-Conquer

q solves a problem’s instance by dividing it into

several smaller instances, solving each ofthem recursively, and then combining theirsolutions to get a solution to the originalinstance of the problem [LEV07].


Page 2 of 28


3/28


Wh at i s D i v i d e - an d -

Co n q u er A l go r i t h m D esi gnTech n i q u e?

c Example:

q Computing the sum of n numbers

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


Page 3 of 28


4/28


I m p o r t an ce o f D i v i d e- an d -

Co n q u er A l go r i t hm

c Solving difficult problems

c Algorithm efficiency

c Parallelism


Page 4 of 28

c Memory Access


5/28


D i v i d e - a n d - C o n q u e r

i n M er ge So r t

c Merge sort

q combines two files ordered into one orderedfile on the same given key

c Pseudocodeof Merge Sort


Page 5 of 28


6/28



i n M er ge So r t

c Example of Merge Sort:

q Given the list below:

c Solution:

1. Divide the arra :

29 12 14 22 27 15 17 24


Page 6 of 28

2. Sort each sub-array:

3. Merge the sub-arrays:

29 12 14 22 27 15 17 24

12 14 22 29 15 17 24 27


7/28



i n M er ge So r t

c Simulation of the Merge sort algorithm:


Page 7 of 28


8/28



i n M er ge So r t

c Analysis of algorithm of Merge sort using theDivide-and-Conquer approach


Page 8 of 28

c Merge Sort Time complexity


9/28



i n Q u i ck So r t

c Quick sort

q an exchange sort developed by C.A.R. Hoare in

1962q sorting is done by dividing the array into two

partitions and then sorting each partitionrecursively

q a pivot key is placed in its correct position in


Page 9 of 28

widely dispersed across the list


10/28



i n Q u i ck So r t

c Pseudo code for Quick sort algorithm


Page 10 of 28

q Select one element target from the input

q Partition the input into part containingelements not greater than target and partcontaining all bigger elements.

q Sort each part separately.

q Concatenate these sorted parts.


11/28



i n Q u i ck So r t

c Example of Quick sort algorithm

q Given the original unsorted list:

q Solution:

After Pass 2


Page 11 of 28

1. Partition the array :


12/28



i n Q u i ck So r t



q Solution:

After Pass 2


Page 12 of 28

. -


13/28



i n Q u i ck So r t



q Solution:

After Pass 2


Page 13 of 28

3. Merge sub-arrays:


14/28



i n Q u i ck So r t

c Simulation of the Quick sort algorithm:

After Pass 2


Page 14 of 28


15/28


D i v i d e - a n d -C o n q u e r

i n Q u i ck So r t

c Analysis of algorithm of Quick sort using theDivide-and-Conquer technique

q quick sort efficiency:O (nlog2n)

q Analysis of Worst-Case Time Complexity


Page 15 of 28


16/28



i n B i n ar y Sear ch

c Binary search

q used to locate an element in an ordered array

q starts by testing the data in the element at themiddle of the array to determine if the targetis in the first or second half of the list


Page 16 of 28


17/28




c Pseudo code for Binary search algorithm


Page 17 of 28


18/28




c Example:

q Solution:


Page 18 of 28


19/28




c Analysis of algorithm of Binary search usingthe Divide-and-Conquer approach


Page 19 of 28


20/28


D i v i d e- an d - Co n q u e r i n

B i n ar y Tr ee Tr aver sa l s

c Binary Tree

q no node can have more than two subtrees.

c sample binary tree:


Page 20 of 28


21/28




c Balance factor

q height difference between the left subtreeand

the right subtree:B = HL – HR

A A


Page 21 of 28

B C

B = 0

B C

DB = -1

A

B C

D

B = 1


22/28




c Algorithm for defining the height of a binarytree


Page 22 of 28

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


23/28




c Binary tree traversal

q requires that each node of the tree beprocessed once and only once in apredetermined sequence.

c Approaches to the traversal sequence:


Page 23 of 28

q

ep rs raversa• the processing proceeds along a path from theroot through one child to the most distantdescendent of that first child before processinga second child

q Breadth-first traversal

• the processing proceeds horizontally from theroot to all of its children, then to its children’schildren, and so forth until all nodes have beenprocessed


24/28




c Types of Depth-first search Binary Treetraversals :

q Preorder traversal• root – left subtree – right subtree


Page 24 of 28

//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


25/28





q Inordertraversal• left subtree– root – right subtree


Page 25 of 28




1inOrder (root -> leftSubtree)

2 process (root)

3 inOrder(root -> rightSubtree)

2 end if

return


26/28





q

Postordertraversal• left subtree– right subtree – root


Page 26 of 28




1 postOrder (root -> leftSubtree)

2 postOrder(root -> rightSubtree)

3 process (root)

2 end if return


27/28




c Sample application of Binary Tree Traversals:

q Consider thegivenbinarytreebelow:


Page 27 of 28

q Preorder: A, B, C, D, E, F, G

q Inorder: C, B, D, A, F, E, G

q Postorder: C, D, B, F, G, E, A


28/28


SEATWORK

c Draw a binary tree consists of ten nodes

based on the given:q inorder traversal:ABCEDFJGIH

q preorder traversal: JCBADEFIGH

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

Documents

Transcript of MELJUN CORTES ALGORITHM Divide-And-Conquer Algorithm Design Technique II