Goal. Sort any type of data. Ex 1. Sort random real ...
Algorithms
R OBERT S EDGEWICK | K EVIN W AYNE
2.1 E LEMENTARY S ORTS
? rules of the game
? rules of the game
? selection sort
? selection sort
? insertion sort
Algorithms
F O U R T H
2.1 E LEMENTARY S ORTS
? shuffling
R OBERT S EDGEWICK | K EVIN W AYNE
? shellsort
? shuffling
R OBERT S EDGEWICK | K EVIN W AYNE
? convex hull
? insertion sort
Algorithms
? shellsort
E D I T I O N
? convex hull
Sorting problem
Sample sort client 1
Ex. Student records in a university.
Goal. Sort any type of data.
item
key
Chen
3
A
991-878-4944
308 Blair
Rohde
2
A
232-343-5555
343 Forbes
Gazsi
4
B
766-093-9873
101 Brown
101 Brown
Furia
1
A
766-093-9873
Kanaga
3
B
898-122-9643
22 Brown
Andrews
3
A
664-480-0023
097 Little
Battle
4
C
874-088-1212
121 Whitman
Ex 1. Sort random real numbers in ascending order.
seems artificial, but stay tuned for an application
public class Experiment
{
public static void main(String[] args)
{
int N = Integer.parseInt(args[0]);
Double[] a = new Double[N];
for (int i = 0; i < N; i++)
a[i] = StdRandom.uniform();
Insertion.sort(a);
for (int i = 0; i < N; i++)
StdOut.println(a[i]);
}
}
Sort. Rearrange array of N items into ascending order.
Andrews
3
A
664-480-0023
097 Little
Battle
4
C
874-088-1212
121 Whitman
Chen
3
A
991-878-4944
308 Blair
Furia
1
A
766-093-9873
101 Brown
101 Brown
Gazsi
4
B
766-093-9873
Kanaga
3
B
898-122-9643
22 Brown
Rohde
2
A
232-343-5555
343 Forbes
3
% java Experiment 10
0.08614716385210452
0.09054270895414829
0.10708746304898642
0.21166190071646818
0.363292849257276
0.460954145685913
0.5340026311350087
0.7216129793703496
0.9003500354411443
0.9293994908845686
4
Sample sort client 2
Sample sort client 3
Goal. Sort any type of data.
Goal. Sort any type of data.
Ex 2. Sort strings from file in alphabetical order.
Ex 3. Sort the files in a given directory by filename.
public class StringSorter
{
public static void main(String[] args)
{
String[] a = In.readStrings(args[0]);
Insertion.sort(a);
for (int i = 0; i < a.length; i++)
StdOut.println(a[i]);
}
}
import java.io.File;
public class FileSorter
{
public static void main(String[] args)
{
File directory = new File(args[0]);
File[] files = directory.listFiles();
Insertion.sort(files);
for (int i = 0; i < files.length; i++)
StdOut.println(files[i].getName());
}
}
% more words3.txt
bed bug dad yet zoo ... all bad yes
% java FileSorter .
Insertion.class
Insertion.java
InsertionX.class
InsertionX.java
Selection.class
Selection.java
Shell.class
Shell.java
ShellX.class
ShellX.java
% java StringSorter words3.txt
all bad bed bug dad ... yes yet zoo
5
6
Callbacks
Callbacks: roadmap
Goal. Sort any type of data.
client
object implementation
import java.io.File;
public class FileSorter
{
public static void main(String[] args)
{
File directory = new File(args[0]);
File[] files = directory.listFiles();
Insertion.sort(files);
for (int i = 0; i < files.length; i++)
StdOut.println(files[i].getName());
}
}
Q. How can sort() know how to compare data of type Double, String, and
java.io.File without any information about the type of an item's key?
Callback = reference to executable code.
?Client passes array of objects to sort() function.
?The sort() function calls back object's compareTo() method as needed.
public class File
implements Comparable
{
...
public int compareTo(File b)
{
...
return -1;
...
return +1;
...
return 0;
}
}
Implementing callbacks.
?Java: interfaces.
?C: function pointers.
?C++: class-type functors.
?C#: delegates.
?Python, Perl, ML, Javascript:
Comparable interface (built in to Java)
public interface Comparable
{
public int compareTo(Item that);
}
first-class functions.
key point: no dependence
on File data type
7
sort implementation
public static void sort(Comparable[] a)
{
int N = a.length;
for (int i = 0; i < N; i++)
for (int j = i; j > 0; j--)
if (a[j].compareTo(a[j-1]) < 0)
exch(a, j, j-1);
else break;
}
8
Total order
Comparable API
A total order is a binary relation ¡Ü that satisfies:
Implement compareTo() so that pareTo(w)
?Antisymmetry: if v ¡Ü w and w ¡Ü v, then v = w.
