Transcript Lecture 6 - 2-4 Trees

More Trees
Multiway Trees
and
2-4 Trees
Motivation of Multi-way Trees

Main memory vs. disk
◦ Assumptions so far:
◦ We have assumed that we
can store an entire data
structure in the main
memory of a computer.
◦ What if we have more
data than can fit in main
memory?
◦ Meaning that we must have
the data structure reside on
disk.
◦ The rules of the game
change, because the Big-Oh
model doesn’t apply if all
operations are not equal.
Motivation of Multi-way Trees

Main memory vs. disk
◦ Disk Accesses are
incredibly expensive.
◦ 1 disk access is worth
about 4,000,000
instructions.
◦ (See the book for
derivation)
◦ So we’re willing to do lots
of calculations just to save
disk accesses.
Motivation of Multi-way Trees

For example:
◦ Suppose we want to
access the driving
records of the citizens
of Florida.
 10 million items.
 Assume doesn’t fit in
main memory.
 Assume in 1 sec, can
execute 25 million
instructions or perform
6 disk accesses.


The Unbalanced Binary tree
would be a disaster.
 In the worst case, it has linear
depth and could require 10
mil disk accesses.
An AVL Tree
 In the typical case, it has a
depth close to log N, log 10
mil ≈ 24 disk accesses,
requiring 4 sec.
The point is…

Reduce the # of disk
accesses to a very small
constant,
 Such as 3 or 4
 And we are willing to write
complicated code to do this,
because in comparison machine
instructions are essentially free.
 As long as we’re not ridiculous.

We cannot go below
log N using a BST
◦ Even an AVL

Solution??
◦ More branching, less
height.
◦ Multiway Tree
Multiway Tree (or B-tree in book)

Multiway tree is similar to BST
◦ In a BST
 We need 1 key to decide which of 2 branches to take.
◦ In an Multiway tree
 We need M-1 keys to decide which branch to take, where M is the
order of the Multiway tree.
 Also need to balance,
 Otherwise like a BST it could degenerate into a linked list.
◦ Here is a Multiway tree of Order 5:
Specifications of a MultiwayTree
I2
10
20
30
All values
in S2 < I2,
but ≥ to I1.
All values
in S1 < I1.
S1
3
1
_
_
_
I1
If a node contains k items (I1, I2.. Ik),
it will contain k+1 subtrees.
Specifically subtrees S1 thru Sk+1
I3
7
3
4
5
_
7
8
_
_
All values in
S3 < I3, but ≥ I2.
S2
_
15
10
12
_
_
S3
_
15
17
_
_
All
values in
Sk+1 >Ik.
_
22
20
21
__
28
22
26
_
_
Sk+1
40
50
_
28
29
_
_
30
__
_
40
45
_
_
60
55
_
_
_
65
70
_
_
2-4 Trees

2-4 Trees
◦ Specific type of multitree.
1) Every node must have in between 2 and 4 children.
(Thus each internal node must store in between 1 and
3 keys)
2) All external nodes (the null children of leaf nodes) and
leaf nodes have the same depth.
Example of a 2-4 tree
10
3
1
_
_
_
7
3
4
5
_
_
7
8
_
_
15
10
12
_
_
_
15
17
_
_
20
_
30
22
20
21
__
28
22
26
_
_
_
28
29
_
_
40
30
__
_
50
40
45
_
_
60
55
_
_
_
65
70
_
_
Insert

Insert 4 into the 2-4 tree below.
Compare 4 to vals in root node.
4 < 10,
4 goes in the subtree to left of 10.
10
20
30
4 > 3 and 4 < 7,
4 goes in the subtree to
right of 3 and left of 7
3
1
_
_
7
4
5
_
5
_
_
8
_
_
15
12
_
_
_
17
_
_
_
22
21
__
__
28
26
_
_
_
28
29
__
40
_
_
_
50
45
_
_
60
55
_
_
65
70
_
Problems with Insert

What if the node
that a value gets
inserted into is full?
◦ We could just insert
the value into a new
level of the tree.
◦ BUT then not ALL of
the external nodes will
have the same depth
after this.

Insert 18 into the
following tree:
10
3
7
_
20
13
15
17
_
22
_
_
Problems with Insert

What if the node
that a value gets
inserted into is full?
◦ We could just insert
the value into a new
level of the tree.
◦ BUT then not ALL of
the external nodes will
have the same depth
after this.

