Name

Tree::Multi - Multi-way tree in Pure Perl with an even or odd number of keys per node.

Synopsis

Construct and query a multi-way tree in 100% Pure Perl with a choice of an odd or an even numbers of keys per node: local $Tree::Multi::numberOfKeysPerNode = 4; # Number of keys per node - can be even

 my $t = Tree::Multi::new;                     # Construct tree
    $t->insert($_, 2 * $_) for reverse 1..32;  # Load tree in reverse

 is_deeply $t->print, <<END;
15 21 27
  3 6 9 12
    1 2
    4 5
    7 8
    10 11
    13 14
  18
    16 17
    19 20
  24
    22 23
    25 26
  30
    28 29
    31 32
END

 ok  $t->height     ==  3;                     # Height

 ok  $t->find  (16) == 32;                     # Find by key
     $t->delete(16);                           # Delete a key
 ok !$t->find  (16);                           # Key no longer present


 ok  $t->find  (17) == 34;                     # Find by key
 my @k;
 for(my $i = $t->iterator; $i->more; $i->next) # Iterator
  {push @k, $i->key unless $i->key == 17;
  }

 $t->delete($_) for @k;                        # Delete

 ok $t->find(17) == 34 && $t->size == 1;       # Size

Description

Multi-way tree in Pure Perl with an even or odd number of keys per node.

Version "20210602".

The following sections describe the methods in each functional area of this module. For an alphabetic listing of all methods by name see Index.

Multi-way Tree

Create and use a multi-way tree.

root($tree)

Return the root node of a tree.

   Parameter  Description
1  $tree      Tree

Example:

  local $numberOfKeysPerNode = 3; my $N = 13; my $t = new;

  for my $n(1..$N)
   {$t->insert($n, $n);
   }

  is_deeply $t->leftMost ->keys, [1, 2];
  is_deeply $t->rightMost->keys, [13];
  ok $t->leftMost ->leaf;
  ok $t->rightMost->leaf;

  ok $t->root == $t;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


  ok T($t, <<END);
 6
   3
     1 2
     4 5
   9 12
     7 8
     10 11
     13
END

leaf($tree)

Confirm that the tree is a leaf.

   Parameter  Description
1  $tree      Tree

Example:

  local $numberOfKeysPerNode = 3; my $N = 13; my $t = new;

  for my $n(1..$N)
   {$t->insert($n, $n);
   }

  is_deeply $t->leftMost ->keys, [1, 2];
  is_deeply $t->rightMost->keys, [13];

  ok $t->leftMost ->leaf;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


  ok $t->rightMost->leaf;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

  ok $t->root == $t;

  ok T($t, <<END);
 6
   3
     1 2
     4 5
   9 12
     7 8
     10 11
     13
END

find($root, $key)

Find a key in a tree returning its associated data or undef if the key does not exist

   Parameter  Description
1  $root      Root of tree
2  $key       Key

Example:

  local $Tree::Multi::numberOfKeysPerNode = 4;                                  # Number of keys per node - can be even

  my $t = Tree::Multi::new;                                                     # Construct tree
     $t->insert($_, 2 * $_) for reverse 1..32;                                  # Load tree in reverse

  is_deeply $t->print, <<END;
 15 21 27
   3 6 9 12
     1 2
     4 5
     7 8
     10 11
     13 14
   18
     16 17
     19 20
   24
     22 23
     25 26
   30
     28 29
     31 32
END

  ok  $t->height     ==  3;                                                     # Height


  ok  $t->find  (16) == 32;                                                     # Find by key  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

      $t->delete(16);                                                           # Delete a key

  ok !$t->find  (16);                                                           # Key no longer present  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲




  ok  $t->find  (17) == 34;                                                     # Find by key  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

  my @k;
  for(my $i = $t->iterator; $i->more; $i->next)                                 # Iterator
   {push @k, $i->key unless $i->key == 17;
   }

