Mergesort and Quicksort are two efficient sorting algorithms that run in O(n log n) time. Mergesort uses a divide-and-conquer approach, recursively splitting the list into halves until individual elements remain, then merging the sorted halves back together. Quicksort chooses a pivot element and partitions the list into elements less than or greater than the pivot, then recursively sorts the sublists until the entire list is sorted.
Quick sort is an internal algorithm which is based on divide and conquer strategy. In this:
The array of elements is divided into parts repeatedly until it is not possible to divide it further.
It is also known as “partition exchange sort”.
It uses a key element (pivot) for partitioning the elements.
One left partition contains all those elements that are smaller than the pivot and one right partition contains all those elements which are greater than the key element.divide and conquer strategy. In this:
The elements are split into two sub-arrays (n/2) again and again until only one element is left.
Merge sort uses additional storage for sorting the auxiliary array.
Merge sort uses three arrays where two are used for storing each half, and the third external one is used to store the final sorted list by merging other two and each array is then sorted recursively.
At last, the all sub arrays are merged to make it ‘n’ element size of the array. Quick Sort vs Merge Sort
Partition of elements in the array : In the merge sort, the array is parted into just 2 halves (i.e. n/2). whereas In case of quick sort, the array is parted into any ratio. There is no compulsion of dividing the array of elements into equal parts in quick sort.
Worst case complexity : The worst case complexity of quick sort is O(n^2) as there is need of lot of comparisons in the worst condition. whereas In merge sort, worst case and average case has same complexities O(n log n).
Usage with datasets : Merge sort can work well on any type of data sets irrespective of its size (either large or small). whereas The quick sort cannot work well with large datasets.
Additional storage space requirement : Merge sort is not in place because it requires additional memory space to store the auxiliary arrays. whereas The quick sort is in place as it doesn’t require any additional storage.
Efficiency : Merge sort is more efficient and works faster than quick sort in case of larger array size or datasets. whereas Quick sort is more efficient and works faster than merge sort in case of smaller array size or datasets.
Sorting method : The quick sort is internal sorting method where the data is sorted in main memory. whereas The merge sort is external sorting method in which the data that is to be sorted cannot be accommodated in the memory and needed auxiliary memory for sorting.
Stability : Merge sort is stable as two elements with equal value appear in the same order in sorted output as they were in the input unsorted array. whereas Quick sort is unstable in this scenario. But it can be made stable using some changes in code.
Preferred for : Quick sort is preferred for arrays. whereas Merge sort is preferred for linked lists.
Locality of reference : Quicksort exhibits good cache locality and this makes quicksort faster than merge sort (in many cases like in virtual memory environment).
Quick sort is an internal algorithm which is based on divide and conquer strategy. In this:
The array of elements is divided into parts repeatedly until it is not possible to divide it further.
It is also known as “partition exchange sort”.
It uses a key element (pivot) for partitioning the elements.
One left partition contains all those elements that are smaller than the pivot and one right partition contains all those elements which are greater than the key element.divide and conquer strategy. In this:
The elements are split into two sub-arrays (n/2) again and again until only one element is left.
Merge sort uses additional storage for sorting the auxiliary array.
Merge sort uses three arrays where two are used for storing each half, and the third external one is used to store the final sorted list by merging other two and each array is then sorted recursively.
At last, the all sub arrays are merged to make it ‘n’ element size of the array. Quick Sort vs Merge Sort
Partition of elements in the array : In the merge sort, the array is parted into just 2 halves (i.e. n/2). whereas In case of quick sort, the array is parted into any ratio. There is no compulsion of dividing the array of elements into equal parts in quick sort.
Worst case complexity : The worst case complexity of quick sort is O(n^2) as there is need of lot of comparisons in the worst condition. whereas In merge sort, worst case and average case has same complexities O(n log n).
Usage with datasets : Merge sort can work well on any type of data sets irrespective of its size (either large or small). whereas The quick sort cannot work well with large datasets.
Additional storage space requirement : Merge sort is not in place because it requires additional memory space to store the auxiliary arrays. whereas The quick sort is in place as it doesn’t require any additional storage.
Efficiency : Merge sort is more efficient and works faster than quick sort in case of larger array size or datasets. whereas Quick sort is more efficient and works faster than merge sort in case of smaller array size or datasets.
