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    • B-Tree Insertion Guide
        +
      • Pre-emptive Splitting
      • Case 1a
      • Case 1b
      • Case 2a
      • @@ -65,6 +66,7 @@
      • B-Tree Deletion Guide -
        -

        B-Tree Insertion

        - -
        -

        What is Preemptive Splitting?

        -
          -
        • Preemptive splitting is a type of insertion method that we use for inserting into B-trees. This method ensures that insertions within a B-Tree can be done within a single traversal to the insertion point.
        • -
        • To preemptively split a B-Tree, while traversing the tree to the insertion point, we check every single node that we traverse to see if it needs to be preemptively split. This is done regardless of whether or not the insertion occurs at that node.
        • -
        • To determine if a node needs to be split, we look at the number of keys within that node. If the number of keys within that node is equal to the maximum number of allowable keys then it needs to undergo splitting.
        • -
        • To perform the split, take the median key of that node and place it into its parent node, then update the parent’s node child pointers to point to the left half of the split node and the right half of the split node in the appropriate locations.
        • -
        + +
        + +

        Insertion Guide:

        + + +
        +

        What is Preemptive Splitting?

        +
          +
        • Preemptive splitting is a type of insertion method that we use for inserting into B-trees. This method ensures that insertions within a B-Tree can be done within a single traversal to the insertion point.
        • +
        • To preemptively split a B-Tree, while traversing the tree to the insertion point, we check every single node that we traverse to see if it needs to be preemptively split. This is done regardless of whether or not the insertion occurs at that node.
        • +
        • To determine if a node needs to be split, we look at the number of keys within that node. If the number of keys within that node is equal to the maximum number of allowable keys then it needs to undergo splitting.
        • +
        • To perform the split, take the median key of that node and place it into its parent node, then update the parent’s node child pointers to point to the left half of the split node and the right half of the split node in the appropriate locations.
        • +
        +
        +
          +
        • To demonstrate this: we will look at an example:
        • +
        • Example: Insert 10
        • +
          + Image 2 + +
          +
        • On insertion we traverse the tree to the correct location.
        • +
          + Image 2 + +
          +
        • Every node that we traverse, we check if that node has the maximum amount of keys in it
        • +
        • If a node has the maximum amount of keys in it, during traversal then we split it.
        • +
        • We can see that during traversal in this example, we need to split the node with keys [13,20,28]
        • +
          + Image 2 + +
          +
        • To perform the split, take the median key of that node and place it into its parent node, then update the parent’s node child pointers to point to the left half of the split node and the right half of the split node in the appropriate locations.
        • +
          + Image 2 + Image 2 + +
          +
        • When we have successfully split the node, we can now carry on the insertion as normal by following the cases that are described in the user guide for insertion.
        • +
          + + + Image 2 + Image 2 + Image 2 + Image 2 +
          +
        +
        + +
        +
          +
        • Please also note that pre-emptive splitting also occurs at the root node.
        • +
        • This is depicted in the example below:
        • +
        • Example: Insert 10:
        • +
          + + + Image 2 + Image 2 + Image 2 + Image 2 +
          +
          + + + Image 2 + Image 2 + Image 2 + Image 2 +
          +
        +
        + + +
        - + +
        +

        Inserting into a B-Tree

        + +
        +

        Cases for Insertion:

