Sorting algorithms

Sorting algorithms
Eleonora Ciceri, Politecnico di Milano
Email: eleonora.ciceri@polimi.it

What is sorting
 Sorting is any process of arranging items systematically,
with two distinct meanings:
 Ordering: arranging items in a sequence ordered by some
criterion
 Categorizing: grouping items with similar properties

What is a sorting algorithm
 A sorting algorithm is an algorithm that puts elements of a
list in a certain order
78
26
4
12
90
4
12
26
78
90

Why do we need to sort?
 Sorted lists / sequences are useful in the following cases:
 1) Efficient lookup and search
56 31 2 47 54 19 64 85 23
56 31 2 47 54 19 64 85 23
56 31 2 47 54 19 64 85 23
56 31 2 47 54 19 64 85 23
56 31 2 47 54 19 64 85 23
56 31 2 47 54 19 64 85 23
Finding “19”
Non-ordered
list: 6 accesses

 1) Efficient lookup and search
2 19 23 31 47 54 56 64 85
2 19 23 31 47 54 56 64 85
Finding “19”
Ordered list:
2 accesses

 2) Merge sequences
2 5 9 13 20
1 4 8 21
1 2 4 5 8 9 13 20 21

What do we sort?
“Workers sort
parcels in a postal
facility”
(from: Wikipedia,
the free
encyclopedia)

What do we sort?
 We are going to order collections of data
 A couple of examples:
 Arrays
 Linked lists

Compare and swap to sort elements

Common building block of sorting
algorithms
 There are several algorithms one could use to order a list of
elements
 Although they are all different, they share a common building
block:
 They compare pairs of elements to decide which is their relative
order
 When needed, they swap the elements to restore their order
6 8 3 0 5 9Compare:
6 5 3 0 8 9Swap:

How to compare elements:
Lexicographic order
 Elements are usually ordered according to the lexicographic
(or, equivalently, lexicographical) order
 Also known as dictionary order
 The lexicographic order is a generalization of the way the
alphabetical order of words is based on the alphabetical
order of their component letters

How to compare elements:
Some examples
 Some examples
 Order numbers from the smallest to the largest
 Order words alphabetically
 Order people (implemented using a struct that contains name,
surname, age) alphabetically by surname
 People having the same surname are ordered according to their name
 People having the same name and surname are ordered according to their age
 Thus: this proves that comparisons could be based on
complex rules that sort out what to do in case of ties

Sorting algorithms time complexity

Which is the difference between sorting
algorithms?
 Each sorting algorithm is characterized by a particular sorting
strategy
 Are they all equal? NO!
 Some of them are naïve, some of them aren’t
 Naïve algorithms require a larger number of comparisons
 A larger number of comparisons amount to a larger time frame
spent to order elements in the collection

0
2000
4000
6000
8000
10000
12000
10 20 30 40 50 60 70 80 90 100
BubbleSort
MergeSort

(Time) complexity of an algorithm
 In computer science, the time complexity of an algorithm
quantifies the amount of time taken by the algorithm to run
 The time complexity:
 is expressed as a function of the length of the input
 is commonly expressed using big-O notation

Big-O notation
 “The big-O notation describes the limiting behavior of a
function when the number of elements it has to process tends
to infinity”
 We’ll try to simplify the concept in this way:
number of
elements complexity
N O(N2)
e.g., the number of
elements in the array
we want to order
e.g., the number of
comparisons we need to
perform to order the array

Big-O notation:
Which algorithm is “the best”?
 To us, an algorithm is “good” if it allows us to save time
 Let N be the number of elements we want to process
 E.g., number of elements in an array
 What if we have four algorithms whose time complexities are
as follows?
Algorithm
1
O(N)
Algorithm
2
O(N2)
Algorithm
3
O(log(N))

Big-O notation:
 Time complexity can be treated as if it were a mathematical
function
 Being mathematical functions, these curves can be drawn on
a graph
Algorithm
1
O(N) f(N) = N
Algorithm
2
O(N2) f(N) = N2
Algorithm
3
O(log(N)) f(N) = log(N)
Algorithm
4
O(N*log(N)) f(N) = N*log(N)

Big-O notation:
0
100
200
300
400
500
600
700
800
900
1000
1 5 10 15 20 25 30
f(N) = N
f(N) = N
f(N) = log(N)
f(N) = N*log(N)
2

Big-O notation:
0
100
200
300
400
500
600
700
800
900
1000
1 5 10 15 20 25 30
f(N) = N
f(N) = N
f(N) = log(N)
f(N) = N*log(N)
2
The higher the number
of required operation,
the more the time,
the worse the algorithm!

So, how do we select a sorting
algorithm?
 We can identify several “families” of algorithms:
 Simple sorts
 Efficient sorts
 BubbleSort and variants
 Distribution sorts

algorithm?
 Simple sorts
 Efficient sorts
Simple sorting algorithms are efficient on small data amounts (due to
low overhead), but generally do not perform well on large lists
 Insertion sort
 Selection sort

algorithm?
 Simple sorts
 Efficient sorts
Efficient sorting algorithms are those algorithms whose average
complexity is the best you can find (O(N*log(N)))
 Merge sort
 Heap sort
 Quick sort

algorithm?
 Simple sorts
 Efficient sorts
Bubble sort algorithm is very simple, and the same characteristics is
inherited by all its variants. However, it is highly inefficient (i.e., its time
complexity is very high: O(N2))
 Bubble sort
 Shell sort
 Comb sort

algorithm?
 Simple sorts
 Efficient sorts
These algorithms distribute the input to intermediate structures, which are
gathered and placed on the output. They are useful in case of very large
data sets that do not fit in memory (since the intermediate structures can
be deployed on different machines)
 Counting sort
 Bucket sort
 Radix sort

0
2000
4000
6000
8000
10000
12000
10 20 30 40 50 60 70 80 90 100
BubbleSort
MergeSort
One of the most
inefficient algorithms
(O(N2))
One of the most
efficientalgorithms
(O(N*log(N)))

A quick note on time complexity
 An algorithm performance may vary with different input of the
same sizes
 [1 2 3 4 5] does not require any swap
 [5 4 3 2 1] requires a lot of swaps!
 We commonly attribute to each algorithm:
 A worst-case complexity (maximum amount of time it may require)
 An average-case complexity (averaged on all possible inputs)
 We won’t go into the details, since this is just an introduction to
sorting algorithms 

An inefficient sorting algorithm:
the Bubble Sort algorithm

Bubble sort: the idea
 The bubble sort algorithm repeatedly steps through the list of
elements to be sorted:
 Comparing each pair of adjacent elements
 Swapping them if they are in wrong order
 The algorithm stops when, by going through the whole array,
we do not require any swap (i.e., the array is fully ordered)

Bubble sort: the running example
 Array:
 [5 1 4 2 8]
 5 > 1, thus: swap required
Swapped something? False
End of iteration? False

 Array:
 [1 5 4 2 8]
Swapped something? Yes

 Array:
 [1 4 5 2 8]

 Array:
 [1 4 2 5 8]
 5 < 8, thus: swap not required
End of iteration? Yes

 Array:
 [1 4 2 5 8]
 We need to proceed, since in the current iteration we swapped
something
 Following operations:
 Exclude “8” from next iteration
 Restart with a new iteration

 Array:
 [1 4 2 5 8]
Swapped something? No
End of iteration? No

 Array:
 [1 4 2 5 8]

 Array:
 [1 2 4 5 8]

 Array:
 [1 2 4 5 8]
 We need to proceed, since in the current iteration we swapped
something
 Following operations:
 Exclude “5” from next iteration
 Restart with a new iteration

 Array:
 [1 2 4 5 8]

 Array:
 [1 2 4 5 8]
 In the current iteration nothing was swapped
 The algorithm terminates

Bubble sort: the pseudo-code
procedure bubbleSort( A : list of sortable items )
n = length(A)
repeat
swapped = false
for i = 1 to n-1 inclusive do
if A[i-1] > A[i] then
swap(A[i-1], A[i])
swapped = true
end if
end for
n = n - 1
until not swapped
end procedure
(from: Wikipedia, the free encyclopedia)

Bubble sort in C++:
Iterate over the array
 We will use the concept of iterators over the array
 To iterate over the array:
int* first int* last
while (first < last) {
// do what you want
first++;
}

Bubble sort in C++:
The algorithm
void bubble_sort_int(int* first, int* last) {
bool swapped;
do {
swapped = false;
int* current = first + 1;
while (current < last) {
if (*(current-1) > *current) {
swap(current-1, current);
swapped = true;
}
current++;
}
last--;
} while (swapped);
}

Bubble sort in C++:
Swapping elements
void swap(int* first, int* second) {
int temp = *first;
*first = *second;
*second = temp;
}

An efficient sorting algorithm:
the Merge Sort algorithm

Merge sort: the idea
 The merge sort is a divide-and-conquer algorithm where:
 The unsorted list of N elements is divided into N sub-lists, each
containing one element
 Sub-lists are repeatedly merged to produce new sorted sub-lists

How to merge two sub-lists
 Sub-lists are merged so as produce a longer sorted sub-list
 At each step, we:
 compare the first element of sub-list A and the first element of
sub-list B
 select the smallest element
 insert it into the resulting list

How to merge two sub-lists:
An example
 How to merge these:
 to obtain this?
1 6 10
4 5 27
1 4 5 6 10 27

An example
1 6 10
4 5 27
1
1)

An example
1 6 10
4 5 27
1
6 10
4 5 27
1 4
1) 2)

An example
1 6 10
4 5 27
1
6 10
4 5 27
1 4
6 10
5 27
1 4 5
1) 2) 3)

An example
1 6 10
4 5 27
1
6 10
4 5 27
1 4
6 10
5 27
1 4 5
6 10
27
1 4 5 6
1) 2) 3) 4)

An example
1 6 10
4 5 27
1
6 10
4 5 27
1 4
6 10
5 27
1 4 5
6 10
27
1 4 5 6
10
27
1 4 5 6 10
1) 2) 3) 4)
5)