  $t->delete($_) for @k;                                                        # Delete


  ok $t->find(17) == 34 && $t->size == 1;                                       # Size  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

leftMost($tree)

Return the left most node below the specified one

   Parameter  Description
1  $tree      Tree

Example:

  local $numberOfKeysPerNode = 3; my $N = 13; my $t = new;

  for my $n(1..$N)
   {$t->insert($n, $n);
   }


  is_deeply $t->leftMost ->keys, [1, 2];  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

  is_deeply $t->rightMost->keys, [13];

  ok $t->leftMost ->leaf;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

  ok $t->rightMost->leaf;
  ok $t->root == $t;

  ok T($t, <<END);
 6
   3
     1 2
     4 5
   9 12
     7 8
     10 11
     13
END

rightMost($tree)

Return the right most node below the specified one

   Parameter  Description
1  $tree      Tree

Example:

  local $numberOfKeysPerNode = 3; my $N = 13; my $t = new;

  for my $n(1..$N)
   {$t->insert($n, $n);
   }

  is_deeply $t->leftMost ->keys, [1, 2];

  is_deeply $t->rightMost->keys, [13];  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

  ok $t->leftMost ->leaf;

  ok $t->rightMost->leaf;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

  ok $t->root == $t;

  ok T($t, <<END);
 6
   3
     1 2
     4 5
   9 12
     7 8
     10 11
     13
END

height($tree)

Return the height of the tree

   Parameter  Description
1  $tree      Tree

Example:

local $Tree::Multi::numberOfKeysPerNode = 3;

my $t = new;      ok $t->height == 0; ok $t->leftMost->depth == 0; ok $t->size == 0;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(1, 1); ok $t->height == 1; ok $t->leftMost->depth == 1; ok $t->size == 1;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(2, 2); ok $t->height == 1; ok $t->leftMost->depth == 1; ok $t->size == 2;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(3, 3); ok $t->height == 1; ok $t->leftMost->depth == 1; ok $t->size == 3;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(4, 4); ok $t->height == 2; ok $t->leftMost->depth == 2; ok $t->size == 4;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

depth($tree)

Return the depth of a node within a tree

   Parameter  Description
1  $tree      Tree

Example:

local $Tree::Multi::numberOfKeysPerNode = 3;

my $t = new;      ok $t->height == 0; ok $t->leftMost->depth == 0; ok $t->size == 0;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(1, 1); ok $t->height == 1; ok $t->leftMost->depth == 1; ok $t->size == 1;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(2, 2); ok $t->height == 1; ok $t->leftMost->depth == 1; ok $t->size == 2;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(3, 3); ok $t->height == 1; ok $t->leftMost->depth == 1; ok $t->size == 3;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(4, 4); ok $t->height == 2; ok $t->leftMost->depth == 2; ok $t->size == 4;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

delete($root, $key)

Find a key in a tree, delete it, return the new tree

   Parameter  Description
1  $root      Tree root
2  $key       Key

Example:

  local $Tree::Multi::numberOfKeysPerNode = 4;                                  # Number of keys per node - can be even

  my $t = Tree::Multi::new;                                                     # Construct tree
     $t->insert($_, 2 * $_) for reverse 1..32;                                  # Load tree in reverse

  is_deeply $t->print, <<END;
 15 21 27
   3 6 9 12
     1 2
     4 5
     7 8
     10 11
     13 14
   18
     16 17
     19 20
   24
     22 23
     25 26
   30
     28 29
     31 32
END

  ok  $t->height     ==  3;                                                     # Height

  ok  $t->find  (16) == 32;                                                     # Find by key

      $t->delete(16);                                                           # Delete a key  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

  ok !$t->find  (16);                                                           # Key no longer present


  ok  $t->find  (17) == 34;                                                     # Find by key
  my @k;
  for(my $i = $t->iterator; $i->more; $i->next)                                 # Iterator
   {push @k, $i->key unless $i->key == 17;
   }