        +
          +
        1. Case 1: Inserting a key into a leaf node where the leaf node has less than the maximum amount of keys ie. (2t-1) keys. +
            +
          • 1.a) Without preemptive splitting
          • +
              +
            • Traverse the tree to the correct leaf node.
            • +
            • Simply insert the key into the correct position within the leaf node.
            • +
            • Example: Insert 7
            • +
              + Image 1 + Image 2 + Image 3 + Image 3 + +
              +
            +
          • 1.b) With pre-emptive splitting
          • +
              +
            • Traverse the tree to the correct leaf node.
            • +
            • If any of the nodes along the traversal that are not the leaf node have the maximum amount of keys, then that specific node needs pre-emptive splitting.
            • +
            • Split that node by taking the median of that node and inserting it into it's parent node.
            • +
            • The left half of the split node and the right half of the split node become the new children of the parent node.
            • +
            • Update the children to correspond to the correct node.
            • +
            • Example: Insert 49
            • +
              + Image 1 + Image 2 + Image 3 + Image 3 + +
              + +
            • Now complete the traversal to the correct leaf node.
            • +
            • Simply insert the key into the correct position within the leaf node.
            • + +
              + Image 1 + Image 2 + Image 3 + Image 3 + +
              +
            +
          +
        2. +
        3. Case 2: Inserting a key into a leaf node where the leaf node has the maximum amount of keys ie. (2t-1) keys. +
            +
          • 2.a) Without preemptive splitting
          • +
              +
            • Traverse the tree to the correct leaf node.
            • +
            • Example: Insert 69
            • +
              + Image 1 + Image 2 + Image 3 + + +
              +
            • If the leaf node where we want to insert the new key, has the maximum amount of keys, then split that leaf node.
            • +
              + Image 3 + +
              +
            • Split the leaf node by taking the median key of that node and moving it to the correct position in its parent node.
            • +
            • The left half of the split node and the right half of the split node become the new children of the parent node.
            • +
            • Update the children to correspond to the correct node.
            • +
              + Image 1 + + +
              + + +
            • Now that the leaf node is appropriately split:
            • +
            • Insert the key into the leaf node that corresponds to the correct traversal of the tree.
            • +
              + Image 2 + Image 3 + Image 3 + +
              + +
            +
          • 2.b) With preemptive splitting
          • +
              +
            • Traverse the tree to the correct leaf node.
            • +
            • If any of the nodes along the traversal that are not the leaf node have the maximum amount of keys, then that specific node needs pre-emptive splitting.
            • +
            • Example: Insert 18
            • +
              + Image 1 + Image 2 + Image 3 + + +
              +
            • Split that node by taking the median of that node and inserting it into it's parent node.
            • +
            • The left half of the split node and the right half of the split node become the new children of the parent node.
            • +
            • Update the children to correspond to the correct node.
            • +
              + Image 3 + +
              + +
            • Now complete the traversal to the correct leaf node.
            • + +
              + Image 1 + Image 2 + Image 3 + + +
              + + +
            • If the leaf node where we want to insert the new key, has the maximum amount of keys, then split that leaf node.
            • + +
              + Image 3 + + + +
              + +
            • Split the leaf node by taking the median key of that node and moving it to the correct position in its parent node.
            • +
            • The left half of the split node and the right half of the split node become the new children of the parent node.
            • +
            • Update the children to correspond to the correct node.
            • + +
              + Image 1 + +
              + +
            • Now that the leaf node is appropriately split:
            • +
            • Insert the key into the leaf node that corresponds to the correct traversal of the tree.
            • + +
              + Image 2 + Image 3 + Image 3 + Image 3 + +
              +
            +
          +
        4. + +
        + +
        + +
        -

        Inserting into a B-Tree with Preemptive Splitting

        -

        Let's go through the steps of inserting a new key into a B-tree with preemptive splitting:

        -
          -
        1. Start at the Root Node: Begin the insertion process at the root node of the B-tree.
        2. -
        3. Set the root node to be the node that we are currently inspecting. Let’s call this “Current Node”.
        4. -
        5. Check the Current Node for preemptive splitting: -
            -
          • If the current node has within it a number of keys that are equal to (2*t-1), then split the node.
          • -
          • This is done by taking the median of the current node and moving the median key to the appropriate position within the parent node of the current node. If the current node is the root node, perform splitting by making the median key of the root node the new root node.
          • -
          • Update the connections within the relationship between the parent node and current node to properly represent the correct child relationships.
          • -
          -
        6. - -
        7. Insert the Key: Once you reach the appropriate leaf node, insert the new key into the node. -
            -
          • If the node is not full (has fewer than 2*t-1 keys), insert the key directly.
          • -
          • If the node is full then split the leaf node by moving the median key of the leaf node to its parent node. Now insert the key into the newly created child nodes where appropriate.
          • -
          -
        8. -
        9. Finished: Your B-tree is now updated with the new key, and it remains balanced.
        10. -
        11. Case 1: Inserting a key into a leaf node where the leaf node has less than the maximum amount of keys ie. (2t-1) keys. +
          +

          Deletion Guide:

          +
          +

          B-Tree Properties

          +

          Before we delve into B-tree insertions with preemptive splitting, let's ensure we have a solid understanding of what a B-tree is. A B-tree is a self-balancing search tree data structure that maintains sorted data and enables efficient insertion, deletion, and search operations. It's widely used in database systems and file systems because of its balanced structure and efficient performance characteristics.