Sorting method : The quick sort is internal sorting method where the data is sorted in main memory. whereas The merge sort is external sorting method in which the data that is to be sorted cannot be accommodated in the memory and needed auxiliary memory for sorting.
Stability : Merge sort is stable as two elements with equal value appear in the same order in sorted output as they were in the input unsorted array. whereas Quick sort is unstable in this scenario. But it can be made stable using some changes in code.
Preferred for : Quick sort is preferred for arrays. whereas Merge sort is preferred for linked lists.
Locality of reference : Quicksort exhibits good cache locality and this makes quicksort faster than merge sort (in many cases like in virtual memory environment).
Sorting plays a very important role in data structure.
Different type of sorting is given that is most popular and mostly used. It is described in very easy and brief way.
Sorting plays a very important role in data structure.
Different type of sorting is given that is most popular and mostly used. It is described in very easy and brief way.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Delivering Micro-Credentials in Technical and Vocational Education and TrainingAG2 Design
Explore how micro-credentials are transforming Technical and Vocational Education and Training (TVET) with this comprehensive slide deck. Discover what micro-credentials are, their importance in TVET, the advantages they offer, and the insights from industry experts. Additionally, learn about the top software applications available for creating and managing micro-credentials. This presentation also includes valuable resources and a discussion on the future of these specialised certifications.
For more detailed information on delivering micro-credentials in TVET, visit this https://tvettrainer.com/delivering-micro-credentials-in-tvet/
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
2. Sorting algorithms
• Insertion, selection and bubble sort have
quadratic worst-case performance
• The faster comparison based algorithm ?
O(nlogn)
• Mergesort and Quicksort
3. Merge Sort
• Apply divide-and-conquer to sorting problem
• Problem: Given n elements, sort elements into
non-decreasing order
• Divide-and-Conquer:
– If n=1 terminate (every one-element list is already
sorted)
– If n>1, partition elements into two or more sub-
collections; sort each; combine into a single sorted list
• How do we partition?
4. Partitioning - Choice 1
• First n-1 elements into set A, last element set B
• Sort A using this partitioning scheme recursively
– B already sorted
• Combine A and B using method Insert() (= insertion
into sorted array)
• Leads to recursive version of InsertionSort()
– Number of comparisons: O(n2)
• Best case = n-1
• Worst case = 2
)
1
(
2
n
n
i
c
n
i
5. Partitioning - Choice 2
• Put element with largest key in B, remaining elements
in A
• Sort A recursively
• To combine sorted A and B, append B to sorted A
– Use Max() to find largest element recursive
SelectionSort()
– Use bubbling process to find and move largest element to
right-most position recursive BubbleSort()
• All O(n2)
6. Partitioning - Choice 3
• Let’s try to achieve balanced partitioning
• A gets n/2 elements, B gets rest half
• Sort A and B recursively
• Combine sorted A and B using a process
called merge, which combines two sorted
lists into one
– How? We will see soon
9. Static Method mergeSort()
Public static void mergeSort(Comparable []a, int left,
int right)
{
// sort a[left:right]
if (left < right)
{// at least two elements
int mid = (left+right)/2; //midpoint
mergeSort(a, left, mid);
mergeSort(a, mid + 1, right);
merge(a, b, left, mid, right); //merge from a to b
copy(b, a, left, right); //copy result back to a
}
}
12. Solution
By Substitution:
T(n) = 2T(n/2) + c2n
T(n/2) = 2T(n/4) + c2n/2
T(n) = 4T(n/4) + 2 c2n
T(n) = 8T(n/8) + 3 c2n
T(n) = 2iT(n/2i) + ic2n
Assuming n = 2k, expansion halts when we get T(1) on right side;
this happens when i=k T(n) = 2kT(1) + kc2n
Since 2k=n, we know k=logn; since T(1) = c1, we get
T(n) = c1n + c2nlogn;
thus an upper bound for TmergeSort(n) is O(nlogn)
13. Quicksort Algorithm
Given an array of n elements (e.g., integers):
• If array only contains one element, return
• Else
– pick one element to use as pivot.