?Transitivity: if v ¡Ü w and w ¡Ü x, then v ¡Ü x.
?Totality: either v ¡Ü w or w ¡Ü v or both.
?Is a total order.
?Returns a negative integer, zero, or positive integer
if v is less than, equal to, or greater than w, respectively.
?Throws an exception if incompatible types (or either is null).
Ex.
?Standard order for natural and real numbers.
?Chronological order for dates or times.
?Alphabetical order for strings.
?¡
w
v
w
less than (return -1)
v
w
equal to (return 0)
v
greater than (return +1)
violates totality: (Double.NaN that.month)
if (this.day
< that.day )
if (this.day
> that.day )
return 0;
}
Less. Is item v less than w ?
private static boolean less(Comparable v, Comparable w)
{ return pareTo(w) < 0; }
only compare dates
to other dates
Exchange. Swap item in array a[] at index i with the one at index j.
return
return
return
return
return
return
-1;
+1;
-1;
+1;
-1;
+1;
private static void exch(Comparable[] a, int i, int j)
{
Comparable swap = a[i];
a[i] = a[j];
a[j] = swap;
}
}
11
12
Testing
Goal. Test if an array is sorted.
private static boolean isSorted(Comparable[] a)
{
for (int i = 1; i < a.length; i++)
if (less(a[i], a[i-1])) return false;
return true;
}
2.1 E LEMENTARY S ORTS
? rules of the game
? selection sort
? insertion sort
Algorithms
? shellsort
? shuffling
R OBERT S EDGEWICK | K EVIN W AYNE
Q. If the sorting algorithm passes the test, did it correctly sort the array?
? convex hull
A.
13
Selection sort demo
Selection sort
?In iteration i, find index min of smallest remaining entry.
?Swap a[i] and a[min].
Algorithm. ¡ü scans from left to right.
Invariants.
?Entries the left of ¡ü (including ¡ü) fixed and in ascending order.
?No entry to right of ¡ü is smaller than any entry to the left of ¡ü.
initial
in final order
15
¡ü
16
Selection sort inner loop
Selection sort: Java implementation
To maintain algorithm invariants:
public class Selection
{
public static void sort(Comparable[] a)
{
int N = a.length;
for (int i = 0; i < N; i++)
{
int min = i;
for (int j = i+1; j < N; j++)
if (less(a[j], a[min]))
min = j;
exch(a, i, min);
}
}
?Move the pointer to the right.
i++;
in final order
¡ü
?Identify index of minimum entry on right.
int min = i;
for (int j = i+1; j < N; j++)
if (less(a[j], a[min]))
min = j;
in final order
¡ü
¡ü
private static boolean less(Comparable v, Comparable w)
{ /* as before */ }
?Exchange into position.
private static void exch(Comparable[] a, int i, int j)
{ /* as before */ }
}
exch(a, i, min);
in final order
¡ü
¡ü
17
Selection sort: mathematical analysis
18
Selection sort: animations
Proposition. Selection sort uses (N ¨C 1) + (N ¨C 2) + ... + 1 + 0 ~ N 2 / 2 compares
20 random items
and N exchanges.
i min
a[]
5 6
0
1
2
3
4
7
8
9 10
S
O
R
T
E
X
A
M
P
L
E
0
1
6
4
S
A
O
O
R
R
T
T
E
E
X
X
A
S
M
M
P
P
L
L
E
E
2
3
4
5
10
9
7
7
A
A
A
A
E
E
E
E
R
E
E
E
T
T
L
L
O
O
O
M
X
X
X
X
S
S
S
S
M
M
M
O
P
P
P
P
L
L
T
T
E
R
R
R
6
7
8
9
10
8
10
8
9
10
A
A
A
A
A
E
E
E
E
E
E
E
E
E
E
L
L
L
L
L
M
M
M
M
M
O
O
O
O
O
S
P
P
P
P
X
X
R
R
R
P
S
S
S
S
T
T
T
T
T
R
R
X
X
X
A
E
E
L
M
O
P
R
S
T
X
entries in black
are examined to find
the minimum
entries in red
are a[min]
entries in gray are
in final position
algorithm position
Trace of selection sort (array contents just after each exchange)
in final order
not in final order
Running time insensitive to input. Quadratic time, even if input is sorted.
Data movement is minimal. Linear number of exchanges.
19
20
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