          +

          A B-tree of degree t is defined by the following properties:

            -
          • Without preemptive splitting
          • -
          • With preemptive splitting
          • +
          • Every node has at most 2*t-1 keys. Ex: A tree of degree 2 has a maximum of (2*2)-1 = 3 keys for each node.
          • +
          • Every node (except the root) has at least t-1 keys. Ex: A tree of degree 2 has a minimum of (2)-1 = 1 key for each node.
          • +
          • The root has at least 1 key.
          • +
          • All keys within a node in the tree must be sorted in increasing order.
          • +
          • If a node has n keys then that node must have n+1 children.
          • +
          • All leaves have the same depth, ensuring balanced search operations.
          -
        12. -
        13. Case 2: Inserting a key into a leaf node where the leaf node has the maximum amount of keys ie. (2t-1) keys. +
        + +
        +

        Introduction to B-Tree Deletion

          -
        • Without preemptive splitting
        • -
        • With preemptive splitting
        • +
        • B-Tree deletion is more complicated than insertion because operations not only occur at the leaf nodes but we can also delete from the internal nodes.
        • +
        • As with insertion, we must perform the operation where we do not violate the properties of a B-Tree.
        • +
        • Just as with insertions we are going to use a method called preemptive merging to ensure that B-Tree deletions can be performed with a single traversal of the tree.
        -
      • -
      • Note: Generally, there should be 1 split per insert.
      • -
- + + + + +
+ + +

What is Preemptive Merging?

+
    +
  • Preemptive merging, similar to preemptive splitting, is a type of deletion method that we use for deleting from B-trees. This method ensures that deletions from a B-Tree can be done within a single traversal to the deletion point on every occurence of the operation.
  • +
  • To preemptively merge a B-Tree, while traversing the tree to the deletion point, we check every single node that we traverse, besides the root node, to see if it needs to be preemptively merged. This is done regardless of whether or not the deletion occurs at that node.
  • +
  • To determine if a node needs to be merged, we look at the number of keys within that node. If the number of keys within that node is equal to the minimum number of allowable keys then it needs to undergo merging.
  • +
  • Let’s call the node that needs to undergo merging the “merging node”.
  • +
  • To perform the merge there are two cases that we must account for:
  • +
+
+

Case 1: The merging node has an immediate sibling that has more than the minimum amount of keys.

+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Delete 88
  • +
    + Image 2 + +
    +
  • On deletion we traverse the tree to the correct location.
  • + +
  • Every node that we traverse, we check if that node has the minimum amount of keys in it, (excluding the root node)
  • +
  • If a node has the minimum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to merge the node with key [83]
  • +
    + Image 2 + +
    +
  • To perform the merge, borrow the predecessor of the immediate sibling node that shares the same shared parent key.
  • +
  • Take this key and move it into the parent node.
  • +
  • Now move the shared parent key into the node that we are pre-merging
  • +
  • Update the children of all nodes accordingly
  • +
    + Image 2 + +
    +
  • When we have successfully merged the node, we can now carry on the deletion as normal by following the cases that are described in the user guide for deletion.
  • +
    + Image 2 +
    +
+
+ +
+

Case 2: The merging node has no immediate siblings that has more than the minimum amount of keys.

+
+

Case2.a) The merging node has no immediate siblings that has more than the minimum amount of keys and the parent node has the minimum amount of keys.

+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Delete 51
  • +
    + Image 2 + +
    +
  • On deletion we traverse the tree to the correct location.
  • + +
  • Every node that we traverse, we check if that node has the minimum amount of keys in it, (excluding the root node).
  • +
  • If a node has the minimum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to merge the node with key [47]
  • +
    + Image 2 + +
    +
  • We look to the merging node's immediate siblings.
  • +
  • We see that all immediate siblings have the minimum number of keys in them
  • +
    + Image 2 + +
    + +
  • To perform the merge, move the keys from both the merging node and the immediate sibling into their parent node
  • +
  • Update the children of all nodes accordingly
  • +
  • Note that the height of the tree shrinks in this case.
  • +
    + Image 2 + +
    +
  • When we have successfully merged the node, we can now carry on the deletion as normal by following the cases that are described in the user guide for deletion.
  • +
    + Image 2 +
    +
+
+ +
+

Case2.b) The merging node has no immediate siblings that has more than the minimum amount of keys and the parent node has more than the minimum amount of keys.

