Quick Sort By Prof Lili Saghafi

Prof. Lili Saghafi
proflilisaghafi@gmail.com
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Sorting Algorithms
Algorithms
and Data Structures
Quick Sort
By : Prof. Lili Saghafi
@Lili_PLS

Prof. Lili Saghafi
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Objectives
–To study and analyze time complexity of
various sorting algorithms
–To design, implement, and analyze
selection sort, insertion sort
–To design, implement, and analyze bubble
sort To design, implement, and analyze
merge sort
–– To design, implement, and analyzeTo design, implement, and analyze
quick sortquick sort

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Why study sorting?
Sorting is a classic subject in computer science.
There are three reasons for studying sorting algorithmssorting algorithms.
– First, sorting algorithms illustrate many creative
approaches to problem solving
– Second, sorting algorithms are good for practicing
fundamental programming techniques using
selection statements, loops, methods, and
arrays.
– Third, sorting algorithms are excellent examples to
demonstrate algorithm performance.

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What data to sort?
The data to be sorted might be integers,
doubles, characters, or objects.
For “Sorting Arrays,” for simplicity lets
assume:
• data to be sorted are integers,
• data are sorted in ascending order, and
• data are stored in an array.
• The programs can be easily modified to sort
other types of data, to sort in descending order,
or to sort data in an ArrayList or a LinkedList.

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Sorting Arrays
• Sorting, like searching, is also a common task in computer
programming.
• Sorting would be used, for instance, if you wanted to display
the grades in alphabetical order.
• Many different algorithms have been developed for
sorting.
• Selection Sort | Insertion Sort | Bubble Sort | Radix
Sort | Merge Sort | Merge two sorted lists | QuickQuick
SortSort | Partition in quick sort
• Quicksort is a little more complex than selection and bubble
• Best choice for general purpose and in memory sorting
• Used to be the standard algorithm for sorting of arrays of
primitive types in Java

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Quick Sort
•Quick sort, developed by C. A. R. Hoare (1962),
works as follows:
– The algorithm selects an element, called the pivot, in the
array.
– Divide the array into two parts such that all the elements
in the first part are less than or equal to the pivot and
all the elements in the second part are greater than the
pivot.
– Recursively apply the quick sort algorithm to the first part
and then the second part.
Sir Charles Antony Richard Hoare
1980 Turing Award

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Quick Sort
• Quicksort honoured as one of top 10
algorithms of 20th century in science and
engineering.
Basic plan.
• Shuffle the array.
• Partition so that, for some j
– entry a[j] is in place
– no larger entry to the left of j
– no smaller entry to the right of j
– Sort each piece recursively.

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Important factor in Quick Sort is Pivot
PIVOT needs to have following condition
after we sort it

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What we need to choose in next
step
We need to swap these two
Now we need to repeat , choose
item from left and item from right
and if needed swap the two.
Quicksort is
Recursive sort

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• We repeat until we see that item from left has a greater index
than item from right, so we know we're done .( LOW to HIGH , If
HIGH is less than LOW and pivot )
• we STOP and swap item from left ( HIGH) with our pivot 3 , our
pivot is now in its correct spot
We swap item from left
with are PIVOT

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Characteristic of PIVOT
Quicksort is Recursive , we
choose larger partition
We choose 7 as are pivot
and move it to the end
So we swap 8 & 6
then we swap
8 and pivot
Now we have our PIVOT 7 in
the right place
We leave Recursion handle
the rest
We move pivot to the
end
STOP When
item from left
has a greater
index than
item from right

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how do we choose the pivot
Choose the median of the array
•One popular method is called median of three
•In this method we look at the first middle and last elements of the
array we sort them properly and choose the middle item as our
pivot
•We're making the guess that the middle of these three items
could be close to the median of the entire array and as you can
see it's not too far off

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Why does it work?
• On the partition step, algorithm divides thealgorithm divides the
array into two partsarray into two parts and every element a from
the left part is less or equal than every
element b from the right part.
• Also a and b satisfy a ≤ pivot ≤ b inequality.
• After completion of the recursion calls both of
the parts become sorted and, taking into account
arguments stated above, the whole array is
sorted.

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Quick Sort Partition
How it works
(Quick Sort Partition )
• In the beginning , First element as PIVOT
• Element after PIVOT as LOW the last element as HIGH
• Is LOW less than PIVOT ? Yes , Then move LOW to next element
– If NOT then we move HIGH one to the right since list[high] > pivot,
high was moved backward.
– Is LOW greater than PIVOT ? Yes ,
– and IF HIGH is lower than PIVOT then we swap the low and high
– If NOT then we move HIGH one to the right since list[high] > pivot,
high was moved backward.
– REPEAT until LOW and HIGH are equal .
– If it is MORE than PIVOT then we move high to the right.
– If it is LESS than Pivot then we swap high and pivot
– Now the partition is complete and PIVOTE makes the array into two sub
arrays for sorting .
• The only thing we care about for now is that all these elements to the right
of the wall are bigger than the pivot.
• No actual order is assumed yet.

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Quick Sort
How it works
Quick Sort
• Pivot is determined
• All elements less than PIVOTE in one subarray
• All elements more than PIVOTE in one subarray
• Recursively apply Quiksort to first subarray until
all elements sort
• Recursively apply Quiksort to the second
subarray until all elements sort
• Now the entire array/list is sorted

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Analysis
• Each time we repeat the process on the left and right
sides of the array. we choose a pivot and do the
comparisons and create another level of left and right
subarrays.
• This recursive call will continue until we reach the end
when we've divided up the overall array into just
subarrays of length 1.
• From there, we know the array is sorted because every
element has, at some point, been a pivot.
• In other words, for every element, all the numbers to
the left are lesser values than PIVOT and all the
numbers to the right have greater values than PIVOT.