– Partition elements into two sub-arrays:
• Elements less than or equal to pivot
• Elements greater than pivot
– Quicksort two sub-arrays
– Return results
15. Pick Pivot Element
There are a number of ways to pick the pivot element. In this
example, we will use the first element in the array:
40 20 10 80 60 50 7 30 100
16. Partitioning Array
Given a pivot, partition the elements of the array
such that the resulting array consists of:
1. One sub-array that contains elements >= pivot
2. Another sub-array that contains elements < pivot
The sub-arrays are stored in the original data array.
Partitioning loops through, swapping elements
below/above pivot.
45. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• What is best case running time?
– Recursion:
1. Partition splits array in two sub-arrays of size n/2
2. Quicksort each sub-array
46. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• What is best case running time?
– Recursion:
1. Partition splits array in two sub-arrays of size n/2
2. Quicksort each sub-array
– Depth of recursion tree?
47. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• What is best case running time?
– Recursion:
1. Partition splits array in two sub-arrays of size n/2
2. Quicksort each sub-array
– Depth of recursion tree? O(log2n)
48. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• What is best case running time?
– Recursion:
1. Partition splits array in two sub-arrays of size n/2
2. Quicksort each sub-array
– Depth of recursion tree? O(log2n)
– Number of accesses in partition?
49. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• What is best case running time?
– Recursion:
1. Partition splits array in two sub-arrays of size n/2
2. Quicksort each sub-array
– Depth of recursion tree? O(log2n)
– Number of accesses in partition? O(n)
50. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• Best case running time: O(n log2n)
51. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• Best case running time: O(n log2n)
• Worst case running time?
52. Quicksort: Worst Case
• Assume first element is chosen as pivot.
• Assume we get array that is already in
order:
2 4 10 12 13 50 57 63 100
pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index
53. 1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100
pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index
54. 1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100
pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index
55. 1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100
pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index
56. 1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100
pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index
57. 1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100
pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index
58. 1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100
pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
too_big_index too_small_index
59. 1. While data[too_big_index] <= data[pivot]
++too_big_index
2. While data[too_small_index] > data[pivot]
--too_small_index
3. If too_big_index < too_small_index
swap data[too_big_index] and data[too_small_index]
4. While too_small_index > too_big_index, go to 1.
5. Swap data[too_small_index] and data[pivot_index]
2 4 10 12 13 50 57 63 100
pivot_index = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
> data[pivot]
<= data[pivot]
60. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• Best case running time: O(n log2n)
• Worst case running time?
– Recursion:
1. Partition splits array in two sub-arrays:
• one sub-array of size 0
• the other sub-array of size n-1
2. Quicksort each sub-array
– Depth of recursion tree?
61. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• Best case running time: O(n log2n)
• Worst case running time?
– Recursion:
1. Partition splits array in two sub-arrays:
• one sub-array of size 0
• the other sub-array of size n-1
2. Quicksort each sub-array
– Depth of recursion tree? O(n)
62. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• Best case running time: O(n log2n)
• Worst case running time?
– Recursion:
1. Partition splits array in two sub-arrays:
• one sub-array of size 0
• the other sub-array of size n-1
2. Quicksort each sub-array
– Depth of recursion tree? O(n)
– Number of accesses per partition?
63. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• Best case running time: O(n log2n)
• Worst case running time?
– Recursion:
1. Partition splits array in two sub-arrays:
• one sub-array of size 0
• the other sub-array of size n-1
2. Quicksort each sub-array
– Depth of recursion tree? O(n)
– Number of accesses per partition? O(n)
64. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• Best case running time: O(n log2n)
• Worst case running time: O(n2)!!!
65. Quicksort Analysis
• Assume that keys are random, uniformly
distributed.
• Best case running time: O(n log2n)
• Worst case running time: O(n2)!!!
• What can we do to avoid worst case?
66. Improved Pivot Selection
Pick median value of three elements from data array:
data[0], data[n/2], and data[n-1].
Use this median value as pivot.
67. Improving Performance of
Quicksort
• Improved selection of pivot.
• For sub-arrays of size 3 or less, apply brute
force search:
– Sub-array of size 1: trivial
– Sub-array of size 2:
• if(data[first] > data[second]) swap them
– Sub-array of size 3: left as an exercise.