+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Delete 4
  • +
    + Image 2 + +
    +
  • On deletion we traverse the tree to the correct location.
  • + +
  • Every node that we traverse, we check if that node has the minimum amount of keys in it, (excluding the root node).
  • +
  • If a node has the minimum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to merge the node with key [23]
  • +
    + Image 2 + +
    +
  • We look to the merging node's immediate siblings.
  • +
  • We see that all immediate siblings have the minimum number of keys in them
  • +
  • We also look to the parent node of the merging node and see that it has more than the minimum amount of keys in it.
  • + + +
  • To perform the merge, move the shared parent key of both the merging node and the immediate sibling node, and use this as the new median of the merged merging node and immediate sibling node.
  • +
  • Update the children of all nodes accordingly
  • +
    + Image 2 + +
    +
  • When we have successfully merged the node, we can now carry on the deletion as normal by following the cases that are described in the user guide for deletion.
  • +
    + Image 2 + Image 2 + Image 2 +
    +
+
+ +
+ + + +
+ + +
+

Deleting from a B-Tree

+ +
+

Cases for Deletion:

+
    +
  1. Case 1: Deleting a key at the leaf node where the leaf node has more than the minimum amount of keys in it, ie more than (t-1) keys +
      +
        +
      • Locate the key to delete.
      • +
      • Example: Delete 24
      • +
        + Image 1 +
        + +
      • If the key to delete lies in a leaf node and the leaf node contains more than the minimum amount of keys, ie. more than (t-1) keys:
      • +
      • Simply delete the key from the leaf node.
      • + +
        + + Image 2 + Image 3 +
        +
      +
    +
  2. + +
  3. Case 2: Deleting a key at the leaf node where the leaf node has the minimum amount of keys in it, ie (t-1) keys +
      +
    • 2.a) Deleting a key at the leaf node where the leaf node has the minimum amount of keys in it, ie (t-1) keys and The leaf node has a left immediate sibling node with more than the minimum amount of keys ie. more than (t-1) keys
    • +
        +
      • Example: Delete 86
      • + +
      • Locate the key to delete.
      • +
      • If the key to delete lies in a leaf node and the leaf node contains the minimum amount of keys, ie. (t-1) keys:
      • + +
        + Image 1 +
        + +
      • Look to the immediate left sibling node of the shared parent key
      • + +
        + Image 1 +
        + + +
      • If the immediate left sibling node of the shared parent key has more than (t-1) keys:
      • +
      • Borrow the shared parent key's predecessor by moving it into the parent node and moving the shared parent key into the node we want to delete from
      • + +
        + Image 1 +
        + +
      • The leaf node now has more than the minimum amount of keys so:
      • +
      • We simply delete the key from the leaf node.
      • + +
        + Image 1 + Image 1 +
        + +
      + +
    • 2.b) Deleting a key at the leaf node where the leaf node has the minimum amount of keys in it, ie (t-1) keys and The leaf node’s left immediate sibling node has the minimum amount of keys ie (t-1) keys, but the leaf node’s right immediate sibling node has more than the minimum amount of keys ie. more than (t-1) keys
    • +
        +
      • Example: Delete 47
      • + +
      • Locate the key to delete.
      • +
      • If the key to delete lies in a leaf node and the leaf node contains the minimum amount of keys, ie. (t-1) keys:
      • + +
        + Image 1 +
        + +
      • First look to the immediate left sibling node of the shared parent key
      • +
      • If the immediate left sibling node of the shared parent key also has the minimum amount of keys, ie (t-1) keys:
      • + +
        + Image 1 +
        + +
      • Look to the immediate right sibling node of the shared parent key
      • + +
        + Image 1 +
        + +
      • If the immediate right sibling node of the shared parent key has more than the minimum amount of keys, ie. (t-1) keys:
      • +
      • Borrow the shared parent key's successor by moving it into the parent node and moving the shared parent key into the node we want to delete from
      • + +
        + Image 1 +
        + +
      • The leaf node now has more than the minimum amount of keys so:
      • +
      • We simply delete the key from the leaf node.
      • + +
        + Image 1 + Image 1 + Image 1 + Image 1 +
        + +
      + +
    • 2.c) Deleting a key at the leaf node where the leaf node has the minimum amount of keys in it, ie (t-1) keys and All immediate siblings have the minimum amount of keys ie. (t-1) keys
