AVL search Tree L-Deletions

L CA CATEGORY DELETIONS DELETIONS IN AVL SEARCH TREES

AVL TREE DELETION 

Similar to insertion but more complex  ±

 ±

Rotations and double rotations needed to rebalance Imbalance Imbalanc e may propagate upward upward so that many rotations may be needed.

TYPES OF DELETION There are two types of deletion, 

Rotation free deletions

 Deletion with

rotations rotations

L CA CATEGO TEGORY RY

R CA CATEG TEGORY ORY

L0

R0

L1

R1

L-1

R-1

L CA CATEG TEGORY ORY ROTA ROTATION TIONS S Let A be the node whose balance factor is affected

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For the L classification, If bf(A) = -2 (where as in R classification bf(A) = +2) then it should have have been -1 before deletion and A should have a right subtree with root B. Based on bf(B) being either 0 or +1 or -1, classify the L rotations as L0, L1, L-1.

GENERIC GENERIC REPRESEN REPRESENT TATION OF L0 L0 ROTATION (+1) 

(-2)

(-1)

B

A

A

(-1)

AL

AL

(0)

(0)

A

B

B

h

h-1

h

h-1

h

h

h

AL BL

BR

Balanced tree before deletion

BL

BR

Unbalanced tree after deletion

BR

BL

Balanced tree after deletion

EXAMPLE (1) (-2)

DELETE 20

(-1)

BEFORE DELETION

30

(-1)

(-1) (0) 25

UNBALANCE

(0)

(1)

AFTER DELETION

35

(0)

(0)

(1)

(-1)

(0) 20

15

(

40

(0) 10

R0 ROTATION

(0) 45

BALANCED TREE

GENERIC GENERIC REPRESEN REPRESENT TATION OF L1 L1 ROTATION (-2) (-1)

A AL

B (0)

h- 1

(0)

A

C

AL

(+1)

h-1

BR

(0) C

C

h-1

Balanced tree before deletion

A

B

h-1

h -1

CR

(0)

h-1

h-1

CL

(0)

B (+1) BR

CL

h-1

AL

CL

CR

CR

Unbalanced tree after deletion

Balanced tree after deletion

BR

EXAMPLE (0) DELETE 49

(-2)

50

BEFORE DELETION

(-1) UNBALANCED TREE

(0) (0)

AFTER DELETION

(1)

(-1) 45

55

(0) L1 ROTATION

(0)

(0)

49

(0)

(0)

51

53

56

60

(0)

(0)

BALANCED TREE

GENERIC GENERIC REPRESEN REPRESENT TATION OF L-1 ROTATION (0) (-2)

(-1)

B

A

A AL

AL

(-1)

(0)

(-1)

A

B

B

h

h-1

h h-1

h

BL

BR

Balanced tree before deletion

h-1

h- 1

h- 1

h

BL

BR

Unbalanced tree after deletion

AL

BL

BR

Balanced tree after deletion

EXAMPLE (0) (-2)

DELETE 72

75

BEFORE DELETION

(-1) UNBALANCE AFTER

(+1)

(0) (-

DELETION

(-1)

60

80

1)

L-1 ROTATION

(+1) (0) (0)

72

(0)

(0)

76

90

(0) (0)

85

BALANCED TREE

ARGUMENTS FOR AVL TREES

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Search is O(log N) since AVL trees are always balanced. Insertion and deletions are also O(logn) The height balancing adds no more than a constant constant factor factor to the speed of insertion.

ARGUMENTS AGAINST USING AVL TREES 

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Difficult to program & debug; more space for balance factor. Asymptotically faster but rebalancing costs time. Most large searches are done in database systems on disk and use other structures (e.g. B-trees). May be OK to have O(N) for a single operation if total run time for many consecutive operations is fast (e.g. Splay trees).