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Inserting into a (2,4)-tree

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• Recall: properties of the (2,4)-tree
  • Important: properties of the (2,4)-tree

    Size property: Each internal node has at most 4 children nodes Depth (height) property: Let E be the set of internal nodes which only have external children nodes All external node of the nodes of E have the same height (depth)

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• Insert location
  • Inserting an new entry:

    A new entry is always inserted in a node that contains only external nodes In other words: A new entry is always inserted into a leaf node

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  • Example 1:

    Before insert(18): After insert(18):

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  • Example 2: we may need to move some elements inside a node !!!

    Before insert(7): After insert(7):

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• The simplest case of the insert operation
  • Simplest case of the insert operation:

    When an entry is inserted into a (leaf) node that contains ≤ 3 entries, the new entry is simply added in that node

    Note:

    We have seen 2 examples already (see above !!!)

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  • Teaching note:

    I will now do a "evolving" example: I start with an empty 2,4-tree I will keep inserting entries into the 2,4-tree and see how the 2,4-tree changes

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  • Examples: inserting into a 2,4-tree starting with an empty tree

    Comments:

    The first insert (insert(4)) will create a node The next 2 insert operations will simply add entries to this node These are what I call easy cases: when the (leaf) node has space to hold one entry, you just add the entry in the node.

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• Handling overflow in the root noode
  • Overflow in the root node:

    The root node has 3 entries and will overflow when another entry is inserted Example:

    Comment:

    A node with 4 entries does not exist. I drew a node with 4 entries to show you how the keys must be distributed !!!

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  • Handling an overflow situation:

    Let the keys and links in the overflow root node be as follows: (Notice that p1, p2, p3, p4 and p5 can be references to subtrees !!!) Create a new nodes and copy into this new node: p4 k4 p5 while leaving the entry k3 out: Create a new root node using the entry k3 and make the other 2 nodes the subtrees:

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  • This operation is called splitting.... (quite obvious)

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  • Note:

    The height of the 2,4-tree is increased by 1 when the root node is split !!! In fact, the only time that the height of of a 2,4-tree will increased is when the root node is split

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• Example: root node overflow and splitting the root node
  • Example:

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• Splitting a non-root node
  • We need to insert more entries to get to another node split:

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  • When we do an insert(5), we need to split the non-root node (3,4,6):

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  • Splitting a non-root node (pay special attention to the node references):

    Node p overflows and needs to be split: Create a new node and move (p4, e4, p5) into the new node while leaving the entry e3 out: Note: The subtrees p4 and p5 have a new parent node !!! Insert the left over entry e3 into the parent node along with the new node as it's right subtree:

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  • Summary:

    Insert(5): 2,4-tree after node split: Repeated warning: Notice that subtrees p4 and p5 have a new parent node !!!

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  • Comment:

    You have now learn everything you need to insert a new entry into a 2,4-tree I will explain cascaded node splitting next --- which is not a new technique.... You just repeat the node splitting at a high level in the tree. Here a summary on what's ahead: You will always insert a new entry into a leaf node If the leaf node has ≤ 2 entries, you insert the new entry and you are done Otherwise: Split the node If you splitted a node in the previous step, an entry would have been inserted into the parent node !!! If the parent node has ≤ 2 entries, you insert the new entry and you are done Otherwise: Split the parent node !!! And so on !!! (Can you smell recursion ???)

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• Let's fill up the tree some more to trigger a cascaded split...
  • This is like the calm before the storm...

  • Let's fill the the tree some more...

    These insertion are of the simple type:

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• We are preparing for a cascaded split by filling up the parent (= root) node
  • We node do insert(11), causing a node to split and fill up its parent node:

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  • The split operation: (we have seen this twice already...)

    Create a new node and split the entriess and links (while leaving the middle entry out) as follows: Insert the left over entry and its right subtree into the parent node: (Notice that "insert an entry" into a node requires that the entries in the array remains sorted !!!)

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  • Notice that:

    We added one new node The new node is the right child of the key 10 that was moved upward (into the parent node) The 2,4 - tree has a full internal node (the root node with the 3 entries (5, 10, 12)) and it is ripe for a cascaded split !!!

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• Cascaded split: when the parent node overflows...
  • OK, now the fire works...

    Cascading splits/insertions:

    An insert operation that causes an overflow not only split the overflow node, but will also: Perform an insert operation in the parent node of the overflow node The insert operation in the parent node can again cause an overflow in the parent node This will cause: The parent node to split Perform an insert operation in the parent's parent node of the overflow node And so on...

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  • Note:

    The process will eventually stop: When a node that receives a new entry has ≤ 2 entries Or if all parent nodes towards the root nodes are all filled (= 3 entries), the split process will reach the (full) root node and causes the root node to split Recall that when the root node splits, we create a new root node: The new root node will only contain one key... and the height of the 2,4-tree is increased by 1 !!!

  • Repeated fact:

    The height of a 2,4-tree is increased (by 1) only when the root node is split !!!

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  • Example: cascaded splits/insertions

    insert(17) causes a node to overflow: Perform a split operation: which causes entry 15 to be inserted in the parent node: ....and the parent node overflows !!!

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  • We apply the same split technique on the parent node:

    Let's start with the overflow: Perform a split operation on the overflow node: which now causes entry 12 to be inserted in the parent node. Because there is no parent node (the overflow node is the root node !!!), we create a new root node: ....and we are done !!!