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;
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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
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       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
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       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
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       7 8
       10 11
       13 14
     18
       16 17
       19 20
     24
       22 23
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     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);
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  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);
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  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
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       1 2
       4 5
       7 8
       10 11
       13 14
     18
       16 17
       19 20
     24
       22 23
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     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
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       13 14
     18
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  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
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       1 2
       4 5
       7 8
       10 11
       13 14
     18
       16 17
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     24
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     30
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       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

more

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

Copyright (c) 2016-2021 Philip R Brenan.

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