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2-4 tree. Definition search Insertion deletion. Office Hour: Monday 2:00-3:00 pm Thursday 10:00-11:00 am CSB 113. Definition. A 2-4 tree is an m-way search tree T in which an ordering is imposed on the set of keys which reside in each node such that:

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2 4 tree
2-4 tree
  • Definition
  • search
  • Insertion
  • deletion
office hour monday 2 00 3 00 pm thursday 10 00 11 00 am csb 113
Office Hour:

Monday 2:00-3:00 pm

Thursday 10:00-11:00 am

CSB 113

definition
Definition
  • A 2-4 tree is an m-way search tree T in which an ordering is imposed on the set of keys which reside in each node such that:
  • Each node has a maximum of 4 children and between 1 and 3 keys.
  • The keys in each node appear in ascending order.
  • The keys in the first i children are smaller than the ith key.
  • The keys in the last m-1 children are larger than the ith key.
  • All external nodes have the same depth.
example

6

12

7

8

5

10

15

3

4

11

13

14

17

Example
search
Search
  • The keys in the first i children are smaller than the ith key.
  • The keys in the last m-i children are larger than the ith key
search6

12

6

7

8

15

11

17

5

10

3

4

13

14

Search
insertion
Insertion
  • Maintain the size and depth of the tree
insertion8
Insertion

Algorithm inserItem(k,x)

1. We search for key k to locate the insertion node v

2. We add the new item (k,x) at node v

3. While overflow(v)

if isRoot(v)

Create a new empty root above v

vsplit(v)

split
split
  • Replace node v with two nodes v1 and v2 where:

v1 is a 3-node with children v1, v2, v3 storing keys k1 and k2.

v2 is a 2-node with children v4, v5, storing key k4.

  • If v was the root of the tree, create a new root node u; otherwise, let u be the parent of v.
  • Insert key k3 into u and make v1 and v2 children of u, so that if v was the ith child of u, then v1 and v2 become children i and i+1 of node u respectively.
split10

u

v = u2

u3

u1

k1

k2

k3

k4

v1

v2

v3

v4

v5

h1

h1

h2

h2

u

u

h1

k3

h2

v = u2

u3

u1

v2

v1

k1

k2

k3

k4

u3

u1

k1

k2

k4

v1

v2

v3

v4

v5

v1

v2

v3

v4

v5

split
example11

4

4

6

4

6

12

(a)

(b)

(c)

12

4

6

12

15

4

6

15

(d)

(e)

Example
  • Sequence of insertions is: 4, 6, 12, 15, 3, 5, 10, and 8.
slide12

12

12

3

4

6

15

3

4

5

6

15

(f)

(g)

5

12

5

12

3

4

6

15

3

4

6

10

15

(h)

(h)

5

12

3

4

6

8

10

15

(i)

example13
Example
  • http://www.cs.mcgill.ca/~cs251/ClosestPair/BalancedTreeApplet/BalancedTreeApplet.html
deletion
Deletion
  • Reduce deletion of an item to the case where the item is at the node with leaf children
  • Otherwise, we replace the item with its inorder successor(or, equivalently, with its inorder predessor) and delete the latter item
underflow fusion transfer
Underflow fusion transfer
  • Deleting an item from a node v may cause an underflow, where node v cecomdes a 1-node with one child and no keys
  • To handle an underflow at node v with parent u, we consider two cases
  • Case 1: the adjacent siblings of v are 2-nodes
    • Fusion operation: We merge v with an adjacent sibling w

and move an item from u to the merge v’

    • After fusion, the underflow may progagate to the parent u
  • Case 2: an adjacent sibiling w of v is a 3-node or a 4-node
    • Transfer :
      • 1. we move a child w to v
      • 2. we move an item from u to v
      • 3. we move an item from w to u
    • After a transfer, no underflow occurs
fusion example

12

8

6

6

13

13

13

14

10

14

10

14

10

15

5

8

11

17

(a) Deletion of 12 causing an underflow at 11.

11

u

15

v

5

8

17

(b) Fusion operation. Move key of u to v. Merge v and sibling.

11

u

15

6

v

5

17

(c) 2-4 Tree after fusion operation completes.

Fusion example
transfer example

u

5

7

5

25

25

25

v

4

7

30

30

30

7

37

15

37

37

15

58

18

18

58

58

(a) Deletion of 4 causes underflow at node v (it becomes a 1-node)

u

w

v

(b) Transfer operation. 3-node sibling w is identified.

Key movement identified.

u

w

v

5

15

18

(c) 2-4 Tree after transfer operation completes.

Transfer example
exercise
Exercise
  • Sequence of insertions is: 5, 3, 18, 15, 32, 6, 12, 46,23,77

Sequence of deletion is: 12,32,5,3,18,6,46,77,15,23