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Solving underflow with merging

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• Underflow that cannot be solved by transfer
  • Fact:

    If both immediate sibling nodes of an underflow (has less than 1 subtree after deletion) node have only one entry in them, you cannot use the transfer technique to resolve the underflow condition

    Example:

    Transfering an entry form the sibling node will cause an underflow at the sibling node itself !

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• Solving underflow by merging ---- when both siblings have 2 (= minimum number) subtrees
  • Technique:

    If an underflow condition occurs in a delete operation, and BOTH immediate sibling nodes have 2 subtrees, we must solve the underflow condition by: A 3-way merge operation using All the subtrees and entries from the sibling node + The (1 remaining) subtree in underflow node + the parent key entry (to provide search direction !!!)

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• The left merge operation
  • Merging with your left sibling:

    What happens in a merge operation with a left sibling: The merge operation with a left sibling: Explanation: You move the sandwiched entry from the parent node into the left sibling to: Provide correct search direction for the subtree in the underflow node !!! The right child reference (in magenta color) is deleted because it reference to a node that will be removed In general, there will be a subtree reference (denoted red rectangle) that must be included as the right subtree of the entry from the parent node

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  • Notice that:

    After a merge operation, the subtree under the red reference has a different parent node !!! Example: We must update the parent reference of that subtree after the merge operation...

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  • Example:

    remove(6): Underflow (less than 2 subtrees): Perform a left merge operation: Result:

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• The right merge operation

  • Merging with your right sibling:

    What happens in a merge operation with a right sibling: The merge operation with a right sibling:

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  • Notice that:

    The subtree under the red reference has a different parent node !!! (No picture, you can refer to the one above - it is similar)

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  • Example:

    remove(6): Underflow (less than 2 subtrees) : Perform a right merge operation: Result: