algorithm & data structure lec2 (bet)

7/29/2019 Algorithm & Data Structure Lec2 (BET)

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Algorithm & Data Structure CSC-112Fall 2011

Syed Muhammad Raza


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Efficiency of an Algorithm

The Amount of Computer Time and Space that an AlgorithmRequires

We are more interested in time complexity. Why?

Efficiency of algorithms helps comparing different methods of

solution for the same problem

Algorithm Analysis Should be Independent of

Specific Implementation languages,

Computers and

Data

Efficiency is a concern for large problems (large data size)


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Algorithm Growth Rate

Algorithm growth rate is a measure of its efficiency Growth Rate: Measure an Algorithms Time Requirement as a

Function of the Problem Size (N)

How quickly the Algorithms Time Requirement Grows as a Function

of the Problem Size

The Way to Measure a Problems Size Depends on the Application

Number of Nodes in a Linked List

The Size of an Array

Number of Items in the Stack

Etc.


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Example

Suppose we have two algorithms A and B for solving the sameproblem. Which one do we use?

Suppose

Algorithm A Requires Time Proportional to N2 (expressed as

O(N2 ). This is Big O Notation.)

Algorithm B Requires Time Proportional to N , i.e. B is O(N)

Obviously B is a better choice


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Big O Notation

If Algorithm A Requires Time Proportional to f(N), Algorithm A is Saidto be Order f(N), Which is Denoted as O(f(N));

f(N) is Called Algorithms Growth-Rate Function


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Growth Rates Comparison


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Growth Rates Comparison (Tabular)


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Common Growth Rates

Constant : O(1)

Time Requirement is Constant

Independent of Problem Size

Example: Accessing an array element

Logarithmic : O(log N)

Time Requirement Increases Slowly for a Logarithmic Function

Typical algorithm: Solves a Problem by Solving a Smaller Constant Fraction

of the Problem

Example: Binary search

Linear : O(N)

Time Requirement Increases Directly with the size of the Problem

Example: Retrieving an element from a linked list


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Common Growth Rates

O(N log N)

Time Requirement Increases More Rapidly Than a Linear Algorithm

Typical algorithms : Divide a Problem into Smaller Problems That are

Each Solved Separately

Quadratic : O(N2)

Increases Rapidly with the Size of the Problem

Typical Algorithms : Algorithms that Use Two Nested Loops

Practical for Only Small Problems

Cubic : O(N3)

Increases Rapidly with the Size of the Problem

Typical Algorithms : Algorithms that Use Three Nested Loops


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Common Growth Rates

Exponential time: O( 2N

)

Very, very, very bad, even for small N

Usually Increases too Rapidly to be Practical


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Sorting Algorithms

Sorting is the process of rearranging your data elements/Item inascending or descending order

Unsorted Data

Sorted Data (Ascending)

Sorted Data (Descending)

4 3 2 7 1 6 5 8 9

1 2 3 4 5 6 7 8 9

9 8 7 6 5 4 3 2 1


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Sorting Algorithms

They are many

Bubble Sort

Selection Sort

Insertion Sort

Shell sort

Comb Sort

Merge Sort

Heap Sort

Quick Sort

Counting Sort

Bucket Sort

Radix Sort

Distribution Sort

Time SortSource: Wikipedia


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Bubble Sort

Compares Adjacent Items and Exchanges Them if They are Out ofOrder

When You Order Successive Pairs of Elements, the Largest Element

Bubbles to the Top(end) of the Array

Bubble Sort (Usually) Requires Several Passes Through the Array


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Bubble Sort

29 10 14 37 13

10 29 14 37 13

10 14 29 37 13

10 14 29 37 13

10 14 29 13 37

Pass 1

10 14 29 13 37

10 14 29 13 37

10 14 13 29 37

10 14 29 13 37

Pass 2


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Selection Sort

To Sort an Array into Ascending Order, First Search for the Largest Element

Because You Want the Largest Element to be in the Last Position of the Array,

You Swap the Last Item With the Largest Item to be in the Last Position of the

Array, You Swap the Last Item with the Largest Item, Even if These Items Appear

to be Identical

Now, Ignoring the Last (Largest) Item of the Array, search Rest of the Array For

Its Largest Item and Swap it With Its Last Item, Which is the Next-to-Last Item in

the original Array

You Continue Until You Have Selected and Swapped N-1 of the N Items in the

Array

The Remaining Item, Which is Now in the First Position of the Array, is in its

Proper Order


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Selection Sort

29 10 14 37 13

29 10 14 13 37

13 10 14 29 37

13 10 14 29 37

10 13 14 29 37

Initial Array

After4th Swap

After1st Swap

After2nd Swap

After3rd Swap


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Insertion Sort

Divide Array Into Sorted Section At Front (Initially Empty), UnsortedSection At End