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Analysis continue…
• This method works very well, if the value of the pivot
chosen is approximately in the middle range of the
list values.
• This would mean that, after we move elements
around, there are about as many elements to the left
of the pivot as there are to the right.
• And the dividethe divide--andand--conquerconquer nature of the Quicksort
algorithm is then taken full advantage of it .
RUNTIME COMPLEXITY
• Best case: split in the middle — Θ(n log n)
• Worst case: sorted array! — Θ(n2)
• Average case: random arrays — Θ(n log n)
Considered the method of choice for internal sorting of
large files (n ≥ 10000)

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Runtime
• This creates a runtime of O (n log n,) the n
because we have to do n minus 1 comparisons
on each generation, and log n because we
have to divide the list log n times.
• However, in the worst cases, this algorithm can
actually be O (n squared.)
• Suppose on each generation, the pivot just so
happens to be the smallest or the largest of
the numbers we're sorting.
• This would mean breaking down the list,
n times and making n minus 1 comparisons
every single time. Thus, Big O of n squared is
runtime.

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The importance of good Algorithms

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Hoare’s Partitioning Algorithm

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Pseudocode for Quicksort

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Quick Sort
RunQuickSort

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References
• Algorithms and Data Structures with implementations in Java and C++
• Introduction to Programming with C++ 3/E Y. Daniel Liang, Armstrong
State University
• C++ Without Fear: A Beginner's Guide That Makes You Feel Smart, 3/E
• Problem Solving and Programming Concepts, 9/E , Maureen Sprankle,
Jim Hubbard
• C++ Language http://www.cplusplus.com/doc/tutorial/
• Cormen, Leiserson, Rivest. Introduction to algorithms. (Theory)
• Aho, Ullman, Hopcroft. Data Structures and Algorithms. (Theory)
• Robert Lafore. Data Structures and Algorithms in Java. (Practice)
• Mark Allen Weiss. Data Structures and Problem Solving Using
C++. (Practice)

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Sorting algorithms
ANY QUESTION ?
Algorithms and Data
Structures
Quick Sort
By : Prof. Lili Saghafi
proflilisaghafi@gmail.com @Lili_PLS

Prof. Lili Saghafi
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Quicksort
• Quicksort has worst case time complexity, a Big O of
N squared
• If a pivot is chosen properly it can be shown to have
an average case of ,Big O and n log N
Improvements:
1. better pivot selection: median of three partitioning
2. switch to insertion sort on small subfiles
3. elimination of recursion
These combine to 20-25% improvement

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How can we improve the method?
• One good way to improve the method is to cut
down on the probability that the runtime is ever
actually O of n squared.
• Remember this worst worst case scenario can
only happen when the pivot chosen is always
the highest or lowest value in the array.
• To ensure this is less likely to happen, we can
find the pivot by choosing multiple elements and
taking the median value as we described
earlier.

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Quicksort
• Quicksort is a fast sorting algorithm,
which is used not only for educational
purposes, but widely applied in practice.
• On the average, it has O(n log n)
complexity, making quicksort suitable for
sorting big data volumes.

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Algorithm
• The idea of the algorithm is quite simple
• The divide-and-conquer strategy is used in quicksort.
Below the recursion step is described:
• Choose a pivot value. We take the value of the middle
element as pivot value, but it can be any value, which is
in range of sorted values, even if it doesn't present in the
array.
• Partition. Rearrange elements in such a way, that all
elements which are lesser than the pivot go to the left
part of the array and all elements greater than the pivot,
go to the right part of the array. Values equal to the pivot
can stay in any part of the array. Notice, that array may
be divided in non-equal parts.
• Sort both parts. Apply quicksort algorithm recursively to
the left and the right parts.

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Complexity analysis
• On the average quicksort has O(n log n)
complexity
• In worst case, quicksort runs O(n2) time
• On the most "practical" data it works just
fine and outperforms other O(n log n)
sorting algorithms.

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Worst-Case Time
1. To partition an array of n elements, it takes n
comparisons and n moves in the worst case.
2. In the worst case, each time the pivot divides
the array into one big subarray with the other
empty.
3. The size of the big subarray is one less than
the one before divided.
4. The algorithm requires time:
)(12...)2()1( 2
nOnn =+++−+−
Suppose on each generation, the pivot just so happens to be the smallest or the
largest of the numbers we're sorting.This would mean breaking down the list,
n times and making n minus 1 comparisons every single time. Thus, Big O of
n squared is runtime.

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Best-Case Time
1. In the best case, each time the pivot
divides the array into two parts of about
the same size.
2. Let T(n) denote the time required for
sorting an array of elements using
quick sort.
3. So,
n
n
T
n
TnT ++= )
2
()
2
()(

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Average-Case Time
1. On the average, each time the pivot will
not divide the array into two parts of the
same size nor one empty part.
2. Statistically, the sizes of the two parts
are very close.
3. So the average time is O(nlog n).
4. Be reminded that the exact average-case
analysis is beyond the scope of this
course. we have to do n minus 1 comparisons on each generation,
and log n because we have to divide the list log n times.

Quick Sort By Prof Lili Saghafi

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