  $t->delete($_) for @k;                                                        # Delete  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


  ok $t->find(17) == 34 && $t->size == 1;                                       # Size

insert($tree, $key, $data)

Insert a key and data into a tree

   Parameter  Description
1  $tree      Tree
2  $key       Key
3  $data      Data

Example:

  local $Tree::Multi::numberOfKeysPerNode = 4;                                  # Number of keys per node - can be even

  my $t = Tree::Multi::new;                                                     # Construct tree

     $t->insert($_, 2 * $_) for reverse 1..32;                                  # Load tree in reverse  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


  is_deeply $t->print, <<END;
 15 21 27
   3 6 9 12
     1 2
     4 5
     7 8
     10 11
     13 14
   18
     16 17
     19 20
   24
     22 23
     25 26
   30
     28 29
     31 32
END

  ok  $t->height     ==  3;                                                     # Height

  ok  $t->find  (16) == 32;                                                     # Find by key
      $t->delete(16);                                                           # Delete a key
  ok !$t->find  (16);                                                           # Key no longer present


  ok  $t->find  (17) == 34;                                                     # Find by key
  my @k;
  for(my $i = $t->iterator; $i->more; $i->next)                                 # Iterator
   {push @k, $i->key unless $i->key == 17;
   }

  $t->delete($_) for @k;                                                        # Delete

  ok $t->find(17) == 34 && $t->size == 1;                                       # Size

iterator($tree)

Make an iterator for a tree

   Parameter  Description
1  $tree      Tree

Example:

  local $numberOfKeysPerNode = 3; my $N = 256; my $e = 0;  my $t = new;

  for my $n(0..$N)
   {$t->insert($n, $n);

    my @n; for(my $i = $t->iterator; $i->more; $i->next) {push @n, $i->key}  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

    ++$e unless dump(\@n) eq dump [0..$n];
   }

  is_deeply $e, 0;

  local $Tree::Multi::numberOfKeysPerNode = 4;                                  # Number of keys per node - can be even

  my $t = Tree::Multi::new;                                                     # Construct tree
     $t->insert($_, 2 * $_) for reverse 1..32;                                  # Load tree in reverse

  is_deeply $t->print, <<END;
 15 21 27
   3 6 9 12
     1 2
     4 5
     7 8
     10 11
     13 14
   18
     16 17
     19 20
   24
     22 23
     25 26
   30
     28 29
     31 32
END

  ok  $t->height     ==  3;                                                     # Height

  ok  $t->find  (16) == 32;                                                     # Find by key
      $t->delete(16);                                                           # Delete a key
  ok !$t->find  (16);                                                           # Key no longer present


  ok  $t->find  (17) == 34;                                                     # Find by key
  my @k;

  for(my $i = $t->iterator; $i->more; $i->next)                                 # Iterator  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

   {push @k, $i->key unless $i->key == 17;
   }

  $t->delete($_) for @k;                                                        # Delete

  ok $t->find(17) == 34 && $t->size == 1;                                       # Size

Tree::Multi::Iterator::next($iter)

Find the next key

   Parameter  Description
1  $iter      Iterator

Example:

local $numberOfKeysPerNode = 3; my $N = 256; my $e = 0;  my $t = new;

for my $n(0..$N)
 {$t->insert($n, $n);
  my @n; for(my $i = $t->iterator; $i->more; $i->next) {push @n, $i->key}
  ++$e unless dump(\@n) eq dump [0..$n];
 }

is_deeply $e, 0;

reverseIterator($tree)

Create a reverse iterator for a tree

   Parameter  Description
1  $tree      Tree

Example:

local $numberOfKeysPerNode = 3; my $N = 64;  my $e = 0;

for my $n(0..$N)
 {my $t = new;
  for my $i(0..$n)
   {$t->insert($i, $i);
   }
  my @n;

  for(my $i = $t->reverseIterator; $i->less; $i->prev)  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

