deleting from b-trees

8/3/2019 Deleting From B-Trees

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Deleting from B-

Trees

Eric Nevala


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Objective: Teach you the concept

of deleting keys from a B-Tree

1. The concept of deleting from a B-Tree

2. A graphical illustration of all the cases ofdeletion

3. Look at some implementations in C++code

4. Do some practical exercises for practice(cards -> whiteboard)


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The Concept

You can delete a key entry from any node.

->Therefore, you must ensure that

before/after deletion, the B-Tree maintainsits properties.

When deleting, you have to ensure that a

node doesnt get too small(minimum nodesize is T 1). We prevent this bycombiningnodes together.


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Run Time of a B-Tree

The height grows in O(Lg N) time.

t = degree

h = height

n = number of keys

Run time is influenced by CPU speed andDisk I/O speeds

Space used: O(2n) + sizeof(class/struct)

Worst Case:2

1log

n

h t


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Lets look at an example:Were given this valid B-Tree

Source: Introduction to Algorithms, Thomas H. Cormen

Note: T = 2


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Simple DeletionCase 1: We delete F


Result: We remove F from the leaf node. No further

action needed.


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When to just delete a key from a

node: When a key is in a leaf node

When deleting the key doesnt make the

leaf node smaller then the minimumallowable size (t-1)


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Deleting and shiftingCase 2a: We deleted M


Result: We remove M from the parent node. Since

there are four nodes and two letters, we move L to

replace M. Now, the N O node has a parent again.


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The rule:

You musthave n+1 child nodes.

If you delete an intermediate node, you

need to replace the missing node bypromoting a child key.

If you dont go below the minimum ton the

right, then take the right most node fromthe left branch, otherwise try the right leftmost node on the right branch.


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Combining and DeletingCase 2c: Now, we delete G


Result: First, we combine nodes DE and JK. Then,

we push down G into the DEJK node and delete it as

a leaf.


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The rule

Condition: You must delete a key. The key

has a minimum number of nodes on the

left and right branches. Action: Take the key to be deleted, move it

down, and then combine the left and right

branches. Then delete the key.


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Check this out!Case 2b: Now, we delete D


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The rule

The recursion cant descend down to node CL

because it only has two keys, so P is pushed

down to the end of CL and merged with TX toform CLPTX; Then, D is deleted as in the first

rule.


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Deleting BBefore:

After: Deleted B, Demoted C, Promoted E


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The rule

Conditions:

Deleting the key from the node would result in a node

smaller then the minimum allowable size The neighboring node has extra keys

Actions:

Take the parent key and overwrite the node youre

deleting Take the left most node from the right neighbor and

promote it to the parent key

Otherwise: combine nodes


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Any questions so far?

Next: C++ code implementation of B-Tree

delete


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The remove class member

Source: Data Structures and Program Design in C++, Robert L. Kruse;


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Recursive_remove class member


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Remove_data

Copy_in_predecessor


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Restoring node structure


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Move_left class member


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Move_right class member


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Combination Illustration


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Combine class member


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Quick ExerciseRemove C, P and V from this B-Tree.

Work with the guy next to you to compare and check your work. Whichever pair can correctly remove all three the fastest is awesome!

You have five minutes


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Pending any questions, well move

on to the white board.


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