An example
1 6 10
4 5 27
1
6 10
4 5 27
1 4
6 10
5 27
1 4 5
6 10
27
1 4 5 6
10
27
1 4 5 6 10
27
/
1 4 5 6 10 27
1) 2) 3) 4)
5) 6)

Howtomergetwo
sub-lists:
Thepseudocode
function merge(left, right)
// merge lists until at least one of them is empty
while notempty(left) and notempty(right)
if first(left) <= first(right)
append first(left) to result
left = rest(left)
else
append first(right) to result
right = rest(right)
// left has elements left
while notempty(left)
append first(left) to result
left = rest(left)
// right has elements left
while notempty(right)
append first(right) to result
right = rest(right)
return result
(from: Wikipedia, the free encyclopedia)

Merge sort: the step-by-step algorithm
 We start with this list…
 …and, according to the algorithm, we need to keep dividing it
in “left sub-list” and “right sub-list”…

 …and split again, since sub-lists length is greater than 1…

 … and again.

 Now it’s time to merge the lists, in pairs, so that the resulting
list is ordered
 We start from the bottom, i.e., from the shorter sub-lists we
extracted before
This is now
ordered

 Merge…
This is now
ordered

 …and merge.
This is now
ordered

Trying to generalize: Which operations
are needed at each iteration?
 Split phase:
 If the current list has length 1, stop with the split phase
 If the current list has length > 1, split it in two parts (left, right)
 Merge phase:
 Merge the left and right parts into a unique, ordered list

Trying to generalize: What happens to a list?
 Given a list:
Split in
two parts
(left and right)
Split in
two parts
(left and right)
Do something
to order the sub-list
Do something
to order the sub-list
Merge the
parts in a
single list
Merge the
parts in a
single list

 Algorithmically:
[left, right] = split(list)
left = order(left)
right = order(right)
list = merge(left, right)

 Algorithmically:
left = mergeSort(left)
right = mergeSort(right)
This is a
recursive call
to the merge sort function

 Now we finalize the algorithm:
function mergeSort(list) {
if (length(list) == 1)
return list
return list
}

 Now we finalize the algorithm:
function mergeSort(list) {
if (length(list) == 1)
return list
return list
}
This function splits the
list in two parts having
the same length
This function merges
the sub-lists so that the
result is ordered

Merge sort in C++:
// do what you want
first++;
}

Merge sort in C++:
// do what you want
first++;
}
Remember:
last always points
outside the array

Merge sort in C++:
Split the array in two parts
 If the length of the array is even:
 The sub-lists are as follows:
 Left: from first (included) to middle (excluded)
 Right: from middle (included) to last (excluded)
int* first int* lastint* middle

Merge sort in C++:
Split the array in two parts
 If the length of the array is odd:
 The sub-lists are as follows:
 Left: from first (included) to middle (excluded)
 Right: from middle (included) to last (excluded)
 In this case: right is longer than left
int* first int* lastint* middle

Merge sort in C++:
The recursive algorithm
void merge_sort_int(int* first, int* last) {
int N = last - first;
if (N <= 1)
return;
int* middle = first + N/2;
merge_sort_int(first, middle);
merge_sort_int(middle, last);
merge(first, middle, last);
}

MergesortinC++:
Themergefunction
void merge(int* first, int* middle, int* last) {
std::size_t N = last - first;
int *result = new int[N],
*result_current = result,
*left_current = first,
*right_current = middle;
while (left_current < middle && right_current <
last) {
if (*left_current <= *right_current) {
*result_current = *left_current;
left_current++;
}
else {
*result_current = *right_current;
right_current++;
}
result_current++;
}
fill_with_remaining_elements(result_current,
left_current, middle);
right_current, last);
for (std::size_t i = 0; i < N; i++)
first[i] = result[i];
delete[] result;
}

MergesortinC++:
Themergefunction
void merge(int* first, int* middle, int* last) {
std::size_t N = last - first;
int *result = new int[N],
*result_current = result,
*left_current = first,
*right_current = middle;
while (left_current < middle && right_current <
last) {
if (*left_current <= *right_current) {
*result_current = *left_current;
left_current++;
}
else {
*result_current = *right_current;
right_current++;
}
result_current++;
}
left_current, middle);
right_current, last);
for (std::size_t i = 0; i < N; i++)
first[i] = result[i];
delete[] result;
}
result is a temporary
buffer that contains
the ordered array
At the end of the function,
result is restored in the
original array and deleted

Merge sort in C++:
The merge function
void fill_with_remaining_elements(int*& result_current,
int* sub_list_current, int* sub_list_last) {
while (sub_list_current < sub_list_last) {
*result_current = *sub_list_current;
sub_list_current++;
result_current++;
}
}

References
 https://en.wikipedia.org/wiki/Sorting
 https://en.wikipedia.org/wiki/Sorting_algorithm
 https://en.wikipedia.org/wiki/Time_complexity
 https://en.wikipedia.org/wiki/Big_O_notation
 https://en.wikipedia.org/wiki/Bubble_sort
 https://en.wikipedia.org/wiki/Merge_sort

References
 http://stackoverflow.com/questions/24650626/how-to-
implement-classic-sorting-algorithms-in-modern-c

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