Insert 18 into the
following tree:
10
3
7
_

20
13
15
17
18
_
22
_
_
The node has too many values.
 You can send one of the
values to the parent (the
book’s convention is to sent
the 3rd value.
Problems with Insert

What if the node
that a value gets
inserted into is full?
◦ We could just insert
the value into a new
level of the tree.
◦ BUT then not ALL of
the external nodes will
have the same depth
after this.

Insert 18 into the
following tree:
10
3
7
_

17
13
15
_
20
18
__
__
22
_
_
Moving 17, forces you
to “split” the other
three values.
Other Problems with Insert

In the last example,
the parent node was
able to “accept” the
17.
◦ What if the parent
root node becomes
full?

Insert 12 into the 2-4
Tree below:
10
5
_
20
11
14
17
_
30
25
__
__
32
37
_
_
Other Problems with Insert

In the last example,
the parent node was
able to “accept” the
17.

Insert 12 into the 2-4
Tree below:
10
◦ What if the parent
root node becomes
full?
5
_

20
11
12
14
17
30
25
__
__
32
37
_
_
Using the rule from
before, let’s move 12
up.
Other Problems with Insert

In the last example,
the parent node was
able to “accept” the
17.

Insert 12 into the 2-4
Tree below:
10
◦ What if the parent
root node becomes
full?
5
_

14
11
12
20
17
__
__
30
25
__
__
32
37
_
_
Using the rule from
before, let’s move 14
up.
Other Problems with Insert

In the last example,
the parent node was
able to “accept” the
17.
◦ What if the parent
root node becomes
full?


Insert 12 into the 2-4
Tree below:
10
5
_
14
11
12
20
25
__
__
30
17
__
__
32
37
_
_
Now this has too many
parent nodes AND subtrees!
 We can just repeat the process
and make a new root.
Other Problems with Insert

In the last example,
the parent node was
able to “accept” the
17.

Insert 12 into the 2-4
Tree below:
20
◦ What if the parent
root node becomes
full?
10
5
_

30
14
11
12
17
__
__
25
__
__
32
37
_
_
Now this has too many
parent nodes AND subtrees!
 We can just repeat the process
and make a new root.
Deletion from a 2-4 Tree

Delete a non-leaf
value
◦ Replace that value
with the largest value
in its left subtree
 Or smallest value in its
right subtree.
Deletion from a 2-4 Tree

Delete a leaf node
◦ In the standard case:
 a value can simply be
removed from a leaf
node that stores more
than one value.
 Requires no structural
change.
10
20
30
10
20
30
Delete 5
5
7
_
12
17
__
23
27
35
35
_
_
7
_
12
17
__
23
27
35
35
_
_
Deletion from a 2-4 Tree

BUT what if the leaf node has ONLY one value?
◦ If you get rid of it,
 then it would violate the 2-4 property that all leaf nodes MUST be on the
same height of the tree.

Break up into 2 cases:
1) An adjacent sibling has more than one value stored in its node.
2) An adjacent sibling does NOT have more than one value stored
in its node, and a fusion operation MUST be performed.
Deletion from a 2-4 Tree

Case 1:
Take the 10 to replace the
5,
 And then simply replace
the 10 with the smallest
value in its right subtree.

◦ Consider deleting 5 from
the following tree:
 An adjacent sibling has more
than one value stored in its
node.
10
20
◦ This is okay, since there is
more than one value at this
subtree.
30
12
20
30
Delete 5
5
_
_
12
17
__
23
27
_
35
_
_
10
_
__
17
_
__
23
27
_
35
_
_
Deletion from a 2-4 Tree

Case 2:
◦ If an adjacent sibling does NOT have more than one value stored in its node,
and a fusion operation MUST be performed.
◦ The fusion operation is a little more difficult since it may result in needing
another fusion at a parent node.
10
20
30
10
20
30
10
Delete 5
5
_
_
15
_
__
25
_
_
35
_
_
20
30
Fuse empty
node with 15
_
_
15
_
__
25
_
_

35
_
_
15
_
__
25
_
_
35
_
_
We have 3 child nodes when we
should have 4. Thus we can drop
a value, we will drop 10.
Deletion from a 2-4 Tree

Case 2:
◦ If an adjacent sibling does NOT have more than one value stored in its node,
and a fusion operation MUST be performed.
◦ The fusion operation is a little more difficult since it may result in needing
another fusion at a parent node.
10
20
30
20
30
Drop one parent into
fused node
15
_
__
25
_
_
35
_
_
10
15
__

25
_
_
35
_
_
We have 3 child nodes when we
should have 4. Thus we can drop
a value, we will drop 10.
Examples on the Board