   {push @n, $i->key;
   }
  ++$e unless dump(\@n) eq dump [reverse 0..$n];
 }

is_deeply $e, 0;

Tree::Multi::ReverseIterator::prev($iter)

Find the previous key

   Parameter  Description
1  $iter      Iterator

Example:

local $numberOfKeysPerNode = 3; my $N = 64;  my $e = 0;

for my $n(0..$N)
 {my $t = new;
  for my $i(0..$n)
   {$t->insert($i, $i);
   }
  my @n;
  for(my $i = $t->reverseIterator; $i->less; $i->prev)
   {push @n, $i->key;
   }
  ++$e unless dump(\@n) eq dump [reverse 0..$n];
 }

is_deeply $e, 0;

print($tree, $i)

Print the keys in a tree optionally marking the active key

   Parameter  Description
1  $tree      Tree
2  $i         Optional index of active key

Example:

  local $Tree::Multi::numberOfKeysPerNode = 4;                                  # Number of keys per node - can be even

  my $t = Tree::Multi::new;                                                     # Construct tree
     $t->insert($_, 2 * $_) for reverse 1..32;                                  # Load tree in reverse


  is_deeply $t->print, <<END;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

 15 21 27
   3 6 9 12
     1 2
     4 5
     7 8
     10 11
     13 14
   18
     16 17
     19 20
   24
     22 23
     25 26
   30
     28 29
     31 32
END

  ok  $t->height     ==  3;                                                     # Height

  ok  $t->find  (16) == 32;                                                     # Find by key
      $t->delete(16);                                                           # Delete a key
  ok !$t->find  (16);                                                           # Key no longer present


  ok  $t->find  (17) == 34;                                                     # Find by key
  my @k;
  for(my $i = $t->iterator; $i->more; $i->next)                                 # Iterator
   {push @k, $i->key unless $i->key == 17;
   }

  $t->delete($_) for @k;                                                        # Delete

  ok $t->find(17) == 34 && $t->size == 1;                                       # Size

size($tree)

Count the number of keys in a tree

   Parameter  Description
1  $tree      Tree

Example:

local $Tree::Multi::numberOfKeysPerNode = 3;

my $t = new;      ok $t->height == 0; ok $t->leftMost->depth == 0; ok $t->size == 0;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(1, 1); ok $t->height == 1; ok $t->leftMost->depth == 1; ok $t->size == 1;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(2, 2); ok $t->height == 1; ok $t->leftMost->depth == 1; ok $t->size == 2;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(3, 3); ok $t->height == 1; ok $t->leftMost->depth == 1; ok $t->size == 3;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲


$t->insert(4, 4); ok $t->height == 2; ok $t->leftMost->depth == 2; ok $t->size == 4;  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

Tree::Multi Definition

Iterator

Output fields

count

Counter

data

Data at this position

key

Key at this position

keys

Array of key items for this node

less

Iteration not yet finished

node

Current node within tree

number

Number of the node for debugging purposes

pos

Current position within node

tree

Tree we are iterating over

up

Parent node

Private Methods

new()

Create a new multi-way tree node.

Example:

  local $Tree::Multi::numberOfKeysPerNode = 4;                                  # Number of keys per node - can be even


  my $t = Tree::Multi::new;                                                     # Construct tree  # 𝗘𝘅𝗮𝗺𝗽𝗹𝗲

     $t->insert($_, 2 * $_) for reverse 1..32;                                  # Load tree in reverse

  is_deeply $t->print, <<END;
 15 21 27
   3 6 9 12
     1 2
     4 5
     7 8
     10 11
     13 14
   18
     16 17
     19 20
   24
     22 23
     25 26
   30
     28 29
     31 32
END

  ok  $t->height     ==  3;                                                     # Height

  ok  $t->find  (16) == 32;                                                     # Find by key
      $t->delete(16);                                                           # Delete a key
  ok !$t->find  (16);                                                           # Key no longer present


  ok  $t->find  (17) == 34;                                                     # Find by key
  my @k;
  for(my $i = $t->iterator; $i->more; $i->next)                                 # Iterator
   {push @k, $i->key unless $i->key == 17;
   }