Step Iteratively Through Array, Moving Each Element To Proper

Place In Sorted Section

Sorted Unsorted

N-10After i Iterations

.. ..

i


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Insertion Sort

29 29 14 37 13

10 29 29 37 13

10 14 29 37 13

10 14 29 37 13

10 14 14 29 37

Initial Array

Sorted Array

Copy 10

10 29 14 37 13

29 10 14 37 13

10 13 14 29 37

Shift 29

Insert 10, Copy 14

Shift 29

Insert 14; Copy 37

Insert 37 on Itself

Copy 13

Shift 14, 29, 37

Insert 13


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Shell Sort

Improved and efficient version of insertion sort

It iterates the data elements/items like insertion sort, but instead of

shifting it makes the swaps.


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Shell Sort

3

5

1

2

4

1

5

3

2

4

1

2

3

5

4

1

2

3

4

5


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Merge Sort

Divide and Conquer Algorithm

Recursively split the input in half

Then recursively merge pairs of pieces

Recursive Steps:

Divide the Array in Half

Sort the Halves

Merge The Halves inside a Temporary Array

Copy Temporary Array to the appropriate locations in original array


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Merge Sort

The Recursive calls Continue to divide the Array into Pieces UntilEach Piece Contains Only One Item

An Array of One Item is Obviously Sorted

The Algorithm Then Merges These small Pieces Until One Sorted

Array Results

Merge Sort


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Merge Sort

38

38

38

38

38

38

38

16

16

16

16

16

16

16

27

27

27

27

27

27

27

27

27

271239

12

12

12

12

12

12

39

39

39

39

39

39Merge

Steps

Recursive

Calls

to

Mergesort


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Quick Sort

Divide and Conquer algorithm

Quicksort Works by Partitioning the Array into Two Pieces Separated

by a Single Element That is Greater Than all the Elements in the Left

part and Smaller Than all the Elements in the right part

This Guarantees That, the Single Element , Called the Pivot

Element, is in its Correct position

Then the Algorithm Proceeds, Applying the Same Method to the Two

parts Separately


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Quick Sort

Partition (Divide) Choose a pivot

Find the position for the pivot so that

all elements to the left are less

all elements to the right are greater

>= pivotpivot< pivot


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Quick Sort

Conquer Apply the same algorithm to each half

< pivot >= pivot

pivot< p p > p < p p >= p


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Partitioning Method

Must Arrange Items Into Two regions S1, the Set of Items Less Than thePivot, and S2, the Set of Items Greater Than or Equal to Pivot

Different algorithms for Choice of a Pivot

Retain Pivot in A[F] position

The Items That await Placement are in Another Region , Called theUnknown Region

At Each Step of the partition Algorithm you Examine One Item from

Unknown Region and Place it in S1 or S2

P

=P


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To Move A[FirstUnknown] into S1

Swap A[FirstUnknown] With the First Item of S2, Which is A[LastS1+1], and Then

Increment S1 by 1

Thus the Item That Was in A[FirstUnknown] will be at the Rightmost Position of S1

Item of S2 That was Moved to A[FirstUnknown]: If you Increment FirstUnknown by 1,

That Item Becomes the Rightmost Member of S2

Thus, Statements for the Following Steps are Required

Swap A [FirstUnknown] with A[lastS1+1]

Increment LastS1

Increment FirstUnknown

P >=P

=P


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To Move A[FirstUnknown] into S2

Region S2 and Unknown are Adjacent

Simply Increment FirstUnknown by 1, S2 Expands to the Right

To Move Pivot Between S1 and S2

Interchange A[LastS1], the Rightmost Item in S1 with Pivot

Thus, Pivot Would be in its Actual Location

P

=P ?