    • +
        +
      • Example: Delete 40
      • + +
      • Locate the key to delete.
      • +
      • If the key to delete lies in a leaf node and the leaf node contains the minimum amount of keys, ie. (t-1) keys:
      • + +
        + Image 1 +
        + +
      • First look to the immediate left sibling node of the shared parent key
      • +
      • If the immediate left sibling node of the shared parent key also has the minimum amount of keys, ie (t-1) keys:
      • + +
        + Image 1 +
        + +
      • Look to the immediate right sibling node of the shared parent key
      • +
      • If the immediate right sibling node of the shared parent key also has the minimum amount of keys, ie (t-1) keys:
      • + +
        + Image 1 +
        + +
      • Merge the key to delete within either the immediate left sibling node or the immediate right sibling node by merging the sibling node and the shared parent key to the node to delete from.
      • +
      • Please note that you can merge with either sibling node and in this example we are merging with the left sibling node. (Merging with the right will also be fine)
      • + +
        + Image 1 +
        + +
      • The leaf node now has more than the minimum amount of keys so:
      • +
      • We simply delete the key from the leaf node.
      • + +
        + Image 1 + Image 1 + +
        + +
      + + + +
    +
  4. + +
  5. Case 3: Deleting a Key at an Internal Node +
      +
    • 3.a) Deleting a key at an internal node where the left child node has the more than the minimum amount of keys ie. more than (t-1) keys
    • +
        +
      • Example: Delete 49
      • + +
      • Locate the key to delete.
      • +
      • If the key to delete lies in an internal node:
      • + +
        + Image 1 +
        + +
      • Look to the left child node of the key to delete
      • + +
        + Image 1 +
        + + +
      • If the left child node of the key to delete has more than (t-1) keys:
      • +
      • Borrow the key-to-delete's predecessor by moving it into the same node as the node where the key-to-delete is
      • + +
        + Image 1 +
        + +
      • The internal node now has more than the minimum amount of keys so:
      • +
      • We simply delete the key from the internal node.
      • + +
        + Image 1 + Image 1 +
        + +
      + +
    • 3.b) Deleting a key at an internal node where the left child node has the minimum amount of keys ie. (t-1) keys and the right child node has more than the minimum amount of keys ie. more than (t-1) keys
    • +
        +
      • Example: Delete 72
      • + +
      • Locate the key to delete.
      • +
      • If the key to delete lies in an internal node:
      • + +
        + Image 1 +
        + +
      • Look to the left child node of the key to delete
      • + +
        + Image 1 +
        + +
      • If the left child node of the key-to-delete has the minimum amount of keys, ie (t-1) keys:
      • +
      • Look to the right child node of the key-to-delete
      • + +
        + Image 1 +
        + +
      • If the right child node of the key-to-delete has more than the minimum amount of keys, ie more than (t-1) keys:
      • +
      • Borrow the key-to-delete's successor by moving it into the same node as the node where the key-to-delete is
      • + +
        + Image 1 +
        + +
      • The internal node now has more than the minimum amount of keys so:
      • +
      • We simply delete the key from the internal node.
      • + +
        + Image 1 + Image 1 +
        + +
      + +
    • 3.c) Deleting a key at an internal node where both children have the minimum amount of keys in them, ie (t-1) keys
    • +
        +
      • Example: Delete 72
      • + +
      • Locate the key to delete.
      • +
      • If the key to delete lies in an internal node:
      • + +
        + Image 1 +
        + +
      • Look to the left child node of the key to delete
      • + +
        + Image 1 +
        + +
      • If the left child node of the key-to-delete has the minimum amount of keys, ie (t-1) keys:
      • +
      • Look to the right child node of the key-to-delete
      • + +
        + Image 1 +
        + +
      • If the right child node of the key-to-delete has the minimum amount of keys, ie (t-1) keys:
      • +
      • Merge the left child node and the right child node together
      • + +
        + Image 1 +
        + +
      • The internal node now has the same amount of keys as it has children so:
      • +
      • We simply delete the key from the internal node.
      • + +
        + Image 1 + Image 1 +
        + +
      + + + +
    +
  6. + +
+ +
+ +
+ + + @@ -989,13 +1623,79 @@

B-Tree Properties

-

What is Preemptive Splitting?