  $t->delete($_) for @k;                                                        # Delete

  ok $t->find(17) == 34 && $t->size == 1;                                       # Size

minimumNumberOfKeys()

Minimum number of keys per node.

maximumNumberOfKeys()

Maximum number of keys per node.

maximumNumberOfNodes()

Maximum number of children per parent.

full($tree)

Confirm that a node is full.

   Parameter  Description
1  $tree      Tree

halfFull($tree)

Confirm that a node is half full.

   Parameter  Description
1  $tree      Tree

separateKeys($node)

Return ([lower], center, [upper]) keys.

   Parameter  Description
1  $node      Node to split

separateData($node)

Return ([lower], center, [upper]) data

   Parameter  Description
1  $node      Node to split

separateNode($node)

Return ([lower], [upper]) children

   Parameter  Description
1  $node      Node to split

reUp($node, @children)

Reconnect the children to their new parent

   Parameter  Description
1  $node      Node
2  @children  Children

splitNode($node)

Split a full node in half assuming it has a non full parent

   Parameter  Description
1  $node      Node to split

splitRootNode($node)

Split a full root

   Parameter  Description
1  $node      Node to split

splitFullNode($node)

Split a full node

   Parameter  Description
1  $node      Node to split

splitLeafNode($node)

Split a full leaf node in assuming it has a non full parent

   Parameter  Description
1  $node      Node to split

splitRootLeafNode($node)

Split a full root that is also a leaf

   Parameter  Description
1  $node      Node to split

findAndSplit($root, $key)

Find a key in a tree splitting full nodes along the path to the key

   Parameter  Description
1  $root      Root of tree
2  $key       Key

indexInParent($tree)

Get the index of a node in its parent

   Parameter  Description
1  $tree      Tree

fillFromLeftOrRight($n, $dir)

Fill a node from the specified sibling

   Parameter  Description
1  $n         Node to fill
2  $dir       Node to fill from 0 for left or 1 for right

mergeWithLeftOrRight($n, $dir)

Merge two adjacent nodes

   Parameter  Description
1  $n         Node to merge into
2  $dir       Node to merge is on right if 1 else left

mergeOrFill($tree)

make a node larger than a half node

   Parameter  Description
1  $tree      Tree

deleteLeafKey($tree, $i)

Delete a (key, pair) in a leaf

   Parameter  Description
1  $tree      Tree
2  $i         Index to delete at

deleteKey($tree, $i)

Delete a (key, data) pair in a node

   Parameter  Description
1  $tree      Tree
2  $i         Index to delete at

flat($tree, @title)

Print the keys in a tree in a flat manner

   Parameter  Description
1  $tree      Tree
2  @title     Title

T($tree, $expected)

Write a result to the log file

   Parameter  Description
1  $tree      Tree
2  $expected  Expected print

disordered($n, $N)

Disordered but stable insertions

   Parameter  Description
1  $n         Keys per node
2  $N         Nodes

disorderedCheck($t, $n, $N)

Check disordered insertions

   Parameter  Description
1  $t         Tree to check
2  $n         Keys per node
3  $N         Nodes

randomCheck($n, $N, $T)

Random insertions

   Parameter  Description
1  $n         Keys per node
2  $N         Log 10 nodes
3  $T         Log 10 number of tests

Index

1 delete - Find a key in a tree, delete it, return the new tree

2 deleteKey - Delete a (key, data) pair in a node

3 deleteLeafKey - Delete a (key, pair) in a leaf

4 depth - Return the depth of a node within a tree

5 disordered - Disordered but stable insertions

6 disorderedCheck - Check disordered insertions

7 fillFromLeftOrRight - Fill a node from the specified sibling

8 find - Find a key in a tree returning its associated data or undef if the key does not exist

9 findAndSplit - Find a key in a tree splitting full nodes along the path to the key

10 flat - Print the keys in a tree in a flat manner

11 full - Confirm that a node is full.