F LastS1 FirstUnknown L

S1 S2 Unknown

Quick Sort Example (only one step)


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Quick Sort Example (only one step)

38 12 39 27 16

12 38 39 27 16

12 38 39 27 16

12 38 39 27 16

12 16 39 27 38

38 12 39 27 16

38 12 39 27 16

12 27 39 27 38

27

27

27

27

27

27

16

27

S1

S1

S1

S1

S2

S2

S2

S2

S2

S2S1

Unknown

Pivot

Unknown

Unknown

Unknown

Unknown

Pivot

Pivot

Pivot

Pivot

Pivot

Pivot

Pivot

FirstUnknown = 1(Points to 38

38 Belongs in S2

Place Pivot between S1 and S2

S1 is Empty

12 Belongs in S1, swap38 & 12

39 Belongs in S2

27 Belongs in S2

16 Belongs in S1, Swap 38 & 16

No more Unknown

Choose Pivot, keep it in A[F]


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Radix Sort

3 2 9

4 5 7

6 5 78 3 9

4 3 6

7 2 03 5 5

7 2 0

3 5 5

4 3 64 5 7

6 5 7

3 2 98 3 9

7 2 0

3 2 9

4 3 68 3 9

3 5 5

4 5 76 5 7

3 2 9

3 5 5

4 3 64 5 7

6 5 7

7 2 08 3 9


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Comparison of Sorting Algorithms

Worst Case

Selection Sort N2

Bubble Sort N2

Insertion Sort N2

Mergesort N * log N

Quicksort N2

Radix Sort N


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Searching Algorithms

Searching is the process of determining whether or not a given value

exists in a data structure or a storage media.

We will study two searching algorithms

Linear Search

Binary Search


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Linear Search: O(n)

The linear (or sequential) search algorithm on an array is:

Start from beginning of an array/list and continues until the item is

found or the entire array/list has been searched.

Sequentially scan the array, comparing each array item with the

searched value.

If a match is found; return the index of the matched element;

otherwise return1.

Note: linear search can be applied to both sorted and unsorted

arrays.


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Linear Searchbool LinSearch(double x[ ], int n, double item)

{

for(int i=0;i


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Linear Search Tradeoffs

Benefits

Easy to understand

Array can be in any order

Disadvantages

Inefficient for array of N elementsExamines N/2 elements on average for value in array, N

elements for value not in array


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Binary Search: O(log2 n)

Binary search looks for an item in an array/list using

divide and conquer strategy


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Binary Search

Binary search algorithm assumes that the items in the array being

searched is sorted

The algorithm begins at the middle of the array in a binary

search

If the item for which we are searching is less than the item in the

middle, we know that the item wont be in the second half of the

array

Once againwe examine the middle element

The process continues with each comparison cutting in half the

portion of the array where the item might be


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Binary Search

Binary search uses a recursive method to search an array to find a

specified value

The array must be a sorted array:

a[0]a[1]a[2]. . . a[finalIndex]

If the value is found, its index is returned

If the value is not found, -1 is returned

Note: Each execution of the recursive method reduces the search

space by about a half


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Pseudocode for Binary Search


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Execution of Binary Search


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Execution of Binary Search


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Key Points in Binary Search

1. There is no infinite recursion

On each recursive call, the value offirst is increased, or the

value oflast is decreased

If the chain of recursive calls does not end in some other way,

then eventually the method will be called with first larger thanlast

2. Each stopping case performs the correct action for that case

If first > last, there are no array elements between a[first] and

a[last], so key is not in this segment of the array, and resultis

correctly set to -1

Ifkey == a[mid], result is correctly set to mid


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3. For each of the cases that involve recursion, ifall recursive calls

perform their actions correctly, then the entire case performs

correctly

Ifkey < a[mid], then key must be one of the elements a[first]

through a[mid-1], or it is not in the array

The method should then search only those elements, which it

does

The recursive call is correct, therefore the entire action is

correct


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Ifkey > a[mid], then key must be one of the elements

a[mid+1] through a[last], or it is not in the array

The method should then search only those elements, which it

does

The recursive call is correct, therefore the entire action is

correct

The method search passes all three tests:

Therefore, it is a good recursive method definition


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Efficiency of Binary Search

The binary search algorithm is extremely fast compared to an

algorithm that tries all array elements in order

About half the array is eliminated from consideration right at the

start

Then a quarter of the array, then an eighth of the array, and so

forth


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Efficiency of Binary Search

Given an array with 1,000 elements, the binary search will only need

to compare about 10 array elements to the key value, as compared

to an average of 500 for a serial search algorithm

The binary search algorithm has a worst-case running time that is

logarithmic: O(log n)

A serial search algorithm is linear: O(n)

If desired, the recursive version of the method search can be

converted to an iterative version that will run more efficiently


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Binary Search


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IT IS ALL ABOUT DOING THE RIGHT THING

AT THE

RIGHT TIME

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