-
    -
  • Preemptive splitting is a type of insertion method that we use for inserting into B-trees. This method ensures that insertions within a B-Tree can be done within a single traversal to the insertion point.
  • -
  • To preemptively split a B-Tree, while traversing the tree to the insertion point, we check every single node that we traverse to see if it needs to be preemptively split. This is done regardless of whether or not the insertion occurs at that node.
  • -
  • To determine if a node needs to be split, we look at the number of keys within that node. If the number of keys within that node is equal to the maximum number of allowable keys then it needs to undergo splitting.
  • -
  • To perform the split, take the median key of that node and place it into its parent node, then update the parent’s node child pointers to point to the left half of the split node and the right half of the split node in the appropriate locations.
  • -
+
+ +

What is Preemptive Splitting?

+
    +
  • Preemptive splitting is a type of insertion method that we use for inserting into B-trees. This method ensures that insertions within a B-Tree can be done within a single traversal to the insertion point.
  • +
  • To preemptively split a B-Tree, while traversing the tree to the insertion point, we check every single node that we traverse to see if it needs to be preemptively split. This is done regardless of whether or not the insertion occurs at that node.
  • +
  • To determine if a node needs to be split, we look at the number of keys within that node. If the number of keys within that node is equal to the maximum number of allowable keys then it needs to undergo splitting.
  • +
  • To perform the split, take the median key of that node and place it into its parent node, then update the parent’s node child pointers to point to the left half of the split node and the right half of the split node in the appropriate locations.
  • +
+
+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Insert 10
  • +
    + Image 2 + +
    +
  • On insertion we traverse the tree to the correct location.
  • +
    + Image 2 + +
    +
  • Every node that we traverse, we check if that node has the maximum amount of keys in it
  • +
  • If a node has the maximum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to split the node with keys [13,20,28]
  • +
    + Image 2 + +
    +
  • To perform the split, take the median key of that node and place it into its parent node, then update the parent’s node child pointers to point to the left half of the split node and the right half of the split node in the appropriate locations.
  • +
    + Image 2 + Image 2 + +
    +
  • When we have successfully split the node, we can now carry on the insertion as normal by following the cases that are described in the user guide for insertion.
  • +
    + + + Image 2 + Image 2 + Image 2 + Image 2 +
    +
+
+ +
+
    +
  • Please also note that pre-emptive splitting also occurs at the root node.
  • +
  • This is depicted in the example below:
  • +
  • Example: Insert 10:
  • +
    + + + Image 2 + Image 2 + Image 2 + Image 2 +
    +
    + + + Image 2 + Image 2 + Image 2 + Image 2 +
    +
+
+ + +
@@ -1158,6 +1858,91 @@

Cases for Insertion:

+ + + + + + + + + +
+
+ +

What is Preemptive Splitting?

+
    +
  • Preemptive splitting is a type of insertion method that we use for inserting into B-trees. This method ensures that insertions within a B-Tree can be done within a single traversal to the insertion point.
  • +
  • To preemptively split a B-Tree, while traversing the tree to the insertion point, we check every single node that we traverse to see if it needs to be preemptively split. This is done regardless of whether or not the insertion occurs at that node.
  • +
  • To determine if a node needs to be split, we look at the number of keys within that node. If the number of keys within that node is equal to the maximum number of allowable keys then it needs to undergo splitting.
  • +
  • To perform the split, take the median key of that node and place it into its parent node, then update the parent’s node child pointers to point to the left half of the split node and the right half of the split node in the appropriate locations.
  • +
+
+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Insert 10
  • +
    + Image 2 + +
    +
  • On insertion we traverse the tree to the correct location.
  • +
    + Image 2 + +
    +
  • Every node that we traverse, we check if that node has the maximum amount of keys in it
  • +
  • If a node has the maximum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to split the node with keys [13,20,28]
  • +
    + Image 2 + +
    +
  • To perform the split, take the median key of that node and place it into its parent node, then update the parent’s node child pointers to point to the left half of the split node and the right half of the split node in the appropriate locations.
  • +
    + Image 2 + Image 2 + +
    +
  • When we have successfully split the node, we can now carry on the insertion as normal by following the cases that are described in the user guide for insertion.
  • +
    + + + Image 2 + Image 2 + Image 2 + Image 2 +
    +
+
+ +
+
    +
  • Please also note that pre-emptive splitting also occurs at the root node.
  • +
  • This is depicted in the example below:
  • +
  • Example: Insert 10:
  • +
    + + + Image 2 + Image 2 + Image 2 + Image 2 +
    +
    + + + Image 2 + Image 2 + Image 2 + Image 2 +
    +
+
+ + +
+
+ @@ -1397,6 +2182,137 @@

Introduction to B-Tree Deletion

+ +
+ + +

What is Preemptive Merging?