12 halfFull - Confirm that a node is half full.

13 height - Return the height of the tree

14 indexInParent - Get the index of a node in its parent

15 insert - Insert a key and data into a tree

16 iterator - Make an iterator for a tree

17 leaf - Confirm that the tree is a leaf.

18 leftMost - Return the left most node below the specified one

19 maximumNumberOfKeys - Maximum number of keys per node.

20 maximumNumberOfNodes - Maximum number of children per parent.

21 mergeOrFill - make a node larger than a half node

22 mergeWithLeftOrRight - Merge two adjacent nodes

23 minimumNumberOfKeys - Minimum number of keys per node.

24 new - Create a new multi-way tree node.

25 print - Print the keys in a tree optionally marking the active key

26 randomCheck - Random insertions

27 reUp - Reconnect the children to their new parent

28 reverseIterator - Create a reverse iterator for a tree

29 rightMost - Return the right most node below the specified one

30 root - Return the root node of a tree.

31 separateData - Return ([lower], center, [upper]) data

32 separateKeys - Return ([lower], center, [upper]) keys.

33 separateNode - Return ([lower], [upper]) children

34 size - Count the number of keys in a tree

35 splitFullNode - Split a full node

36 splitLeafNode - Split a full leaf node in assuming it has a non full parent

37 splitNode - Split a full node in half assuming it has a non full parent

38 splitRootLeafNode - Split a full root that is also a leaf

39 splitRootNode - Split a full root

40 T - Write a result to the log file

41 Tree::Multi::Iterator::next - Find the next key

42 Tree::Multi::ReverseIterator::prev - Find the previous key

Installation

This module is written in 100% Pure Perl and, thus, it is easy to read, comprehend, use, modify and install via cpan:

sudo cpan install Tree::Multi

Author

philiprbrenan@gmail.com

http://www.appaapps.com

Copyright

This module is free software. It may be used, redistributed and/or modified under the same terms as Perl itself.

To install Tree::Multi, copy and paste the appropriate command in to your terminal.

cpanm

cpanm Tree::Multi

CPAN shell

perl -MCPAN -e shell
install Tree::Multi

For more information on module installation, please visit the detailed CPAN module installation guide.

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Name

Synopsis

Description

Multi-way Tree

root($tree)

leaf($tree)

find($root, $key)

leftMost($tree)

rightMost($tree)

height($tree)

depth($tree)

delete($root, $key)

insert($tree, $key, $data)

iterator($tree)

Tree::Multi::Iterator::next($iter)

reverseIterator($tree)

Tree::Multi::ReverseIterator::prev($iter)

print($tree, $i)

size($tree)

Tree::Multi Definition

Output fields

count

data

key

keys

less

more

node

number

pos

tree

up

Private Methods

new()

minimumNumberOfKeys()

maximumNumberOfKeys()

maximumNumberOfNodes()

full($tree)

halfFull($tree)

separateKeys($node)

separateData($node)

separateNode($node)

reUp($node, @children)

splitNode($node)

splitRootNode($node)

splitFullNode($node)

splitLeafNode($node)

splitRootLeafNode($node)

findAndSplit($root, $key)

indexInParent($tree)

fillFromLeftOrRight($n, $dir)

mergeWithLeftOrRight($n, $dir)

mergeOrFill($tree)

deleteLeafKey($tree, $i)

deleteKey($tree, $i)

flat($tree, @title)

T($tree, $expected)

disordered($n, $N)

disorderedCheck($t, $n, $N)

randomCheck($n, $N, $T)

Index

Installation

Author

Copyright

Module Install Instructions