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Fondamenti di Informatica
Sorting Algorithms
Prof. Emiliano Casalicchio
[email protected]
Sorting algorithms
n  selection sort
n  Insertion sort
■  incremental
algorithms
■  progressively extend a sorted sub-sequence
until the collection (vector) is sorted
n  bubble sort
■  based
on multiple scans of the array
■  swap adjacent elements until the collection is
sorted
2
Selection sort
[ 1 2 5 9 7 6 8]
i =1 2 3 4 5 6 7
n  Let us suppose the first k elements of a vector (array) are sorted
■  e.g. k=3
n  how can we extend the sorting to the k+1 element?
■  Selection
sort: pick the minimum of the n-k
remaining element and assign it to the k+1
position (with a swap)
3
Selection sort
[ 1 2 5 9 7 6 8]
i =1 2 3 4 5 6 7
n  Let us suppose the first k elements of a vector (array) are sorted
n  how can we extend the sorting to the k+1 element?
■  Selection
sort:
k=3
[ 1 2 5 9 7 6 8]
i =1 2 3 4 5 6 7
à
[1
i =1
2
2
5
3
6
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7
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6
8]
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k=4
4
The Selection Sort Algorithm
n = length of the sequence
1. Take the k-th element of the sorted subsequence
2. Find the minimum of the n-k unsorted elements
3. Swap the k-th elements with the minimum
4. If k<n go to step 1
5. End
5
Selection sort: how it works
(1)
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The Code
function v = SelectionSort( v )
%Sort the vector v using the SelectionSort algorithm
%INPUT: a vector of numbers v
%OUTPUT: a vector v of number sorted in ascending order
for i=1:length(v)-2
m=i;
%the k element of the sorted sub%sequence
for j=i+1:length(v) %find the minimum of the n-k unsorted
%elements
if v(j)<v(m)
m=j;
end
end
t=v(m);
% Swap the k-th elements with the minimum
v(m)=v(i);
v(i)=t;
end
end
7
Homework
n  Modificare il codice in modo tale da ottenere la seguente funzione
■  [v
num_comp num_swap] =
SelectionSort( v )
n 
dove
■  num_comp
contiene il numero di confronti
eseguiti
■  num_swap contiene il numero effettivo di swap
eseguiti
8
Insertion sort
[ 1 2 5 9 7 6 8]
i =1 2 3 4 5 6 7
n  Let us suppose the first k elements of a vector (array) are sorted
■  e.g. k=3
n  how can we extend the sorting to the k+1 element?
■  Insertion
sort: pick the (k+1)-th elements of the
array and insert it into the correct position
respect to the first k elements
9
Insertion sort
[ 1 2 5 9 7 6 8]
i =1 2 3 4 5 6 7
n  Let us suppose the first k elements of a vector (array) are sorted
n  how can we extend the sorting to the k+1 element?
■  Insertion
sort:
k=4
[ 1 2 5 9 7 6 8]
i =1 2 3 4 5 6 7
à
[1
i =1
2
2
5
3
7
4
9
5
6
6
8]
7
10
The Insertion Sort Algorithm
n = length of the sequence
1. Take the (k+1)-th element of the sequence
2. Find the position for the (k+1) element in the sequence
3. If the new position of (k+1) element is less than k+1 then,
is needed a shift of the already sorted elements
4. If k<n go to step 1
5. End
11
Insertion sort: how it works
(1)
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(2)
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(3)
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(4)
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(5)
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1
(6)
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The code
function v = InsertionSort( v )
%Sort the vector v using the InsertionSort algorithm
%INPUT: a vector of numbers v
%OUTPUT: a vector v of number sorted in ascending order
for i=2:length(v)
x=v(i);
%the k+1 element of the sequence (vector)
for j=1:i %find the position for k+1 element in the
%sequense
if v(j)>x
break;
end
end
if j < i %is needed a shift of the already sorted
%elements
for l=i-1:-1:j
v(l+1)=v(l);
end
v(j)=x;
end
end
13
function v = InsertionSort( v )
The code
%Sort the vector v using the InsertionSort
algorithm
%INPUT: a vector of numbers v
%OUTPUT: a vector v of number sorted in
ascending order
for i=2:length(v)
x=v(i);
%the k+1 element of the
sequence (vector)
for j=1:i %find the position for k+1
element in the
%sequense
if v(j)>x
break;
end
end
if j < i %is needed a shift of the
already sorted
%elements
for l=i-1:-1:j
v(l+1)=v(l);
end
v(j)=x;
end
end
(1) 7
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(2) 2
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(6)
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Debug
n  Using debugging tools we can observe the sequence of algorithm steps
and how the sorted sequence is created
15
Homework
n  Modificare il codice in modo tale da ottenere la seguente funzione
■  [v
num_comp num_shift] =
InsertionSort( v )
n 
dove
■  num_comp
contiene il numero di confronti
eseguiti
■  num_shift contiene il numero effettivo di shift
eseguiti
16
Homework
n  Eseguire gli algoritmi InsertionSort e SelectionSort su di un
insieme di 15 vettori generati casualmente e confrontare i risultati,
ovvero: num_comp e num_shift
n  ad es:
for i=1:15
v=rand(1,10)
[~ num_comp_IS(i) num_shift_IS(i)]=InsertionSort(v)
[~ num_comp_SS(i) num_swap_SS(i)]=SelectionSort(v)
end
17
Bubble sort
n  based on multiple scans of the array
n  on each scan, couple of adjacent elements are compared and swapped
(if needed)
n  if no swap are operated, the array is sorted
18
Bubble sort
(1)
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(4)
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(3)
3
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(2)
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the algorithm
function v = BubbleSort( v )
%Sort the vector v using the BubbleSort algorithm
%INPUT: a vector of numbers v
%OUTPUT: a vector v of number sorted in ascending order
for i=1:length(v)-1 %up to n-1 scans (1)
swap=false;
for j=2:length(v)-(i-1) %do swap
if v(j-1)>v(j)
x=v(j);
v(j)=v(j-1);
v(j-1)=x;
swap=true;
end
(2)
end
if ~swap
break;
end
end
end
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Homework
n  Modificare il codice in modo tale da ottenere la seguente funzione
■  [v
num_comp num_swap] =
BubbleSort( v )
n 
dove
■  num_comp
contiene il numero di confroni
eseguiti
■  num_swap contiene il numero effettivo di swap
eseguiti
21
Homework
n  Implementare gli algoritmi di ordinamento con il costrutto while
n  Gli algoritmi visti fino ad ora effettuano un ordinamento crescente.
■  Progettare
ed implementare un algoritmo di
ordinamento decrescente che utilizzi la tecnica
dell’insertion sort
■  Progettare ed implementare un algoritmo di
ordinamento decrescente che utilizzi la tecnica
del selection sort
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