+
    +
  • Preemptive merging, similar to preemptive splitting, is a type of deletion method that we use for deleting from B-trees. This method ensures that deletions from a B-Tree can be done within a single traversal to the deletion point on every occurence of the operation.
  • +
  • To preemptively merge a B-Tree, while traversing the tree to the deletion point, we check every single node that we traverse, besides the root node, to see if it needs to be preemptively merged. This is done regardless of whether or not the deletion occurs at that node.
  • +
  • To determine if a node needs to be merged, we look at the number of keys within that node. If the number of keys within that node is equal to the minimum number of allowable keys then it needs to undergo merging.
  • +
  • Let’s call the node that needs to undergo merging the “merging node”.
  • +
  • To perform the merge there are two cases that we must account for:
  • +
+
+

Case 1: The merging node has an immediate sibling that has more than the minimum amount of keys.

+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Delete 88
  • +
    + Image 2 + +
    +
  • On deletion we traverse the tree to the correct location.
  • + +
  • Every node that we traverse, we check if that node has the minimum amount of keys in it, (excluding the root node)
  • +
  • If a node has the minimum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to merge the node with key [83]
  • +
    + Image 2 + +
    +
  • To perform the merge, borrow the predecessor of the immediate sibling node that shares the same shared parent key.
  • +
  • Take this key and move it into the parent node.
  • +
  • Now move the shared parent key into the node that we are pre-merging
  • +
  • Update the children of all nodes accordingly
  • +
    + Image 2 + +
    +
  • When we have successfully merged the node, we can now carry on the deletion as normal by following the cases that are described in the user guide for deletion.
  • +
    + Image 2 +
    +
+
+ +
+

Case 2: The merging node has no immediate siblings that has more than the minimum amount of keys.

+
+

Case2.a) The merging node has no immediate siblings that has more than the minimum amount of keys and the parent node has the minimum amount of keys.

+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Delete 51
  • +
    + Image 2 + +
    +
  • On deletion we traverse the tree to the correct location.
  • + +
  • Every node that we traverse, we check if that node has the minimum amount of keys in it, (excluding the root node).
  • +
  • If a node has the minimum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to merge the node with key [47]
  • +
    + Image 2 + +
    +
  • We look to the merging node's immediate siblings.
  • +
  • We see that all immediate siblings have the minimum number of keys in them
  • +
    + Image 2 + +
    + +
  • To perform the merge, move the keys from both the merging node and the immediate sibling into their parent node
  • +
  • Update the children of all nodes accordingly
  • +
  • Note that the height of the tree shrinks in this case.
  • +
    + Image 2 + +
    +
  • When we have successfully merged the node, we can now carry on the deletion as normal by following the cases that are described in the user guide for deletion.
  • +
    + Image 2 +
    +
+
+ +
+

Case2.b) The merging node has no immediate siblings that has more than the minimum amount of keys and the parent node has more than the minimum amount of keys.

+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Delete 4
  • +
    + Image 2 + +
    +
  • On deletion we traverse the tree to the correct location.
  • + +
  • Every node that we traverse, we check if that node has the minimum amount of keys in it, (excluding the root node).
  • +
  • If a node has the minimum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to merge the node with key [23]
  • +
    + Image 2 + +
    +
  • We look to the merging node's immediate siblings.
  • +
  • We see that all immediate siblings have the minimum number of keys in them
  • +
  • We also look to the parent node of the merging node and see that it has more than the minimum amount of keys in it.
  • + + +
  • To perform the merge, move the shared parent key of both the merging node and the immediate sibling node, and use this as the new median of the merged merging node and immediate sibling node.
  • +
  • Update the children of all nodes accordingly
  • +
    + Image 2 + +
    +
  • When we have successfully merged the node, we can now carry on the deletion as normal by following the cases that are described in the user guide for deletion.
  • +
    + Image 2 + Image 2 + Image 2 +
    +
+
+ +
+ + + +
+ +

Deleting from a B-Tree

@@ -1682,6 +2598,295 @@

Cases for Deletion:

+ + + + + + + + + + +
+
+ +

What is Preemptive Merging?

+
    +
  • Preemptive merging, similar to preemptive splitting, is a type of deletion method that we use for deleting from B-trees. This method ensures that deletions from a B-Tree can be done within a single traversal to the deletion point on every occurence of the operation.
  • +
  • To preemptively merge a B-Tree, while traversing the tree to the deletion point, we check every single node that we traverse, besides the root node, to see if it needs to be preemptively merged. This is done regardless of whether or not the deletion occurs at that node.
  • +
  • To determine if a node needs to be merged, we look at the number of keys within that node. If the number of keys within that node is equal to the minimum number of allowable keys then it needs to undergo merging.
  • +
  • Let’s call the node that needs to undergo merging the “merging node”.
  • +
  • To perform the merge there are two cases that we must account for:
  • +
+
+

Case 1: The merging node has an immediate sibling that has more than the minimum amount of keys.

+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Delete 88
  • +
    + Image 2 + +
    +
  • On deletion we traverse the tree to the correct location.
  • + +
  • Every node that we traverse, we check if that node has the minimum amount of keys in it, (excluding the root node)
  • +
  • If a node has the minimum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to merge the node with key [83]
  • +
    + Image 2 + +
    +
  • To perform the merge, borrow the predecessor of the immediate sibling node that shares the same shared parent key.
  • +
  • Take this key and move it into the parent node.
  • +
  • Now move the shared parent key into the node that we are pre-merging
  • +
  • Update the children of all nodes accordingly
  • +
    + Image 2 + +
    +
  • When we have successfully merged the node, we can now carry on the deletion as normal by following the cases that are described in the user guide for deletion.
  • +
    + Image 2 +
    +
+
+ +
+

Case 2: The merging node has no immediate siblings that has more than the minimum amount of keys.

+
+

Case2.a) The merging node has no immediate siblings that has more than the minimum amount of keys and the parent node has the minimum amount of keys.

+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Delete 51
  • +
    + Image 2 + +
    +
  • On deletion we traverse the tree to the correct location.
  • + +
  • Every node that we traverse, we check if that node has the minimum amount of keys in it, (excluding the root node).
  • +
  • If a node has the minimum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to merge the node with key [47]
  • +
    + Image 2 + +
    +
  • We look to the merging node's immediate siblings.
  • +
  • We see that all immediate siblings have the minimum number of keys in them
  • +
    + Image 2 + +
    + +
  • To perform the merge, move the keys from both the merging node and the immediate sibling into their parent node
  • +
  • Update the children of all nodes accordingly
  • +
  • Note that the height of the tree shrinks in this case.
  • +
    + Image 2 + +
    +
  • When we have successfully merged the node, we can now carry on the deletion as normal by following the cases that are described in the user guide for deletion.
  • +
    + Image 2 +
    +
+
+ +
+

Case2.b) The merging node has no immediate siblings that has more than the minimum amount of keys and the parent node has more than the minimum amount of keys.

+
    +
  • To demonstrate this: we will look at an example:
  • +
  • Example: Delete 4
  • +
    + Image 2 + +
    +
  • On deletion we traverse the tree to the correct location.
  • + +
  • Every node that we traverse, we check if that node has the minimum amount of keys in it, (excluding the root node).
  • +
  • If a node has the minimum amount of keys in it, during traversal then we split it.
  • +
  • We can see that during traversal in this example, we need to merge the node with key [23]
  • +
    + Image 2 + +
    +
  • We look to the merging node's immediate siblings.
  • +
  • We see that all immediate siblings have the minimum number of keys in them
  • +
  • We also look to the parent node of the merging node and see that it has more than the minimum amount of keys in it.
  • + + +
  • To perform the merge, move the shared parent key of both the merging node and the immediate sibling node, and use this as the new median of the merged merging node and immediate sibling node.
  • +
  • Update the children of all nodes accordingly
  • +
    + Image 2 + +
    +
  • When we have successfully merged the node, we can now carry on the deletion as normal by following the cases that are described in the user guide for deletion.
  • +
    + Image 2 + Image 2 + Image 2 +
    +
+
+ +
+ + +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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